Method for robust estimation of the velocity of a target using a host vehicle

Abstract

A method for estimating a velocity of a target using a host vehicle equipped with a radar system includes determining a plurality of radar detection points, determining a compensated range rate, and determining an estimation of a first component of a velocity profile equation of the target and an estimation of a second component of the velocity profile equation of the target by using an iterative methodology comprising at least one iteration. The estimations and of the first and second components and of the velocity profile equation are not determined from a further iteration if at least one statistical measure representing the deviation of an estimated dispersion of the estimations and of the first and second components, and of a current iteration from a previous iteration and/or the deviation of an estimated dispersion of the residual from a predefined dispersion of the range rate meets a threshold condition.

Claims

1. A method comprising: estimating a velocity of a target in a horizontal plane using a host vehicle equipped with a radar system, the radar system including a radar sensor unit adapted to receive signals emitted from the host vehicle and reflected by the target in one measurement time instance, the estimating of the velocity of the target in a horizontal plane comprising: a) emitting a radar signal and determining, from a plurality of radar detection measurements captured by the radar sensor unit over a plurality of ranges and a plurality of azimuth angles, a plurality of radar detection points, each radar detection point comprising an azimuth angle θ.sub.i and a range rate {dot over (r)}.sub.i, wherein the range rate {dot over (r)}.sub.i represents the rate of change of the distance between the radar sensor unit and the target; b) determining a compensated range rate {dot over (r)}.sub.i,cmp represented by:
{dot over (r)}.sub.i,cmp={dot over (r)}.sub.i+u.sub.s cos θ.sub.i+v.sub.s sin θ.sub.i, wherein u.sub.s represents a longitudinal velocity component of the radar sensor unit and wherein v.sub.s represents a lateral velocity component of the radar sensor unit; c) determining, from the results of step a) and b), an estimation {tilde over (c)}.sub.t of a first component c.sub.t of a velocity profile equation of the target and an estimation {tilde over (s)}.sub.t of a second component s.sub.t of the velocity profile equation by using an iteratively reweighted least squares methodology comprising multiple iterations and applying weights w.sub.i to the radar detection points, wherein the velocity profile equation is represented by:
{dot over (r)}.sub.i,cmp=c.sub.t cos θ.sub.i+s.sub.t sin θ.sub.i; d) determining an estimation {dot over ({circumflex over (r)})}.sub.i,cmp of the velocity profile equation represented by:
{dot over ({circumflex over (r)})}.sub.i,cmp={tilde over (c)}.sub.t cos θ.sub.i+{tilde over (s)}.sub.t sin θ.sub.i, wherein the azimuth angle θ.sub.i is determined from step a) and the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are determined from step c); e) determining a residual e.sub.{dot over (r)},i of the estimation {dot over ({circumflex over (r)})}.sub.i,cmp of the velocity profile equation determined from step d) and the compensated range rate {dot over (r)}.sub.i,cmp determined from step b), wherein the residual e.sub.{dot over (r)},i is represented by a difference of the compensated range rate {dot over (r)}.sub.i,cmp and the estimation {dot over ({circumflex over (r)})}.sub.i,cmp of the velocity profile equation, and further determining the weights w.sub.i with respect to the residual e.sub.{dot over (r)},i; f) determining an estimation of the velocity of the target on the basis of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation determined from step c); and wherein, in step c), the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are not determined from at least one iteration of the multiple iterations of the iteratively reweighted least squares methodology if at least one statistical measure representing: a deviation of an estimated dispersion of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of a current iteration from a previous iteration; or a deviation of an estimated dispersion of the residual e.sub.{dot over (r)},i from a predefined dispersion of the range rate {dot over (r)}.sub.i; meets a first threshold condition; receiving, by a control unit of the host vehicle, the estimation of the velocity of the target; and: in response to the estimation of the velocity of the target: controlling, by the control unit and based on the estimation of the velocity of the target, the host vehicle; or outputting, by the control unit and based on the estimation of the velocity of the target meeting a second threshold condition, a warning signal.

2. The method according to claim 1, wherein, in step c), the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are also not determined from at least one iteration of the multiple iterations of the iteratively reweighted least squares methodology if at least one statistical measure representing a linear dependency of the radar detection points meets a third threshold condition.

3. The method according to claim 2, wherein the statistical measure representing the linear dependency of the radar detection points is based on a determinant of $X^T\mathrm{WX}$ $\mathrm{with}$ $X=\left[\begin{array}{cc}\cos {\theta }_1& \sin {\theta }_1\\ \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\theta }_n& \sin {\theta }_n\end{array}\right]$ and W representing the weights w.sub.i arranged in a diagonal matrix.

4. The method according to claim 1, wherein, in step c), the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are also not determined from at least one iteration of the multiple iterations of the iteratively reweighted least squares methodology if at least one statistical measure representing the weights w.sub.i applied to the radar detection points meets a third threshold condition.

