Patent Publication Number: US-10317249-B2

Title: Method for determining the position of a moving part along an axis, using an inductive sensor

Description:
FIELD OF THE INVENTION 
     The present invention relates to a method for determining the position of a target along an axis, using an inductive position sensor. 
     BACKGROUND OF THE INVENTION 
     This type of sensor has the advantage of allowing the position of a mechanical part, or any other element, to be determined without the need for contact with the part whose position it is desired to know. Because of this advantage, there are numerous applications of these sensors in all types of industry. These sensors are also used in consumer applications, for example in the motor vehicle field, in which the present invention was devised. However, the invention may be used in a variety of other different fields. 
     The operating principle of an inductive sensor is based on the variation of coupling between a primary winding and secondary windings of a transformer operating at high frequency, without the use of a magnetic circuit. The coupling between these windings varies as a function of the position of a moving (electrically) conductive part, usually called a “target”. Currents induced in the target have the effect of modifying the currents induced in the secondary windings. By adapting the configuration of the windings, and given a knowledge of the current injected into the primary winding, the measurement of the current induced in the secondary windings can be used to determine the position of the target. 
     SUMMARY OF THE INVENTION 
     For the purpose of incorporating an inductive sensor of this type into a device, notably an electronic device, there is a known way of forming the aforementioned transformer on a printed circuit card. The primary winding and the secondary windings are then created by tracks formed on the printed circuit card. The primary winding is then, for example, powered by an external source, and the secondary windings then carry currents induced by the magnetic field created by the flow of a current in the primary winding. The target, which is a conductive part, for example a metal part, may have a simple shape. It may, for example, be a part cut out of a sheet. To produce a linear sensor, the cut-out for forming the target is, for example, rectangular, whereas for a rotary sensor this cut-out is, for example, in the shape of an angular sector with a radius and angle adapted to the movement of the part. 
       FIG. 3  shows the inductive sensor  10 , comprising, among other components, an energizing primary winding B 1  and two receiving secondary windings R 1 , R 2 . 
     The target T moves along the inductive sensor, along an axis X, modifying the currents (eddy currents) in the secondary windings R 1 , R 2  which are induced by the electromagnetic flux generated by the primary winding B 1 . 
     Generally, two sets of secondary windings R 1 , R 2  are provided, to form sine and cosine functions, respectively, of the position of the target T over a complete travel of the inductive sensor  10 . 
     The first and second secondary windings R 1 , R 2  generate a first voltage signal V 1  and a second voltage signal V 2  of sine and cosine form (see  FIG. 1 ) at their terminals, as a function of a spatial angle θ, representing the position of the target T along the axis X. 
     These functions (cosine and sine) are well-known and can easily be processed by a control unit  20  incorporated into an electronic system, represented schematically in  FIG. 3 . By finding the ratio of the sine to the cosine and then applying an arctangent function, an image of the position of the target T along the axis X is obtained (see  FIG. 2 ). 
     This is shown in  FIG. 2 , which represents the arctangent function “tan” as a function of the position P of the target T. The argument of the sine and cosine functions is a linear (or affine) function of the position of the target T whose travel then represents a larger or smaller part of the spatial period of these trigonometric functions. 
     As shown in  FIG. 2 , the resulting arctangent function is not strictly linear. 
     For the purpose of establishing a direct relation between the arctangent function and the position of the target T along the axis, in terms of the spatial angle θ for example, there is a method of linearizing the arctangent function which is known from the prior art. 
     For this purpose, a linear regression y=ax+b, that is to say y=a×θ+b, a and b being two constants, is applied to the arctangent function. Said linear regression is applied to consecutive segments of values of said function, spaced apart at identical spatial angle intervals Δθ and equidistant (see  FIG. 2 ). This linear regression is known to those skilled in the art. 
     For each segment Δθ, the linear interpolation y=a×θ+b is applied. This is known to those skilled in the art. 
     This results in a straight linear regression line D L , enabling the position θ of the target T to be known directly for each value of the arctangent function. 
     However, the linearization of the arctangent function creates imprecision at the ends E 1 , E 2  of the travel C of the target (see  FIG. 2 ), where there are significant edge effects. This imperfect linearity affects the precision of the position P of the target T at these ends E 1 , E 2 . 
     As shown in  FIG. 1 , at each end E 1 , E 2 , there is a considerable difference Δ 1 , Δ 2  between the actual position θ of the target T and that indicated by the straight line D L . 
     This imprecision at the ends E 1 , E 2  reduces the useful travel C u  of the target T to about 60% of the length L of the two secondary coils R 1 , R 2  (see  FIG. 2 ). 
     The invention proposes to overcome this problem, and proposes a method for determining the position of a target T along an axis using an inductive position sensor, whereby the imprecision in the position of the target T at the ends E 1 , E 2  of the travel C u  of said target T can be reduced, and the useful travel C u  of the target can therefore be extended by comparison with the prior art. 
     The invention proposes a method for determining the position of a moving part, called a “target”, along an axis, using an inductive sensor, said inductive sensor comprising:
         a primary winding generating an electromagnetic field,   a first secondary winding, generating a first voltage signal, of the sine function type, representing the current induced in said first secondary winding when the target moves in front of the first secondary winding,   a second secondary winding, generating a second voltage signal, of the cosine function type, representing the current induced in said second secondary winding when the target moves in front of the second secondary winding,   a calculation unit,
 
