Abstract:
A sensing system ( 10 ) for an automotive vehicle includes a first radar sensor ( 18 ) generating a first and a second range signal, and a second radar sensor ( 20 ) generating a first and a second range signal corresponding to two sampling time periods. A controller ( 12 ) is coupled to the first radar sensor ( 18 ) and the second radar sensor ( 20 ). The controller ( 12 ) calculates a first position and a second position from the first radar sensor and the second radar sensor range measurements. The controller ( 12 ) generates a first set of points corresponding to the first position and a second set of points corresponding to the second position. The controller ( 12 ) calculates a plurality of calculated range-rate values in response to the first set of points and the second set of points. The controller ( 12 ) compares the plurality of calculated range-rate values to the measured range-rate and selects the closest range-rate from the plurality of calculated range-rate values. A couple of target position points is generated from the first set of points and the second set of points corresponding to the calculated closest range-rate.

Description:
TECHNICAL FIELD 
     The present invention relates to a radar system for automotive vehicles, and more particularly, to a position detection system in a source vehicle for determining the relative position of a target. 
     BACKGROUND 
     Auto manufacturers are investigating radar, lidar, and vision-based pre-crash sensing systems to improve occupant safety. Current vehicles typically employ accelerometers that measure vehicle body decelerations in the event of a crash. In response to accelerometers, airbags or other safety devices are deployed. 
     In certain crash situations it would be desirable to provide information before forces actually act upon the vehicle when a collision is unavoidable. Such information may include the position of a target vehicle relative to a source vehicle. 
     Remote sensing systems using radar are used in adaptive cruise control, collision avoidance and collision warning applications. These systems have characteristic requirements for false detection. Generally, the remote sensing system reliability requirements for pre-crash sensing for automotive safety related systems are more stringent than those for comfort and convenience features, such as, adaptive cruise control. The reliability requirements even for safety related features vary significantly, depending upon the safety countermeasure under consideration. For example, tolerance towards undesirable activations may be higher for activating motorized belt pre-tensioners than for functions such as vehicle suspension height adjustments. Non-reversible safety countermeasures, including airbags, require extremely reliable sensing systems for pre-crash activation. 
     To meet wide-angle coverage requirements for pre-crash sensing purposes, multiple pulsed radar based sensing systems are being investigated for automotive applications. Multiple, pulsed radar sensor based systems with a wide field of coverage are available. Triangulation techniques with individual radar range measurements are used with multiple pulsed radar systems for object position estimation. 
     Triangulation techniques used for object position identification may result in the occurrence of false intersections or ghosts and inaccuracies in object bearing estimations due to individual radar sensor range measurement errors. Ghost elimination is an important aspect of object track initiations and track data associations, and various proprietary techniques are employed by radar tracking system developers for this task. 
     The accuracy with which range can be measured by a radar is limited by the signal bandwidth and signal to noise ratio. Many other error sources particular to individual radars also act to limit range measurement accuracy where inaccuracies cause large variations in object azimuth position estimation with triangulation techniques. Accurate azimuth position estimation is needed for reliable object tracking and threat assessment for automotive safety applications. 
     It would therefore be desirable to provide a reliable method for determining the position of a target vehicle. 
     SUMMARY OF THE INVENTION 
     The present invention provides an improved pre-crash sensing system that improves azimuth measurements and therefore provides more accurate angular position information for a target vehicle. 
     In one aspect of the invention, a sensing system for an automotive vehicle includes a first radar sensor and a second radar sensor generating a first range signal and a second range signal related to the target object, at two separate times. A controller is coupled to the first radar sensor and the second radar sensor. The controller calculates a first position and a second position from the first radar sensor and the second radar sensor range measurements. The controller generates a first set of points corresponding to the first position and a second set of points corresponding to the second position. The controller calculates a plurality of range-rate values in response to the first set of points and the second set of points. The controller compares the plurality of calculated range-rate values to the measured range-rate and selects the closest range-rate from the plurality of range-rate values. A couple of target position points are generated, one from the first set of points and the other from the second set of points that correspond to the closest measured range-rate. 
     In a further aspect of the invention, a method of determining the position of a target vehicle comprises: determining a first range and a second range from a first radar and a second radar sensor; calculating a first position and a second position from the first range and second range measurements; evaluating a measured range-rate; generating a first set of points corresponding to the first position; generating a second set of points corresponding to the second position; calculating a plurality of calculated range-rate values in response to the first set of points and the second set of points; comparing the plurality of calculated range-rate values to the measured range-rate; selecting the closest calculated range-rate from the plurality of calculated range-rate values to the measured range-rate; selecting a couple of target position points from the first set of points and the second set of points corresponding to the closest calculated range-rate to the measured range-rate. 
     One advantage of the invention is that by more accurately determining the relative position of the target vehicle, activation of the proper countermeasure at the proper time is more accurately performed. 
    