5. The method according to claim 4, wherein the statistical measure representing the weights w.sub.i is based on a mean of the weights w.sub.i.

6. The method according to claim 1, wherein the estimated dispersion of the residual e.sub.{dot over (r)},i is represented by a square root of: ${\hat{\sigma }}_{\stackrel{.}{r}}^2=\frac{\underset{i=1}{\overset{n}{.Math.}}{\left(\psi \left(e_{\stackrel{.}{r},i}\right)\right)}^2}{{\left(\underset{i=1}{\overset{n}{.Math.}}{\psi \left(e_{\stackrel{.}{r},i}\right)}^{\prime }\right)}^2\left(n-2\right)}$ $\mathrm{with}$ $\psi \left(e_{\stackrel{.}{r},i}\right)=w_i{e}_{\stackrel{.}{r},i},$ wherein ψ(e.sub.{dot over (r)},i)′ represents a first derivative of ψ(e.sub.{dot over (r)},i) with respect to the residual e.sub.{dot over (r)},i, and wherein n represent a number of radar detection points.

7. The method according to claim 1, wherein the estimated dispersion of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation is represented by a sum of the estimated dispersions of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation.

8. The method according to claim 1, wherein the estimated dispersion of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation is represented by a trace evaluated on the basis of: ${\hat{\sigma }}_{\mathrm{VP}}^2={{\hat{\sigma }}_{\stackrel{.}{r}}^2\left(X^T\mathrm{WX}\right)}^{-1}=\left[\begin{array}{cc}{\hat{\sigma }}_{\mathrm{cc}}^2& {\hat{\sigma }}_{\mathrm{cs}}^2\\ {\hat{\sigma }}_{\mathrm{sc}}^2& {\hat{\sigma }}_{\mathrm{ss}}^2\end{array}\right]$ $\mathrm{with}$ $X=\left[\begin{array}{cc}\cos {\theta }_1& \sin {\theta }_1\\ \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\theta }_n& \sin {\theta }_n\end{array}\right]$ W representing the weights w.sub.i arranged in a diagonal matrix, and {circumflex over (σ)}.sub.cc.sup.2, {circumflex over (σ)}.sub.ss.sup.2, {circumflex over (σ)}.sub.cs.sup.2, {circumflex over (σ)}.sub.sc.sup.2 representing estimated dispersion coefficients, wherein {circumflex over (σ)}.sub.{dot over (r)}.sup.2 represents a square of the estimated dispersion of the residual e.sub.{dot over (r)},i.

9. The method according to claim 1, wherein the predefined dispersion of the range rate {dot over (r)}.sub.i is given by a specification of the radar sensor unit.

10. A system comprising: a radar sensor unit configured to estimate a velocity of a target in a horizontal plane using a host vehicle equipped with the system, the radar sensor unit adapted to receive signals emitted from the host vehicle and reflected by the target in one measurement time instance, in estimating the velocity of the target in the horizontal plane, the radar sensor unit further configured to: emit a radar signal and determine, from a plurality of radar detection measurements captured by the radar sensor unit over a plurality of ranges and a plurality of azimuth angles, a plurality of radar detection points, each radar detection point comprising an azimuth angle θ.sub.i and a range rate {dot over (r)}.sub.i, wherein the range rate {dot over (r)}.sub.i represents the rate of change of the distance between the radar sensor unit and the target; determine a compensated range rate {dot over (r)}.sub.i,cmp represented by:
{dot over (r)}.sub.i,cmp={dot over (r)}.sub.i+u.sub.s cos θ.sub.i+v.sub.s sin θ.sub.i wherein u.sub.s represents a longitudinal velocity component of the radar sensor unit and wherein v.sub.s represents a lateral velocity component of the radar sensor unit; determine, an estimation {tilde over (c)}.sub.t of a first component c.sub.t of a velocity profile equation of the target and an estimation {tilde over (s)}.sub.t of a second component s.sub.t of the velocity profile equation by using an iteratively reweighted least squares methodology comprising multiple iterations and apply weights w.sub.i to the radar detection points, wherein the velocity profile equation is represented by:
{dot over (r)}.sub.i,cmp=c.sub.t cos θ.sub.i+s.sub.t sin θ.sub.i; determine an estimation {dot over (r)}.sub.i,cmp of the velocity profile equation represented by:
{dot over ({circumflex over (r)})}.sub.i,cmp={tilde over (c)}.sub.t cos θ.sub.i+{tilde over (s)}.sub.t sin θ.sub.i, determine a residual e.sub.{dot over (r)},i of the estimation {dot over (r)}.sub.i,cmp of the velocity profile equation and the compensated range rate {dot over (r)}.sub.i,cmp, wherein the residual e.sub.{dot over (r)},i is represented by a difference of the compensated range rate {dot over (r)}.sub.i,cmp and the estimation {dot over ({circumflex over (r)})}.sub.i,cmp of the velocity profile equation, and further determining the weights w.sub.i with respect to the residual e.sub.{dot over (r)},i; determine an estimation of the velocity of the target on the basis of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation; and wherein the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are not determined from at least one iteration of the multiple iterations of the iteratively reweighted least squares methodology if at least one statistical measure representing: a deviation of an estimated dispersion of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of a current iteration from a previous iteration; or a deviation of an estimated dispersion of the residual e.sub.{dot over (r)},i from a predefined dispersion of the range rate {dot over (r)}.sub.i; meets a threshold condition; and a control unit configured to receive the estimation of the velocity of the target and, in response to the estimation of the velocity of the target: control, based on the estimation of the velocity of the target, the host vehicle; or output, based on the estimation of the velocity of the target meeting a second threshold condition, a warning signal.