said method of determination according to the invention comprising the following steps:
   Step 1: calculating an arctangent function on the basis of the first voltage signal and the second voltage signal,   Step 2: calculating an error between the arctangent function calculated in this way and a predetermined straight line,   Step 3: calculating the positions of linearization points of the arctangent function according to the formula       

     
       
         
           
             
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     where:
         i is the index of the linearization points, varying from 1 to n,   θ i  is the position of the linearization point i as a spatial angle,   F is the spacing factor of the linearization points, where F&gt;0 and F MAX  is such that:       

     
       
         
           
             
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             C u  is the useful travel of the sensor along the axis as a spatial angle. 
             Step 4: for each value of the arctangent function, finding an index i of the linearization point such that:
 
θ i &lt;θ&lt;θ i+1  
 
             where: 
             θ i  is the position of the linearization point i as a spatial angle, 
             θ i+1  is the position of the linearization point (i+1) as a spatial angle, 
             θ is a spatial angle. 
             Step 5: calculating a correction to be applied to the arctangent function according to the formula 
           
         
       
    
     
       
         
           
             
               Corr 
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     where:
         Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ,   ε(θ i ) is the error at the spatial angle θ i ,   ε(θ i+1 ) is the error at the spatial angle θ i+1 ,   tan(θ i ) is the value of the arctangent function at the spatial angle θ i ,   tan(θ i+1 ) is the value of the arctangent function at the spatial angle θ i+1 ,   tan(θ) is the value of the arctangent function at the spatial angle θ.   Step 6: calculating a corrected arctangent function:
 
tan corr (θ)=tan(θ)+Corr(tan(θ))
   where:   tan corr (θ) is the corrected arctangent function,   tan(θ) is the arctangent function,   Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ.   Step 7: determining the position of the target along the axis on the basis of the arctangent function corrected in this way.       

     Preferably, in step 2 the predetermined straight line is a linear regression of the arctangent function. 
     The invention also relates to an inductive sensor for sensing the position of a target along an axis, comprising:
         a primary winding generating an electromagnetic field,   a first secondary winding, generating a first voltage signal, of the sine function type, representing the current induced in said first secondary winding when the target moves in front of the first secondary winding,   a second secondary winding, generating a second voltage signal, of the cosine function type, representing the current induced in said second secondary winding when the target moves in front of the second secondary winding,   a calculation unit,
 
according to the invention, the calculation unit comprises:
   first means for calculating the arctangent function on the basis of the first voltage signal and the second voltage signal,   second means for calculating an error between the arctangent function calculated in this way and a predetermined straight line,   third means for calculating the positions of linearization points of the arctangent function (according to the formula       

     
       
         
           
             
               θ 
               i 
             
             = 
             
               
                 
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             where: 
             i is the index of the linearization point, varying from 1 to n, 
             θ i  is the position of the linearization point i as a spatial angle, 
             F is the spacing factor of the linearization points, where F&gt;0 and F MAX  is such that: 
           