    
     Other advantages and features of the present invention will become apparent when viewed in light of the detailed description of the preferred embodiment when taken in conjunction with the attached drawings and appended claims. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagrammatic view of a pre-crash sensing system according to the present invention. 
     FIG. 2 is a top view of an automotive vehicle illustrating the coordinate system used herein. 
     FIG. 3 is a top view of an automotive vehicle illustrating the coordinate system having a target vehicle and a source vehicle at two different times. 
     FIG. 4 illustrates two sets of points determined from two subsequent positions. 
     FIG. 5A is a top view of a source vehicle illustrating parameter in determining the accuracy of the system. 
     FIG. 5B is a contour plot illustrating the accuracy of various angle determinations. 
     FIG. 5C is a side view of FIG. 5B illustrating the accuracy of the average delta theta versus angle theta. 
     FIG. 6 is a plot illustrating a “net” of sets relative to the host vehicle. 
     FIG. 7 is a plot illustrating a “local net in the center” of sets relative to the host vehicle. 
     FIG. 8 is a summarizing flow chart of the operation of the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     In the following figures the same reference numerals will be used to identify the same components. In the following description, the source vehicle is the vehicle performing the processing to perform a countermeasure. The target vehicle is the vehicle being monitored by the source vehicle for a potential collision. 
     Referring now to FIG. 1, a pre-crash system  10  for use in a source vehicle has a controller  12 . Controller  12  is preferably a microprocessor-based controller that is coupled to a memory  14  and a timer  16 . Memory  14  and timer  16  are illustrated as separate components from that of controller  12 . However, those skilled in the art will recognize that memory  14  and timer  16  may be incorporated into controller  12 . 
     Memory  14  may comprise various types of memory including read only memory, random access memory, electrically erasable programmable read only memory, and keep alive memory. Memory  14  is used to store various thresholds and parameters as will be further described below. 
     Timer  16  is a timer such as a clock timer of a central processing unit within controller  12 . Timer  16  is capable of timing the duration of various events as well as counting up or counting down. For example, by measuring a change in speed over a time counted by timer  16 , acceleration may be determined. Of course, acceleration may be measured using a separate accelerometer. 
     A pair of radar sensors  18  and  20  is coupled to controller  12 . Radar sensors  18  and  20  generate an object signal in the presence of an object within its field of view. The radar sensors  18 ,  20  preferably generate both range data and range-rate. However, only one radar sensor needs to generate range-rate. By spacing the radars  18 ,  20  on the vehicle the range data may be used to obtain a position of a target vehicle relative to a source vehicle as will be described below. One type of radar for the present application is a Doppler radar. More specifically, pulsed Doppler radar sensors were used in one constructed embodiment. 
     A speed sensor  32  is also coupled to controller  12 . Speed sensor  32  may be one of a variety of speed sensors known to those skilled in the art. For example, a suitable speed sensor may include a sensor at every wheel that is averaged by controller  12 . Preferably, controller  12  translates the wheel speeds into the speed of the vehicle. Suitable type of speed sensors  32  may include, for example, toothed wheel sensors such as those employed on anti-lock brake systems. 
     Controller  12  is used to control the activation of a countermeasure system  34 . Countermeasure system  34  may include one or several types of countermeasures each having an individual actuator associated therewith. In that case, controller  12  may direct the individual countermeasure actuator to activate the countermeasure. Various types of countermeasure systems will be evident to those skilled in the art. Examples of a countermeasure within countermeasure system  34  include occupant belt pretensioning, bumper height changing, braking, the pre-arming of internal airbags, the deployment of exterior or internal airbags, pedal control, steering column position, head restraint and knee bolster control. Preferably, controller  12  is programmed to activate the appropriate countermeasure in response to the inputs from the various sensors. As will be described below, the controller may choose the countermeasure or countermeasures based on the position and other pertinent data of the source vehicle and an approaching target vehicle. 
     Referring now to FIG. 2, a source vehicle S is illustrated relative to a target vehicle T. This, for example, may be at an intersection of two roads. The angle of the intersection may have any value within the interval ( 0 , π). Source vehicle S and target vehicle T are moving towards an intersection from different directions. It will be assumed that both vehicles are moving in a straight line with constant accelerations. Denote a s  and a t  as absolute values of accelerations for source vehicle S and target vehicle T, respectively. Assumed also is that source vehicle S has two radar sensors symmetrically located at the front of the source vehicle S at the distance d from the centerline. Each radar  18 , 20  measures the distance R from the target vehicle with accuracy ±ΔR and the relative velocity (range-rate). The value Δt is the interval of time between two measurements. Source vehicle S velocity, v s ( 0 )=v s0 , and acceleration a s  are measurable using the speed sensor  32  and timer  16  described above. 
     In a Cartesian coordinate system O XY , the origin, is located at point O which corresponds to the center of the front of source vehicle S. Axis O X  corresponds to the longitudinal axis of source vehicle S. The coordinate system is fixed at time t=0. The coordinates (x 0 ,y 0 ) are the coordinates of the target vehicle T in O XY  at the moment t=0. An auxiliary coordinate system O′ X′Y′ , is used having an origin O′ corresponding to (x 0 ,y 0 ) of coordinate system O XY . Axes O′ X′ . and O′ Y′  are parallel to O X  and O Y , respectively. 
     Two measurements are taken at time t 0  and t 1  by the two radar sensors. 
     First radar: 
     