11. The system according to claim 10, wherein the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are also not determined from at least one iteration of the multiple iterations of the iteratively reweighted least squares methodology if at least one statistical measure representing a linear dependency of the radar detection points meets a third threshold condition.

12. The system according to claim 11, wherein the statistical measure representing the linear dependency of the radar detection points is based on a determinant of $X^T\mathrm{WX}$ $\mathrm{with}$ $X=\left[\begin{array}{cc}\cos {\theta }_1& \sin {\theta }_1\\ \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\theta }_n& \sin {\theta }_n\end{array}\right]$ and W representing the weights w.sub.i arranged in a diagonal matrix.

13. The system according to claim 10, wherein the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation are also not determined from at least one iteration of the multiple iterations of the iteratively reweighted least squares methodology if at least one statistical measure representing the weights w.sub.i applied to the radar detection points meets a third threshold condition.

14. The system according to claim 13, wherein the statistical measure representing the weights w.sub.i is based on a mean of the weights w.sub.i.

15. The system according to claim 10, wherein the estimated dispersion of the residual e.sub.{dot over (r)},i is represented by a square root of: ${\hat{\sigma }}_{\stackrel{.}{r}}^2=\frac{\underset{i=1}{\overset{n}{.Math.}}{\left(\psi \left(e_{\stackrel{.}{r},i}\right)\right)}^2}{{\left(\underset{i=1}{\overset{n}{.Math.}}{\psi \left(e_{\stackrel{.}{r},i}\right)}^{\prime }\right)}^2\left(n-2\right)}$ $\mathrm{with}$ $\psi \left(e_{\stackrel{.}{r},i}\right)=w_i{e}_{\stackrel{.}{r},i}$ wherein ψ(e.sub.{dot over (r)},i)′ represents the first derivative of ψ(e.sub.{dot over (r)},i) with respect to the residual e.sub.{dot over (r)},i, and wherein n represent a number of radar detection points.

16. The system according to claim 10, wherein the estimated dispersion of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation is represented by a sum of the estimated dispersions of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation.

17. The system according to claim 10, wherein the estimated dispersion of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t of the first and second components c.sub.t and s.sub.t of the velocity profile equation is represented by a trace evaluated on the basis of: ${\hat{\sigma }}_{\mathrm{VP}}^2={{\hat{\sigma }}_{\stackrel{.}{r}}^2\left(X^T\mathrm{WX}\right)}^{-1}=\left[\begin{array}{cc}{\hat{\sigma }}_{\mathrm{cc}}^2& {\hat{\sigma }}_{\mathrm{cs}}^2\\ {\hat{\sigma }}_{\mathrm{sc}}^2& {\hat{\sigma }}_{\mathrm{ss}}^2\end{array}\right]$ $\mathrm{with}$ $X=\left[\begin{array}{cc}\cos {\theta }_1& \sin {\theta }_1\\ \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\theta }_n& \sin {\theta }_n\end{array}\right]$ W representing the weights w.sub.i arranged in a diagonal matrix, and {circumflex over (σ)}.sub.cc.sup.2, {circumflex over (σ)}.sub.ss.sup.2, {circumflex over (σ)}.sub.cs.sup.2, {circumflex over (σ)}.sub.sc.sup.2 representing estimated dispersion coefficients, wherein {circumflex over (σ)}.sub.{dot over (r)}.sup.2 represents a square of the estimated dispersion of the residual e.sub.{dot over (r)},i.

18. The system according to claim 10, wherein the predefined dispersion of the range rate {dot over (r)}.sub.i is given by a specification of the radar sensor unit.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) The present invention will now be described, by way of example with reference to the accompanying drawings, in which:

(2) FIG. 1 shows a target coordinate system;

(3) FIG. 2 shows a vehicle coordinate system;

(4) FIG. 3 shows a sensor coordinate system;

(5) FIG. 4 shows a target vehicle with respect to a host vehicle with detection points located on the target vehicle;

(6) FIG. 5 illustrates how to calculate velocity vectors at the location of a detection point;

(7) FIG. 6 illustrates an embodiment of the method as described herein; and

(8) FIG. 7 illustrates an embodiment of the method with regard to controlling the number of iterations.