         
       
    
     
       
         
           
             
               F 
               
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             C u  is the useful travel of the sensor along the axis as a spatial angle. 
             means for finding an index i of the linearization point such that, for each value of the arctangent function:
 
θ i &lt;θ&lt;θ i+1  
 
             where: 
             θ i  is the position of the linearization point i as a spatial angle, 
             θ i+1  is the position of the linearization point (i+1) as a spatial angle, 
             θ is a spatial angle. 
             fourth means for calculating the correction to be applied to the arctangent function according to the formula 
           
         
       
    
     
       
         
           
             
               Corr 
               ⁡ 
               
                 ( 
                 
                   tan 
                   ⁡ 
                   
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             = 
             
               
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             where: 
             Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ, 
             ε(i) is the error at the linearization point i, 
             ε(i+1) is the error at the linearization point (i+1), 
             tan(i) is the value of the arctangent function at the point i, 
             tan(i+1) is the value of the arctangent function at the point (i+1), 
             tan(θ) is the value of the arctangent function at the spatial angle θ. 
             fifth means for calculating the corrected arctangent function:
 
tan corr (θ)=tan(θ)+Corr(tan(θ))
 
             where: 
           
         
       
    
     tan corr (θ) is the corrected arctangent function,
         tan(θ) is the arctangent function,   Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ.   means for determining the position of the target along the axis on the basis of the arctangent function corrected in this way.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention is equally applicable to any motor vehicle comprising an inductive sensor according to the characteristics listed above. 
       Other characteristics and advantages of the invention will be evident from a reading of the following description and from an examination of the appended drawings, in which: 
         FIG. 1 , explained above, shows, according to the spatial angle θ, the first voltage signal V 1  and the second voltage signal V 2  at the terminals of the first secondary winding R 1  and the second secondary winding R 2  respectively, 
         FIG. 2 , explained above, shows the arctangent function according to the position P of the target T along the axis X, 
         FIG. 3 , explained above, shows the inductive sensor  10  according to the prior art, 
         FIG. 4  shows schematically the error on the arctangent function relative to the straight linear regression line D L , as a function of the spatial angle θ, 
         FIG. 5  shows schematically the distribution of the linearization points i according to the invention as a function of the spatial angle θ, 
         FIG. 6  shows the corrected arctangent function according to the invention, 
         FIG. 7  shows schematically the calculation unit  20 ′ according to the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As shown in  FIG. 2 , and explained above, the inductive position sensor  10  for measuring the position of a target T moving along an axis X comprises:
         a primary winding B 1  generating an electromagnetic field,   a first secondary winding R 1 , generating a first voltage signal V 1  as a function of a spatial angle θ (see  FIG. 1 ), of the sine function type, representing the current induced in said first secondary winding R 1  when the target T moves in front of the first secondary winding, along the axis X,   a second secondary winding R 2 , generating a second voltage signal V 2  as a function of a spatial angle θ (see  FIG. 1 ), of the cosine function type, representing the current induced in said second secondary winding R 2  when the target T moves in front of the second secondary winding, along the axis X,   a calculation unit  20 , which supplies a voltage to the primary winding B 1 , and which measures the first voltage signal V 1  and the second voltage signal V 2 , at the terminals of the first secondary winding R 1  and at the terminals of the second secondary winding R 2  respectively, in order to deduce therefrom the position of the target T along the axis X.       

     For clarity, the position of the target T along the axis X will be expressed here as a spatial angle θ. It should be noted that the invention can be applied in a similar manner to determine the position of the target T along the axis X, expressed as a distance x (in cm or mm). 
     According to the prior art, for the purpose of determining the position of the target T along the axis X, there is a known way of calculating the arctangent of the ratio between the sine and the cosine, that is to say between the first voltage signal V 1  and the second voltage signal V 2 , as follows: 
                 tan   ⁡     (   θ   )       =       (       sin   ⁢           ⁢   θ       cos   ⁢           ⁢   θ       )     =     (         V   ⁢           ⁢   1     ⁢               V   ⁢           ⁢   2       )         ,         
and then linearizing the arctangent function found in this way, by applying a linear regression of the y=a×θ+b type to segments of values of said function, spaced apart from one another by an identical and equidistant spatial angle of Δθ (see  FIG. 2 ).
 