       
           t   0 =0 : R   10   , {dot over (R)}   10   
       
     
     
       
           t   1   =Δt: R   11   , {dot over (R)}   11 .  (2.1) 
       
     
     Second radar: 
     
       
           t   0 =0 : R   20   , {dot over (R)}   20   
       
     
     
       
           t   1   =Δt: R   21   , {dot over (R)}   21 .  (2.2) 
       
     
     The first subscript in (2.1) and (2.2), is the radar number (radar  1  is  18 , radar  2  is  20 ), and the second corresponds to the moment in time. 
     In this first example it is presumed that the measurements are absolutely accurate, i.e. ΔR=0. Later, the case for ΔR≠0 is described using the teachings of this first example. Using measurements (2.1), (2.2), the following variables (parameters of the target vehicle T) may be determined: 
     Initial position coordinates of the target—(x 0 ,y 0 ); 
     Initial velocity absolute value of the target—v t ( 0 )=v t0 ; 
     Acceleration absolute value of the target—a t ; 
     Angle corresponding to direction of motion of the target—α 
     At t=0, initial position coordinates (x 0 ,y 0 ) of target vehicle T may be found as two circular intersection point coordinates. These circles have radii R 10 , R 20 , and centers at the points ( 0 , d), and ( 0 , −d), corresponding to the coordinates of the radar locations in coordinate system O XY . 
     The equations governing the circumference give a system                          x   2     +       (     y   -   d     )     2       =     R   10   2                     x   2     +       (     y   +   d     )     2       =     R   20   2             }           (   3.1   )                                
     From the first equation of (3.1) we have 
     
       
           x   2   =R   10   2 −( y−d ) 2 . 
       
     
     From the second 
     
       
           R   10   2 −( y−d ) 2 +( y+d ) 2   R   20   2 . 
       