DETAILED DESCRIPTION

(9) Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the various described embodiments. However, it will be apparent to one of ordinary skill in the art that the various described embodiments may be practiced without these specific details. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.

(10) Generally, a host vehicle 4 (see FIG. 2) is equipped with a radar system 5′ (see FIG. 3) where reflected radar signals from a target 2 (FIG. 1) in the field of view of the radar system 5′ are processed to provide data in order to ascertain the parameters used in the methodology. In order to do this various conditions and requirements are required. The target 2 (rigid body, e.g. vehicle) needs to be an extended target, i.e., the target allows the determination of a plurality of points of reflection 6′ (see FIG. 4) that are reflected from the target 2 in real-time and that are based on raw radar detection measurements. So, as used herein, the term “extended target” is used to refer to targets 2 that are capable of providing multiple, i.e. two, three or more spaced-apart scattering-points 6′ also known as points of reflection 6′. The term “extended target” is thus understood as a target 2 that has some physical size. In this instance it should be noted that the physical size can be selected e.g. in the range of 0.3 m to 20 m in order to be able to detect points of reflection 6′ stemming from e.g., a moving person to a moving heavy goods vehicle or the like.

(11) The various scattering points 6′ are not necessarily individually tracked from one radar scan to the next and the number of scattering points 6′ can be a different between scans. Furthermore, the locations of the scattering points 6′ can be different on the extended target 2 in successive radar scans. Radar points of reflection 6′ can be determined by the host vehicle 4 from radar signals reflected from the target 2, wherein a comparison of a given reflected signal with an associated emitted radar signal can be carried out to determine the position of the radar point of reflection 6′, e.g., in Cartesian or Polar coordinates (azimuth angle, radial range) with respect to the position of a radar-emitting and/or radar-receiving element/unit on the host vehicle, which can be the position of the radar sensor unit.

(12) By using, e.g., Doppler radar techniques, the range rate is also determined as known in the art. It is to be noted that the “raw data” from a single radar scan can provide the parameters θ.sub.i (azimuth angle) and {dot over (r)}.sub.i (raw range rate, i.e., radial velocity) for the i-th point of reflection of n points of reflection. These are the parameters which are used to estimate the velocity of a (moving) target, wherein i=1, . . . , n. It is also to be noted that the term instantaneous radar scan, single radar scan or single measurement instance can include reflection data from a “chirp” in Doppler techniques, which may scan over, e.g., up to 2 ms. In the subsequent description, the following conventions and definitions are used:

(13) World coordinate system: As a convention, a world coordinate system with the origin fixed to a point in space is used—it is assumed that this world coordinate system does not move and does not rotate. Conventionally, the coordinate system is right-handed; the Y-axis, orthogonal to the X-axis, pointing to the right; the Z-axis pointing into the page and a an azimuth angle is defined in negative direction (clock-wise) with respect to the X-axis; see FIG. 1 which shows such a coordinate system with origin 1 and a non-ego vehicle 2. FIG. 1 further shows the extended target 2 in the form of a vehicle, e.g. an object having a length of approximately 4.5 m.

(14) Vehicle coordinate system: FIG. 2 shows a vehicle coordinate system that in the present instance has its origin 3″ located at the center of the front bumper 3 of a host vehicle 4. It should be noted in this connection that the origin 3″ of the vehicle coordinate system can be arranged at different positions at the host vehicle 4. In the present instance the X-axis is parallel to the longitudinal axis of the vehicle 4, i.e. it extends between the front bumper 3 and a rear bumper 3′ and intersects with the center of the front bumper 3 if the origin 3″ is located there. The vehicle coordinate system is right-handed with the Y-axis orthogonal to the X-axis and pointing to the right, the Z-axis pointing into the page. An (azimuth) angle is defined as in the world coordinate system.

(15) Sensor coordinate system: FIG. 3 shows a sensor coordinate system having the origin 5. In the example of FIG. 3 the origin 5 is located at the center of a sensor unit 5′, which can be a radome. The X-axis is perpendicular to the sensor radome, pointing away from the radome. The coordinate system is right-handed: Y-axis orthogonal to the X-axis and pointing to the right; Z-axis pointing into the page. An (azimuth) angle is defined as in the world coordinate system. The velocity and the yaw rate of the host vehicle 4 are assumed to be known from sensor measurements known in the art. The over-the-ground (OTG) velocity vector of the host vehicle 4 is defined as:
V.sub.h=[u.sub.hv.sub.h].sup.T,
where u.sub.h is the longitudinal velocity of the host vehicle 4 (i.e., the velocity in a direction parallel to the X-axis of the vehicle coordinate system) and v.sub.h is lateral velocity of the host vehicle 4 (i.e., the velocity in a direction parallel to the Y-axis of the vehicle coordinate system). In more general terms the longitudinal velocity and the lateral velocity are a first and a second velocity component of the host vehicle 4, respectively.