     However, this prior art method of determining the position T cannot be used for the precise determination of the position of the target T at the ends E 1 , E 2  of the total travel of the target T, that is to say at the ends E 1 , E 2  of the secondary coils R 1 , R 2 . 
     This drawback has the effect of reducing the useful travel C u  of the target T. 
     To overcome this drawback, the invention proposes the following method of determining the position of the target T, explained below. 
     In a first step (step 1), the arctangent function tan(θ) is calculated, as in the prior art determination method, such that: 
     
       
         
           
             
               tan 
               ⁡ 
               
                 ( 
                 θ 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                 
                 ) 
               
               = 
               
                 ( 
                 
                   
                     
                       V 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                     ⁢ 
                     
                         
                     
                   
                   
                     V 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 ) 
               
             
           
         
       
     
     where: 
     V 1  is the first voltage signal, representing the sine function, 
     V 2  is the second voltage signal, representing the cosine function. 
     In a second step (step 2), according to the invention, an error ε(θ) between the arctangent function tan(θ) and a predetermined straight line D L  (see  FIG. 4 ) is calculated.
 
ε(θ)=tan(θ)− D   t  
 
     In a preferred embodiment, said straight line D L  is the linear regression of the arctangent function, having the equation y DL =y=a×θ+b (see  FIG. 2 ), and therefore:
 
ε(θ)=tan(θ)− y   DL  
 
     Said error ε(θ) is shown in  FIG. 4 , and is greater at the ends E 1 , E 2  of the travel of the target T. 
     In a third step (step 3), the position of the linearization points i (that is to say, the linear regression points) of the arctangent function tan(θ), according to the spatial angle θ, are calculated according to the equation: 
                     θ   i     =         (     i   -   1     )     ×       C   u       (     n   -   1     )         -     [       sin   ⁡     (         (     i   -   1     )     ×   2   ×   π       (     n   -   1     )       )       ×   F   ×     C   u       ]               [   1   ]               
where:
 
θ i  is the position of the linearization point i as a spatial angle θ,
 
i is the index of the linearization point, varying from 1 to n; in this example, n is in the range from 5 to 101. It should be noted that n may be greater than 101. F is the spacing factor of the linearization points i, where F&gt;0 and F MAX  is such that:
 
               F   MAX     &lt;         (     i   -   1     )     ×       C   u       (     n   -   1     )         -     [       sin   ⁡     (         (     i   -   1     )     ×   2   ×   π       (     n   -   1     )       )       ×     C   u       ]             
C u  is the useful travel of the sensor  10  along the axis X as a spatial angle θ in the range from 0° to 360°.
 
π: is a constant equal to 3.14.
 
     The essence of the invention lies in the use of a sinusoidal function to distribute the linear regression points i over the arctangent function tan(θ). By distributing the linear regression points i according to the sinusoidal function, the number of linearization points i can be made more dense at the ends E 1 , E 2  of the arctangent function, where the error ε(θ) is greatest (see  FIG. 4 ). 
     It should be noted that the invention may be implemented by using the arctangent function in place of the sine function in equation [1]. 
       FIG. 4  shows 9 linear regression points i; that is to say, n=9. The first three linearization points  0 ,  1 ,  2  are located at the end E 1 , and the last three linearization points  7 ,  8 ,  9  are located at the end E 2 . 
     The distribution of points  0 ,  1 ,  2 ,  7 ,  8 ,  9  at the ends E 1 , E 2  is denser than that of points  4 ,  5 ,  6  on the rest of the useful travel C u  of the target T. 
     In a fourth step (step 4), for each value of the arctangent function tan(θ), the window of linear regression points i containing the abscissa θ of said value is determined. More precisely, for each value tan(θ), the index i of the linearization point is determined, such that:
 