     
     Finally, after further simplification,                  y   0     =         R   20   2     -     R   10   2         4      d         ,           (   3.2   )                                              x   0     =     ±           R   10   2     -       (       y   0     -   d     )     2         .               (   3.3   )                                 
     It is assumed that x 0 &gt;0, in (3.3) it can only be a positive value. 
     Referring now to FIG. 3, the second position of the target vehicle T is determined. In this figure, (x 1 ,y 1 ) represent the second position coordinates of target vehicle T in O XY  at the moment t 1 =Δt. The points (x 1s ,y 1s ), (x 2s ,y 2s ) are the second position coordinates of radars in O XY . 
     Thus,                        x     1      s       =         v   s0        Δ                 t     +       1   2          a   s        Δ                   t   2                       y     1      s       =   d           }           (   3.4   )                                
     and,                        x     2      s       =         v   s0        Δ                 t     +       1   2          a   s        Δ                   t   2                       y     2      s       =     -   d             }           (   3.5   )                                
     Point (x 1 ,y 1 ) is an intersection of two circles with center points (3.4), (3.5). Consider the equations of these circles as a system for determining (x 1 ,y 1 ),                            (     x   -       v   s0        Δ                 t     -                  1   2          a   s        Δ                   t   2         )     2     +       (     y   -   d     )     2       =     R   11   2                       (     x   -       v   s0        Δ                 t     -                  1   2          a   s        Δ                   t   2         )     2     +       (     y   +   d     )     2       =     R   21   2             }           (   3.6   )                                
     The solution of system (3.6) is                  y   1     =         R   21   2     -     R   11   2         4      d         ,           (   3.7   )                                              x   1     =       ±         R   11   2     -       (       y   1     -   d     )     2           +       v   s0        Δ                 t     +       1   2          a   s        Δ                     t   2     .                 (   3.8   )                                 
     It is assumed that x 1 &gt;0, in (3.8) it can only be a positive value. 
     Using the two positions coordinates of the target vehicle determined from (3.2), (3.3), (3.7), and (3.8), angle a may be found uniquely. 
     Consider expressions 
     
       
           x   1   −x   0 =ρ cos α, 
       
     
     
       
           y   1   −y   0 =ρ sin α, 
       
     
     where        ρ   =           (       x   1     -     x   0       )     2     +       (       y   1     -     y   0       )     2                                
     is the distance between points (x 0 ,y 0 ) and (x 1 ,y 1 ). As a result we have                  cos                 α     =         x   1     -     x   0       ρ       ,           (   3.9   )                                              sin                 α     =           y   1     -     y   0       ρ     .             (   3.10   )                                 
     Values cos α and sin α determine the moving direction of target T. Therefore, it is not necessary to find the actual angle α. 
     Angles θ 10 , θ 11  are also determined. θ is the angle between axis O X  and the direction from a radar to the target (the first subscript for θ 10 , and θ 11 , is the number of the radar, and the second one corresponds to the instant in time.)          Case                   y   0              &lt;   0     ,                α   ∈       (       π   2     ,   π     )     .                                
     Thus,                  cos                   θ   10       =       x   0       R   10         ,           (   3.11   )                                
     and                  sin                   θ   10       =         y   0     -   d       R   10         ,           (   3.12   )                                
     where point (x 0 ,y 0 ) is obtained from (3.2), (3.3). Analogously, at t=Δt                  cos                   θ   11       =         x   1     -     x     1      s           R   11         ,           (   3.13   )                                              sin                   θ   11       =           y   1     -     y     1      s           R   11       =           y   1     -   d       R   11       .               (   3.14   )                                 
     Here point (x 1 ,y 1 ) is derived from (3.7), (3.8) and (x 1s ,y 1s ) is derived from (3.4). 
     To calculate the initial velocity of the target v t0  the range-rate {dot over (R)} 10  measured from the radar sensor  18  is used. Consider the well-known equation 
     
       
         
           R{dot over (R)}=x{dot over (x)}+y{dot over (y)}. 
         