(16) The sensor mounting position and boresight angle with respect to the vehicle coordinate system are assumed to be known with respect to the vehicle coordinate system (VCS), wherein the following notations are used:

(17) x.sub.s,VCS—sensor mounting position with respect to longitudinal (X-) coordinate

(18) y.sub.s,VCS—sensor mounting position with respect to lateral (Y) coordinate

(19) γ.sub.s,VCS—sensor boresight angle.

(20) The sensor over-the-ground (OTG) velocities can be determined from the known host vehicle velocity and the known sensor mounting position. It is understood that more than one sensor can be integrated into one vehicle and specified accordingly. The sensor OTG velocity vector is defined as:
V.sub.s=[u.sub.sv.sub.s].sup.T,
wherein u.sub.s is the sensor longitudinal velocity and v.sub.s is the sensor lateral velocity corresponding generally to first and second velocity components in the case of a yaw rate of zero.

(21) At each radar measurement instance (scan) the radar sensor unit captures n (raw) detection points from the target. Each detection point i=1, . . . , n can be described by the following parameters expressed in the sensor coordinate system:

(22) r.sub.i—range (or radial distance),

(23) θ.sub.i—azimuth angle,

(24) {dot over (r)}.sub.i—raw range rate (or radial velocity).

(25) Target planar motion can be described by the target OTG velocity vector at the location of each raw detection:
V.sub.t,i=[u.sub.t,iv.sub.t,i].sup.T,
wherein u.sub.t,i represents the longitudinal velocity of the target at the location of the i-th detection point and v.sub.t,i represents the lateral velocity of the target at the location of the i-th detection point, both preferably but not necessarily with respect to the sensor coordinate system. Target planar motion can be described as well by:
V.sub.t,COR=[ω.sub.tx.sub.t,CORy.sub.t,COR].sup.T,
wherein ω.sub.t represents the yaw rate of the target, x.sub.t,COR the longitudinal coordinate of the center of target's rotation, and y.sub.t,COR the lateral coordinate of the center of target's rotation. The longitudinal and lateral coordinates or components may also be denoted as first and second coordinates or components. These are preferably but not necessarily in orthogonal relation to each other.

(26) FIG. 4 illustrates target velocity vectors as lines originating from a plurality of detection points 6′ illustrated as crosses, wherein the detection points 6′ are all located on the same rigid body target 2 and wherein the detection points 6′ are acquired using a sensor unit of a host vehicle 4.

(27) The general situation is shown in greater detail in FIG. 5 showing three detection points 6 located on a target (not shown) with a center of rotation 7. The vehicle coordinate system with axes X.sub.VCS, Y.sub.VCS is shown in overlay with the sensor coordinate system having axes X.sub.SCS, Y.sub.SCS. The velocity vector of one of the detection points 6 (i=1) is shown together with its components u.sub.t,i, v.sub.t,i. The range rate equation for a single detection point can be expressed as follows:
{dot over (r)}.sub.i+u.sub.s cos θ.sub.i+v.sub.s sin θ.sub.i=u.sub.t,i cos θ.sub.i+v.sub.t,i sin θ.sub.i,
wherein {dot over (r)}.sub.i represents the range rate, i.e., the rate of change of the distance between the origin of the sensor coordinate system and a detection point 6, as illustrated in FIG. 5. The location of the detection point 6 can be described by the azimuth angle θ.sub.i=1 and the value of the radial distance r.sub.i=1 (range of detection point, i.e. distance between origin and the detection point).

(28) To simplify the notation the compensated range rate can be defined as:
{dot over (r)}.sub.i,cmp={dot over (r)}.sub.i+u.sub.s cos θ.sub.i+v.sub.s sin θ.sub.i
with {dot over (r)}.sub.i,cmp representing the range rate of the i-th detection point compensated for the velocity of the host vehicle 4.

(29) The compensated range rate can also be expressed as:
{dot over (r)}.sub.i,cmp=u.sub.t,i cos θ.sub.i+v.sub.t,i sin θ.sub.i.
The compensated range rate can also be expressed in vector notation as:

(30) ${\stackrel{.}{r}}_{i,\mathrm{cmp}}=\left[\begin{array}{cc}\cos {\theta }_i& \sin {\theta }_i\end{array}\right]\left[\begin{array}{c}{u}_{t,i}\\ v_{t,i}\end{array}\right].$
The so called velocity profile equation (or range rate equation) is defined as:
{dot over (r)}.sub.i,cmp=c.sub.t COS θ.sub.i+s.sub.t sin θ.sub.i,
wherein c.sub.t represents the first, e.g. longitudinal, coefficient or component of the range rate and s.sub.t represents the second, e.g. lateral, coefficient or component of the range rate equation. Note that the coefficients c.sub.t, s.sub.t are preferably invariant with respect to the azimuth angle at least for a range of azimuth angles corresponding to the location of the target to which a plurality of detection points refer to and on which basis the coefficients have been determined. This means that the velocity profile equation is assumed to be valid not only for specific detection points but for a range of azimuth angles. Therefore, the range rate can readily be determined for any azimuth angle from a specific angle range using the range rate equation.