θ i &lt;θ&lt;θ i+1  
 
     where: 
     θ i  is the position of the linearization point i as a spatial angle, 
     θ i+1  is the position of the linearization point (i+1) as a spatial angle, 
     θ is a spatial angle. 
     An example is shown in  FIG. 5 . In this example, the value tan(θ) corresponds to a spatial angle θ included in the window of the linear regression points i=7 and (i+1)=8. 
     In step 5, the correction to be applied to the arctangent function tan(θ) is calculated according to the following formula: 
               Corr   ⁡     (     tan   ⁡     (   θ   )       )       =       ɛ   ⁡     (   i   )       +       [       ɛ   ⁡     (     θ     i   +   1       )       -     ɛ   ⁡     (     θ   i     )         ]     ×     [         tan   ⁡     (   θ   )       -     tan   ⁡     (     θ   i     )             tan   ⁡     (     θ     i   +   1       )       -     tan   ⁡     (     θ   i     )           ]               
where:
 
Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ,
 
ε(θ 1 ) is the error at the spatial angle θ i ,
 
ε(θ i+1 ) is the error at the spatial angle θ i+1 ,
 
tan(θ 1 ) is the value of the arctangent function at the spatial angle θ i ,
 
tan(θ i+1 ) is the value of the arctangent function at the spatial angle θ i+1 ,
 
tan(θ) is the value of the arctangent function at the spatial angle θ.
 
     Then, in the sixth step (step 6), the correction calculated in this way is applied to the arctangent function tan(θ)
 
tan corr (θ)=tan(θ)+Corr(tan(θ))
 
where:
 
tan corr (θ) is the corrected arctangent function,
 
tan(θ) is the arctangent function,
 
Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ.
 
     This is shown in  FIG. 6 ; the arctangent function corrected in this way tan corr (θ) is a straight line, different from the straight linear regression line D L  of the prior art. Said corrected arctangent function has errors Δ 1 ′ and Δ 2 ′ at its ends E 1 , E 2  between said corrected function tan corr (θ) and the arctangent function tan(θ) which are smaller than the errors Δ 1  and Δ 2  between the arctangent function tan(θ) and the straight linear regression line D L  of the prior art. 
     Because of the precision provided by the determination method of the invention at the ends E 1 , E 2 , the useful travel C u  of the target T is then considerably elongated. The new useful travel C u′ , found by using the determination method according to the invention, is about 20% longer than the useful travel C u  of the prior art. For example, with the linearization method according to the prior art, the useful travel Cu is equal to 40 mm, while, according to the method of the invention, the new useful travel Cu′ is 48 mm. 
     Steps 1 to 7 can be executed by means of software, using an electronic computer connected electrically to the inductive sensor  10 , or alternatively by using a calculation unit  20 ′ according to the invention (see  FIG. 7 ). 
     The invention also relates to an inductive position sensor  10  comprising:
         a primary winding B 1  generating an electromagnetic field,   a first secondary winding R 1 , generating a first voltage signal V 1 , of the sine function type,   a second secondary winding R 2 , generating a second voltage signal of the cosine function type,   a calculation unit  20 ′,       

     According to the invention, the calculation unit  20 ′ is adapted to execute steps 1 to 7 of the determination method detailed above. 
     For this purpose, the calculation unit  20 ′ according to the invention comprises (see  FIG. 7 ):
         first means (M 1 ) for calculating the arctangent function tan(θ) on the basis of the first voltage signal V 1  and the second voltage signal V 2 ,   second means (M 2 ) for calculating an error ε(θ) between the arctangent function calculated in this way tan(θ) and a predetermined straight line D L ; in the preferred embodiment of the invention, the predetermined straight line D L  is the straight linear regression line of said arctangent function tan(θ),   third means (M 3 ) for calculating the positions of linear regression points i of the arctangent function according to the formula:       

     
       
         
           
             
               θ 
               i 
             
             = 
             
               
                 
                   ( 
                   
                     i 
                     - 
                     1 
                   
                   ) 
                 
                 × 
                 
                   
                     C 
                     u 
                   
                   
                     ( 
                     
                       n 
                       - 
                       1 
                     
                     ) 
                   
                 
               
               - 
               
                 [ 
                 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               i 
                               - 
                               1 
                             
                             ) 
                           
                           × 
                           2 
                           × 
                           π 
                         
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   × 
                   F 
                   × 
                   