       
     
     At t=0, 
     
       
           R   10   {dot over (R)}   10   =x   0   {dot over (x)}   0 +( y   0   −d ){dot over (y)} 0   
       
     
     or                    R   .     10     =             x   0       R   10              x   .     0       +           y   0     -   d       R   10              y   .     0         =     (         e   →       R   10       ·       v   →     0       )         ,           (   3.15   )                                
     where            e   →       R   10       =       (         x   0       R   10       ,       y   0       R   10         )     T                            
     is the identity vector for {right arrow over (R)} 10 , and {right arrow over (v)} 0 =({dot over (x)} 0 ,{dot over (y)} 0 ) T  is the velocity vector of the target at the moment t=0. It follows from (3.15) that the value {dot over (R)} 10  is a velocity vector projection with sign onto the direction defined with vector {right arrow over (R)} 10 . This projection is calculated using information about angles α and θ 10 . Consider equation (3.15) in the following form 
     
       
           {dot over (R)}   10   =−v   s0  cos θ 10   −v   t0  cos(π+θ 10 −α). 
       
     
     The minus signs on the right side of this expression indicate the fact that the distance between vehicles decreases when moving towards the intersection. Using the well known trigonometric identity 
     
       
         cos(π+θ 10 −α)=−cos(θ 10 −α) 
       
     
     The previous equation becomes 
     
       
           {dot over (R)}   10   =−v   s0  cos θ 10   +v   t0  cos(θ 10 −α).  (3.16) 
       
     
     From (3.16)          v   t0     =             R   .     10     +       v   s0        cos                   θ   10           cos        (       θ   10     -   α     )         .                            
     Using the identity 
     
       
         cos(θ 10 −α)=cos θ 10  cos α+sin θ 10  sin α 
       
     
     and therefore,                v   t0     =             R   .     10     +       v   s0        cos                   θ   10             cos                   θ   10        cos                 α     +     sin                   θ   10        sin                 α         .             (   3.17   )                                
     Formula (3.17) uniquely determines the absolute value of initial velocity of the target v t0  using (3.9)-(3.12). 
     To calculate the absolute value a t  of acceleration of the target, the well-known relation          x   1     =       x   0     +       v   t0        cos                 α                 Δ                 t     +       1   2          a   t        cos                 α                 Δ                   t   2                                
     is used. 
     Finally,                  a   t     =     2                       x   1     -     x   0     -       v   t0        cos                 α                 Δ                 t         cos                 α                 Δ                   t   2             ,           (   3.18   )                                
     where x 1  is determined in (3.8). 
     Equation (3.18) uniquely determines the absolute value of acceleration of the target a t  using (3.3), (3.8), (3.9), (3.17). Moreover, the result may be checked using a calculation using the y co-ordinates.          a   t     =     2                         y   1     -     y   0     -       v   t0        sin                 α                 Δ                 t         sin                 α                 Δ                   t   2         .                              
     Equations (3.2), (3.3), (3.9), (3.10), (3.17), (3.18) give the full solution of the kinematic parameters of the target vehicle T. 
     Referring now to FIG. 4, in the preceding it was assumed that ±ΔR=0. For a radar, this is typically not the case. In this part it is assumed that radars R 1  and R 2  measure the distance to the target R, with accuracy ±ΔR≠0. This means that some probability is factored in. Therefore, instead of one point, a set of points is determined. In this case, the real position of the target is an unknown point inside this set. 
     In this example, the analytical description of a set  50  of possible initial positions of the target as well as a set  52  of possible second positions is constructed. As will be further described below a search method for determining a couple of positions: one from each set is created. This couple of position points will be as close as possible to the real positions of target vehicle. 
     The determination of a couple from the sets of possible solutions is important because small errors in radar range accuracies can result in large variations in the target azimuth position estimation with the triangulation method. For example, delta theta, which corresponds to the angle subtended by points A 2  and A 3  in FIG. 4, is very strongly influenced by the bearing location of the target object. 
     FIGS. 5A and 5B show the variations in angle subtended at the origin (delta theta) as a function of radial distance and angular position, when d=36 cm and ΔR=5 cm. As seen from the three-dimensional part of FIG. 5B, delta theta is almost constant for radial distances between 6 and 14 meters. This fact was used to simplify the picture to two dimensions as shown in FIG.  5 C. Accurate azimuth determination is important for object tracking and threat assessment for pre-crash sensing. A technique to accurately determine object azimuth position using the Doppler effect is developed below. 
     It should be noted that in general each radar has its own accuracy of measurement. It is assumed that the accuracies are different: ΔR 1  and ΔR 2 . Obviously, the possible initial positions set of the target is bounded with four intersecting circles. These circles have radii R 10 −ΔR 1 , R 10 +ΔR 1 , R 20 −ΔR 2 , R 20 +ΔR 2 , respectively, and centers at the points ( 0 , d), ( 0 , −d)—coordinates of radars in O XY . The equations of these circles have the following form 
     