(31) As the skilled person understands, in practice, the “true” coefficients c.sub.t, s.sub.t is usually estimated from a plurality of detection points. These estimates are denoted {tilde over (c)}.sub.t and {tilde over (s)}.sub.t and are estimated using an iteratively (re-) weighted least squares methodology.

(32) In the following, a preferred version of the method is described.

(33) Step 1:

(34) In an initial step the method comprises emitting a radar signal and determining, from a plurality of radar detection measurements captured by said radar sensor unit, a plurality of radar detection points at one measurement instance. Each radar detection point comprises at least an azimuth angle θ.sub.i and a range rate {dot over (r)}.sub.i, wherein the range rate {dot over (r)}.sub.i represents the rate of change of the distance between the sensor unit and the target at the location of the i-the detection point (cf. FIG. 4). It is understood that the azimuth angle θ.sub.i describes the angular position of the i-th detection point. It is assumed that the plurality of detection points are located on a single target (such target is usually referred to as a distributed target) as shown in FIG. 4.

(35) Step 2:

(36) The compensated range rate {dot over (r)}.sub.i,cmp is determined as:
{dot over (r)}.sub.i,cmp={dot over (r)}.sub.i+u.sub.s cos θ.sub.i+v.sub.s sin θ.sub.i,
wherein u.sub.s represents the first (e.g. longitudinal) velocity component of the sensor unit and wherein v.sub.s represents the second (e.g. lateral) velocity component of the sensor unit. The compensated range rate is the range rate compensated for the velocity of the host vehicle. Therefore, the compensated range rate can be interpreted as the effective velocity of the target at the location of the i-th detection point.

(37) Step 3:

(38) From the results of steps 1 and 2, an estimation {tilde over (c)}.sub.t of the first component c.sub.t of the velocity profile equation of the target and an estimation {tilde over (s)}.sub.t of the second component s.sub.t of the velocity profile equation of the target are determined by using an iteratively reweighted least squares (IRLS) methodology comprising at least one iteration and applying weights w.sub.i to the radar detection points, wherein the velocity profile equation of the target is represented by:
{dot over (r)}.sub.i,cmp=c.sub.t cos θ.sub.i+s.sub.t sin θ.sub.i.
The IRLS methodology is initialized, e.g., by the ordinary least squares (OLS) solution. This is done by first computing:

(39) $\left[\begin{array}{c}{\tilde{c}}_t\\ {\tilde{s}}_t\end{array}\right]={\left[X^{T}X\right]}^{-1}{X}^{T}W{\stackrel{.}{r}}_{\mathrm{cmp}},$
wherein {dot over (r)}.sub.cmp represents the vector of compensated range rates {dot over (r)}.sub.i,cmp for i=1, 2 . . . n. Using
{dot over ({circumflex over (r)})}.sub.i,cmp={tilde over (c)}.sub.t cos θ.sub.i+{tilde over (s)}.sub.t sin θ.sub.i
an initial solution for {dot over ({circumflex over (r)})}.sub.i,cmp is computed. Then, the initial residual is
e.sub.{dot over (r)},i={dot over (r)}.sub.i,cmp−r.sub.i,cmp
is computed.

(40) The variance of the residual is then computed as:

(41) ${\hat{\sigma }}_{\stackrel{.}{r}}^2=\frac{\underset{i=1}{\overset{n}{.Math.}}{\left(e_{\stackrel{.}{r},i}\right)}^2}{n-2}.$
Next, an estimation of the variance of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t is computed:

(42) ${\hat{\sigma }}_{\mathrm{VP}}^2={{\hat{\sigma }}_{\stackrel{.}{r}}^2\left(X^{T}X\right)}^{-1},\mathrm{wherein}$$X=\left[\begin{array}{cc}\cos {\theta }_1& \sin {\theta }_1\\ \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\theta }_n& \sin {\theta }_n\end{array}\right].$

(43) Furthermore, the trace tr({circumflex over (σ)}.sub.VP) is computed. With the initial solution, weights w.sub.i∈[0; 1] are computed as:

(44) 0$w_i=\{\begin{array}{cc}1,& \mathrm{for}.Math.e_{\stackrel{.}{r},i}.Math.\le K\\ 0,& \mathrm{for}.Math.e_{\stackrel{.}{r},i}.Math.>K\end{array}\mathrm{wherein}K=K_{\sigma }\min \left(k_1{\hat{\sigma }}_{\stackrel{.}{r}},{\sigma }_{\stackrel{.}{r}}\right).$
K.sub.σ represents a calibration parameter, σ.sub.{dot over (r)} represents the predefined accuracy of the range rate measurement and k.sub.1 represents a further calibration parameter, and min( ) is the minimum function which gives the minimum of the arguments. The weights w.sub.i are then arranged in a diagonal matrix W and the estimation of the coefficients of the first iteration is given as:

(45) $\left[\begin{array}{c}{\tilde{c}}_t\\ {\tilde{s}}_t\end{array}\right]={\left[X^T\mathrm{WX}\right]}^{-1}{X}^{T}W{\stackrel{.}{r}}_{\mathrm{cmp}}.$

(46) Step 4:

(47) From the solution of the first iteration an estimation {dot over ({circumflex over (r)})}.sub.i,cmp of the velocity profile is determined represented by:
{dot over ({circumflex over (r)})}.sub.i,cmp={tilde over (c)}.sub.t cos θ.sub.i+ŝ.sub.t sin θ.sub.i,
wherein the azimuth angle θ.sub.i is determined from step 1 and the estimation of the first and second components {tilde over (c)}.sub.t and {tilde over (s)}.sub.t is determined from step 3 (initial solution). A new residual is computed as:
e.sub.{dot over (r)},i={dot over (r)}.sub.i,cmp−{dot over ({circumflex over (r)})}.sub.i,cmp.
The variance of the new residual is then computed as:

(48) ${\hat{\sigma }}_{\stackrel{.}{r}}^2=\frac{\underset{i=1}{\overset{n}{.Math.}}{\left(\psi \left(e_{\stackrel{.}{r},i}\right)\right)}^2}{{\left(\underset{i=1}{\overset{n}{.Math.}}{\psi \left(e_{\stackrel{.}{r},i}\right)}^{\prime }\right)}^2\left(n-2\right)}$$\mathrm{with}$$\psi \left(e_{\stackrel{.}{r},i}\right)=w_i{e}_{\stackrel{.}{r},i},$
wherein  (e.sub.{dot over (r)},i)′ represents the first derivative of ψ(e.sub.{dot over (r)},i) with respect to the residual e.sub.{dot over (r)},i, and wherein n represent the number of detection points.

(49) Next, an estimation of the variance of the estimations {tilde over (c)}.sub.t and {tilde over (s)}.sub.t is computed as:
{circumflex over (σ)}.sub.VP.sup.2={circumflex over (σ)}.sub.{dot over (r)}.sup.2(X.sup.TX).sup.−1,
followed by computing the trace of the square root, i.e. tr({circumflex over (σ)}.sub.VP).

(50) Step 5:

(51) A plausibility check is carried out. In the check it is determined whether each of three statistical measures meets a respective threshold condition according to:

(52) $\mathrm{Ap}.\det \left(X^T\mathrm{WX}\right)>K_{\det }$$\mathrm{Bp}.\frac{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)-\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j}\right)}{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)}>K_{\mathrm{tr}\_\mathrm{plaus}}$$\mathrm{Cp}.\mu \left(\mathrm{diag}\left(W\right)\right)>K_{\mu }$
wherein

(53) j represents the iteration index,

(54) K.sub.det represents a threshold,

(55) K.sub.tr_plaus represents a threshold,

(56) μ(diag(W)) represents the mean of diagonal entries of the matrix W,

(57) K.sub.μ represents a threshold.

(58) If all three conditions Ap, Bp and Cp are fulfilled, the current estimation is considered to be plausible and a convergence check is carried out next. If at least one of the three conditions Ap, Bp and Cp is not fulfilled, the current estimation is considered to be not plausible. In this case the previous estimation is used as a final solution. It is understood that the conditions Ap, Bp and Cp can simply be changed to its respective opposites, that is

(59) ${\mathrm{Ap}}^{\prime }\det \left(X^T\mathrm{WX}\right)<K_{\det }$${\mathrm{Bp}}^{\prime }\frac{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)-\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j}\right)}{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)}<K_{\mathrm{tr}\_\mathrm{plaus}}$${\mathrm{Cp}}^{\prime }\mu \left(\mathrm{diag}\left(W\right)\right)<K_{\mu }.$
So in the latter case, if at least one of the three conditions Ap′, Bp′ and Cp′ is fulfilled, the current estimation is considered to be not plausible and the previous estimation is used as a final solution. No further iteration is carried out.

(60) A convergence check is carried out if the three conditions Ap, Bp and Cp are all fulfilled. When using the alternative conditions Ap′, Bp′ and Cp′, they all need to be not fulfilled in order to allow for the convergence check. In the convergence check it is determined whether each of three statistical measures meets a respective threshold condition according to:

(61) $.Math.\frac{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)-\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j}\right)}{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)}.Math.>K_{\mathrm{tr}\_\mathrm{next}}$$\mathrm{An}.{\hat{\sigma }}_{\stackrel{.}{r}}-{\sigma }_{\stackrel{.}{r}}>K_{{\sigma }_{\stackrel{.}{r}}}\mathrm{Bn}.j<K_{\max ,j}$
wherein

(62) j represents the iteration index,

(63) K.sub.σ.sub.{dot over (r)} represents a threshold,

(64) K.sub.max,j represents a threshold,

(65) K.sub.tr_next represents a threshold.