                     C 
                     u 
                   
                 
                 ] 
               
             
           
         
       
         
         
           
             
               
                 where: 
                 θ i  is the position of the linearization point i as a spatial angle θ, 
                 i is the index of the linearization point, varying from 1 to n; in this example, n is in the range from 5 to 101. 
                 F is the spacing factor of the linearization points i, where F&gt;0 and F MAX  is such that: 
               
             
           
         
       
    
     
       
         
           
             
               F 
               MAX 
             
             &lt; 
             
               
                 
                   ( 
                   
                     i 
                     - 
                     1 
                   
                   ) 
                 
                 × 
                 
                   
                     C 
                     u 
                   
                   
                     ( 
                     
                       n 
                       - 
                       1 
                     
                     ) 
                   
                 
               
               - 
               
                 [ 
                 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               i 
                               - 
                               1 
                             
                             ) 
                           
                           × 
                           2 
                           × 
                           π 
                         
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   × 
                   
                     C 
                     u 
                   
                 
                 ] 
               
             
           
         
       
         
         
           
             
               
                 C u  is the useful travel of the sensor  10  along the axis X as a spatial angle θ in the range from 0° to 360°. 
                 π: is a constant equal to 3.14. 
               
             
             means (M R ) for finding the index i of the linearization point, for each value of the arctangent function, such that:
 
θ i &lt;θ&lt;θ i+1  
 
           
         
       
    
     where: 
     θ i  is the position of the linearization point i as a spatial angle, 
     θ i+1  is the position of the linearization point (i+1) as a spatial angle, 
     
         
         
           
             θ is a spatial angle. 
             fourth means (M 4 ) for calculating the correction to be applied to the arctangent function according to the formula: 
           
         
       
    
     
       
         
           
             
               Corr 
               ⁡ 
               
                 ( 
                 
                   tan 
                   ⁡ 
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 ) 
               
             
             = 
             
               
                 ɛ 
                 ⁡ 
                 
                   ( 
                   
                     θ 
                     i 
                   
                   ) 
                 
               
               + 
               
                 
                   [ 
                   
                     
                       ɛ 
                       ⁡ 
                       
                         ( 
                         
                           θ 
                           
                             i 
                             + 
                             1 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       ɛ 
                       ⁡ 
                       
                         ( 
                         
                           θ 
                           i 
                         
                         ) 
                       
                     
                   
                   ] 
                 
                 × 
                 
                   [ 
                   
                     
                       
                         tan 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                       - 
                       
                         tan 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             i 
                           
                           ) 
                         
                       
                     
                     
                       
                         tan 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             
                               i 
                               + 
                               1 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         tan 
                         ⁡ 
                         
                           ( 
                           
                             θ 
                             i 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
         
         
           
             
               
                 where: 
                 Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ, 
                 ε(θ i ) is the error at the spatial angle 
                 ε(θ i+1 ) is the error at the spatial angle θ i+1 , 
                 tan(θ i ) is the value of the arctangent function at the spatial angle θ 1 , 
                 tan(θ i+1 ) is the value of the arctangent function at the spatial angle θ i+1 , 
                 tan(θ) is the value of the arctangent function at the spatial angle θ. 
               
             
             fifth means (M 5 ) for calculating the corrected arctangent function:
 
tan corr (θ)=tan(θ)+Corr(tan(θ))
           where:   tan corr (θ) is the corrected arctangent function,   tan(θ) is the arctangent function,   Corr(tan(θ)) is the correction on the arctangent function at the spatial angle θ.   
         
             means (M D ) for determining the position θ of the target T along the axis X on the basis of the arctangent function corrected in this way tan corr (θ). 
           
         
       
    
     The first, second, third, fourth, and fifth calculation means, the search means and the determination means M 1 , M 2 , M 3 , M 4 , M 5 , M R , M D  take the form of software integrated into the calculation unit  20 ′ ( FIG. 7 ). 
     The essence of the invention lies in a judicious correction of the arctangent function, which is carried out by inexpensive software means. 
     The invention therefore enables the useful travel of an inductive position sensor to be considerably extended, while improving the precision of the position of the target at the ends of the travel.