       
           x   2 +( y−d ) 2 =( R   10   −ΔR   1 ) 2 ,  (4.1) 
       
     
     
       
           x   2 +( y−d ) 2 =( R   10   +ΔR   1 ) 2 ,  (4.2) 
       
     
     
       
           x   2 +( y+d ) 2 =( R   20   −ΔR   2 ) 2 ,  (4.3) 
       
     
     
       
           x   2 +( y+d ) 2 =( R   20   +ΔR   2 ) 2 ,  (4.4) 
       
     
     If the four equations are solved: 1) (4.1), (4.3); 2) (4.1), (4.4); 3) (4.2), (4.3); 4) (4.2), (4.4), the co-ordinates of the four extreme points A 1 , A 2 , A 3 , A 4 , are obtained for the possible initial positions set of the target. Furthermore, equation (3.6) may be used for the analytical description of the possible second set of positions for the target. 
     In this case the equations of the four circles in the form                      (     x   -       v   s0        Δ                 t     -       1   2          a   s        Δ                   t   2         )     2     +       (     y   -   d     )     2       =       (       R   11     -     Δ                   R   1         )     2       ,           (   4.5   )                                                    (     x   -       v   s0        Δ                 t     -       1   2          a   s        Δ                   t   2         )     2     +       (     y   -   d     )     2       =       (       R   11     +     Δ                   R   1         )     2       ,           (   4.6   )                                                     (     x   -       v   s0        Δ                 t     -       1   2          a   s        Δ                   t   2         )     2     +       (     y   +   d     )     2       =       (       R   21     -     Δ                   R   2         )     2       ,           (   4.7   )                                 
     and,                    (     x   -       v   s0        Δ                 t     -       1   2          a   s        Δ                   t   2         )     2     +       (     y   +   d     )     2       =       (       R   21     +     Δ                   R   2         )     2             (   4.8   )                                
     are obtained. 
     Solving the four systems: 1) (4.5), (4.7); 2) (4.5), (4.8); 3) (4.6), (4.7); 4) (4.6), (4.8) the coordinates of four extreme points B 1 , B 2 , B 3 , B 4 , for the possible second positions set  52  of the target are found. 
     The procedure for determining a solution to multiple sets of points is outlined below. A couple of target positions: one from each set is chosen. Values R 10 , R 11 , R 20 , R 21 , {dot over (R)} 10 , {dot over (R)} 20  are known for this couple of positions so the equations (2.1) to (3.18) may be solved and the corresponding values of cos α, sin α, cos θ 10 , sin θ 10 , v t0 , and a t  found. The range-rate {dot over (R)} 11  is calculated and the calculated range-rate is compared with the measured range-rate from the pulsed Doppler radar sensor  18  at the second time moment. If the calculated and measured values of {dot over (R)} 11  are extremely close to each other for any couple of the target positions then this, couple is the solution. 
     The explicit expression for {dot over (R)} 11  (similar to 3.6) is determined as 
     
       
           {dot over (R)}   11   =−v   s1  cos θ 11   +v   t1  cos(θ 11 −α)  (4.9) 
       
     
     where v s1 =v s0+Δta   s  is the velocity of the source vehicle S at the moment t=Δt, v t1 =v t0 +Δta t  is the velocity of the target vehicle T at the moment t=Δt. 
     To calculate θ 11 , the angle that corresponds to the chosen target positions, the following equations are used 
     
       
           x   1   −x   s1   =R   11  cos θ 11 , 
       
     
     
       
           y   1   −y   s1   =R   11  sin θ 11 . 
       