(66) If all three conditions An, Bn and Cn are fulfilled, i.e., the inequalities are all true, the current estimation is considered to be not close enough to a desired optimum solution and a further iteration of the IRLS methodology is carried out under the assumption that this would deliver an estimation which is closer to the optimum solution. If at least one of the three conditions An, Bn and Cn is not fulfilled, the current estimation is considered to be close enough to a desired optimum (i.e., converging) and the current estimation is used as a final solution. It is understood that the conditions An, Bn and Cn can simply be changed to its respective opposites, that is

(67) ${\mathrm{An}}^{\prime }.Math.\frac{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)-\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j}\right)}{\mathrm{tr}\left({\hat{\sigma }}_{\mathrm{VP},j-1}\right)}.Math.<K_{\mathrm{tr}\_\mathrm{next}}$${\mathrm{Bn}}^{\prime }{\hat{\sigma }}_{\stackrel{.}{r}}-{\sigma }_{\stackrel{.}{r}},<K_{{\sigma }_{\stackrel{.}{r}}}$${\mathrm{Cn}}^{\prime }j>K_{\max ,j}.$
So, if at least one of the three conditions An′, Bn′ and Cn′ is fulfilled, the current estimation is considered to be close enough to a desired optimum (i.e., converging) and the current estimation is used as a final solution. No further iteration is carried out.

(68) It is understood that the combination of the conditions is exemplary and the invention is not limited to this combination and other combinations are also possible to achieve the desired effect, namely to reduce the computational complexity and at the same time to improve the validity of the estimation. Therefore, the estimation determined by one of the embodiments described herein also improves the reliability of automated and autonomous driving applications.

(69) FIG. 6 gives a general overview of an embodiment of the method. Broken lines indicate optional flow of information which depends on the statistical measures used in the undermost block. This block can comprise conditions An′, Bn′, Cn′, Ap′, Bp′, and Cp′, wherein if at least one of these conditions is met no further iteration is carried out, this is that the method proceeds with “yes”.

(70) FIG. 7 shows an alternative for the undermost block of FIG. 6 in line with the plausibility check and convergence check described above. As is understood from FIG. 7, a further iteration (“j+1”) is only carried out when all conditions Ap, Bp, Cp, An, Bn, and Cn are met. The variable j is the iteration index. It is to be noted that carrying out a further iteration corresponds to proceeding with the “yes” branch, which corresponds to the “no” branch from the undermost block of FIG. 6. This is due to the different formulation of the conditions. In FIG. 7, the convergence check is subject to a negative formulation, i.e. the “yes” and “no” branches answer the question whether the current solution is not converging (“further iteration needed?”).

(71) Turning again to FIG. 7, if no further iteration is carried out (i.e. at least one of Ap, Bp, Cp, An, Bn, and Cn is not met) either (i) the estimation of the first and second components of the previous iteration is used as a final solution of the first and second components of the velocity profile equation (“j−1”), or (ii) the estimation from the current iteration is used as a final solution (“j”). Case (i) is reached if at least one of Ap, Bp, Cp is not met. Case (ii) is reached if at least one of An, Bn, and Cn is not met.

(72) While this invention has been described in terms of the preferred embodiments thereof, it is not intended to be so limited, but rather only to the extent set forth in the claims that follow. “One or more” includes a function being performed by one element, a function being performed by more than one element, e.g., in a distributed fashion, several functions being performed by one element, several functions being performed by several elements, or any combination of the above. It will also be understood that, although the terms first, second, etc. are, in some instances, used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first contact could be termed a second contact, and, similarly, a second contact could be termed a first contact, without departing from the scope of the various described embodiments. The first contact and the second contact are both contacts, but they are not the same contact. The terminology used in the description of the various described embodiments herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term “and/or” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It will be further understood that the terms “includes,” “including,” “comprises,” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. As used herein, the term “if” is, optionally, construed to mean “when” or “upon” or “in response to determining” or “in response to detecting,” depending on the context. Similarly, the phrase “if it is determined” or “if [a stated condition or event] is detected” is, optionally, construed to mean “upon determining” or “in response to determining” or “upon detecting [the stated condition or event]” or “in response to detecting [the stated condition or event],” depending on the context.

LIST OF REFERENCE SIGNS

(73) 1 origin of world coordinate system 2, 2′ target vehicle 3 front bumper 3′ rear bumper 3″ origin of vehicle coordinate system 4 host vehicle 5 origin of sensor coordinate system 5′ radar system 6, 6′ detection point 7 center of rotation of the target