     
     As a result                  cos                   θ   11       =         x   1     -     x   s1         R   11         ,           (   4.10   )                                
     and                sin                   θ   11       =           y   1     -     y   s1         R   11       .             (   4.11   )                                
     In (4.10), (4.11) x 1 ,y 1  are determined from (3.7), (3.8), and x 1s ,y 1s  from (3.4). 
     Finally, using identity 
     
       
         cos(θ 11 −α)=cos θ 11  cos α+sin θ 11  sin α 
       
     
     Equation (4.9) may be expressed in the form 
     
       
           {dot over (R)}   11 =−( v   s0   +Δta   s )cos θ 11 +( v   t0   +Δta   t )(cos θ 11  cos α+sin θ 11  sin α),  (4.12) 
       
     
     Referring now to FIG. 6, to effectively use the two set algorithm, it may be desirable organize a retrieval sequence of points from two sets. 
     The first case is referred to as the “net case.” In this case the set of the possible initial positions of the target is illustrated. Any positive integer number no may be used as the partition interval number for this set along radius-vectors {right arrow over (R)} 10  and {right arrow over (R)} 20 . Lengths h 10  and h 20  of each interval may be calculated by using the formulas            h   10     =       2                 Δ                   R   1         n   0         ,                  h   20     =         2                 Δ                   R   2         n   0       .                              
     From this, the minimal lengths of radius-vectors: R min10 =R 10 −ΔR 1 , R min20 =R 20 −ΔR 2  may be determined. A computer program such as Matlab™ may be used assuming the values to be initial and by adding to them h 10  and h 20  in double cycle to organize an exhaustive search of all the net&#39;points. The full number of net points is equal to (n 0 +1) 2  from FIG.  6 . The set of the possible positions of the target at the second moment of time t=Δt also is considered. A net of points for this set is analogously determined. Another positive integer number n 1  is assigned. R min11 =R 11 −ΔR 1 , R min21 =R 21 −ΔR 2  are calculated. Then, lengths h 12  and h 22  may be obtained by the formulas            h   11     =       2                 Δ                   R   1         n   1         ,                  h   21     =         2                 Δ                   R   2         n   1       .                              
     The full number of net points is equal to (n 1 +1) 2 . 
     It should be noted that the full number of possible pairs of target positions is 
     
       
           N =( n   0 +1) 2 ( n   1 +1) 2 . 
       
     
     Referring now to FIG. 7, another way in which to approach the use of sets is a “local net in the center” approach. The case illustrated in FIG. 6 may approach or could quickly lead to a large number of possible pairs of target positions. For example, if n 0 =2 and n 1 =3 (from FIG. 6) then N=(2+1) 2 (3+1) 2 =144 different variants of possible target positions have to be considered. The large number of possible pairs of target positions may make this approach less practical for real time implementation in an actual automotive safety application. 
     To eliminate this potential limitation, it is desirable to decrease the full number of possible pairs of target positions without loosing the method&#39;quality. Obviously, the probability that the real position of the target is located on the boundary of the sets is very small. Therefore, it is not necessary to consider boundary points of the net. Moreover, the formation principle of the net may be changed. 
     It should be noted that the formation principle of the net may be refined by doing experiments with the specific experimental data from the particular type of radar. 
     Consider for example, a local net of five points for the set of initial positions of the target and a local net of seven points for the set of the possible positions of the target at the second moment t=Δt of time (FIG.  7 ). It should be noticed that the full number of possible pairs of target positions will be only 5×7=35 (instead of 144). For a computer program realization, the local nets may be determined as special matrices with dimensions (5×2) and (7×2), respectively. Each row of each matrix contains two values of lengths of two radius-vectors. Then the point of intersection of two circles will be one element of the local net. The co-ordinates of this point may be calculated by using Equations (3.2), (3.3) or (3.7), (3.8). 
     For example, matrices A, B may be determined as the following:        A   =     (                        R   10     -       1   2        Δ                   R   1                 R   20     -       1   2        Δ                   R   2                     R   10     +       1   2        Δ                   R   1                 R   20     -       1   2        Δ                   R   2                   R   10           R   20                 R   10     -       1   2        Δ                   R   1                 R   20     +       1   2        Δ                   R   2                     R   10     +       1   2        Δ                   R   1                 R   20     +       1   2        Δ                   R   2                          )                                  B   =     (                        R   11     -       2   3        Δ                   R   1                 R   21     -       1   3        Δ                   R   2                     R   11     -       2   3        Δ                   R   1                 R   21     +       1   3        Δ                   R   2                   R   11             R   21     -       2   3        Δ                   R   2                   R   11           R   21               R   11             R   21     +       2   3        Δ                   R   2                     R   11     +       2   3        Δ                   R   1                 R   21     -       1   3        Δ                   R   2                     R   11     +       1   3        Δ                   R   1                 R   21     +       1   3        Δ                   R   2                          )                             
     Referring now to FIG. 8, the process is summarized. The process starts in starting block  100 . In block  102  the range and range-rate information when the object is at the first and the second positions is obtained from the radar sensors. 
     In step  104  a value of {overscore (R)} 10  from the closed interval [R 10 −ΔR 1 , R 10 +ΔR 1 ] is chosen based upon the initial range measurement. Also a value of {overscore (R)} 20  from the closed interval [R 20 −ΔR 2 , R 20 +ΔR 2 ] is chosen based upon the initial range measurement. The initial co-ordinates of the target—(x 0 ,y 0 ) using {overscore (R)} 10 , {overscore (R)} 20  and Equations (3.2), (3.3) are calculated. A value of {overscore (R)} 11  is chosen from the closed interval [R 11 −ΔR 1 , R 11 +ΔR 1 ] based upon the second range measurement. A value of {overscore (R)} 21  from the closed interval [R 21 −ΔR 2 , R 21 +ΔR 2 ] is also chosen based upon the second range measurement. The co-ordinates of the target—(x 1 , y 1 ) at the moment t=Δt using {overscore (R)} 11 , {overscore (R)} 21  and formulas (3.7), (3.8) are determined. 
     In step  106 , cos {overscore (α)}, sin {overscore (α)}, cos {overscore (θ)} 10 , sin {overscore (θ)} 10 , {overscore (v)} t0  and {overscore (a)} t0  are calculated using appropriate equations as described before. 
     Also, Values of cos {overscore (θ)} 11 , sin {overscore (θ)} 11  using formulas (4.10), (4.11) are determined. The value of {dot over ({overscore (R)})} 11  using formula (4.12) is determined. The value of                   R   .     11     -         R   .     _     11            ,                          
     is compared with its value for the previous pair of target positions from the set. If the new value is less than the previous one, then this value is stored until a better correlation is found from a new pair of target positions and the related parameters. In this example, {dot over (R)} 11  is the real measurement from one of the radar sensors. In step  108  the target velocity, acceleration and direction of travel (the target trajectory) are predicted based on the best matching couple of target positions. In step  110  the trajectory information is provided to a threat assessment algorithm. In step  112 , the threat assessment algorithm compares the possibility of impact threat against predefined threshold values. If the threat assessment criteria is met in step  112 , step  114  activates the countermeasure and the operation finally ends in step  116 . In step  112 , if the threat assessment threshold is not met the operation returns to step  102  to acquire new range and range-rate data from the radar sensors. 
     It should be noted that the iterative procedure for determining the target position is stopped when a predetermined minimum value for                 R   .     11     -         R   .     _     11                                 
     is reached. 
     The present disclosure describes an algorithm for position estimation of radar targets, which can be implemented in a vehicle equipped with a multi-sensor pulsed radar sensor system (with at least one sensor with Doppler measurement capabilities). If more than one pulsed sensor has Doppler capabilities, it may be used to validate the prediction of the first Doppler radar sensor with some additional computational costs. 
     While particular embodiments of the invention have been shown and described, numerous variations and alternate embodiments will occur to those skilled in the art. Accordingly, it is intended that the invention be limited only in terms of the appended claims.