System for determining the position of the position indicator of a position measuring system
View Patent ↗The invention concerns a method for determining a position ϕ of a position indicator of a position measuring system, in which the sensor signals a 1 to a r of a number of r sensors are recorded as a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), a measurement vector {right arrow over (q)} is formed according to {right arrow over (q)}=g({right arrow over (a)}), a 2-component vector {right arrow over (p)} is calculated according to {right arrow over (p)}=M·{right arrow over (q)}, wherein M is a 2×n matrix, and the position ϕ is determined by means of a predetermined function ƒ({right arrow over (p)}) to ϕ=ƒ({right arrow over (p)}), wherein the function ƒ({right arrow over (p)})=ƒ(p 1 ,p 2 ) is based on the equations p 1 =cos(ϕ) and p 2 =sin(ϕ). The matrix M is determined in a calibration phase in such a manner, that the transformation {right arrow over (p)}=M·{right arrow over (q)} is a linear mapping, which maps a number m of measurement vectors {right arrow over (q)} k on 2-component vectors {right arrow over (p)} k , wherein the tips of the vectors {right arrow over (p)} k substantially lie on a circle or on a circular arc. The method allows the elimination of external fields.
1. A system comprising:
a sensor chip comprising a plurality, r, of magnetic field sensors,
a rotationally moveable or linearly displaceable position indicator having a magnet configured to produce a periodically changing magnetic field at the locations of the magnetic field sensors when the magnet is moved or displaced, respectively, and
a position measuring system operated in two modes, a first mode and a second mode, the first mode being a calibration mode and the second mode being a measurement mode, the position measuring system operated at first in the calibration mode and then in the measurement mode;
the calibration mode of operation configured to determine a calibration matrix M, the calibration matrix M being a 2×n matrix, using one of:
A) (1) detecting the sensor signals a k1 to a kr of the magnetic field sensors for a predetermined number of k=1 to m≥3 positions of the position indicator,
(2) forming m signal vectors {right arrow over (a)} k =(a k1 , . . . , a kr ) from the sensor signals with k=1 to m,
(3) forming m measurement vectors {right arrow over (q)} k with k=1 to m according to a predetermined function g to {right arrow over (q)} k =g({right arrow over (a)} k ),
(4) forming a matrix Q according to
Q
=
(
q
→
1
⋮
q
→
m
)
,
and
(5) determining the matrix M from the matrix Q in such a way, that the transformation {right arrow over (p)}=M·{right arrow over (q)} is a linear mapping, which maps each of the m measurement vectors {right arrow over (q)} k on a 2-component vector {right arrow over (p)} k , wherein the tips of the vectors {right arrow over (p)} k substantially lie on a circle or on a circular arc,
or:
B) (1) setting an index i=0 and configuring a predetermined starting matrix M (0) , the matrix M (0) being a 2×n matrix,
(2) detecting and storing a plurality of measurement vectors q according to the steps A and B of the measurement mode of operation,
(3) repeatedly executing the following steps:
(a) setting the calibration matrix M to M=M (i)
(b) increasing the index i by 1,
(c) calculating the positions ϕ of the stored measurement vectors {right arrow over (q)} according to the steps C and D of the measurement mode of operation,
(d) selecting a number of k=1 to m of the stored measurement vectors {right arrow over (q)} k , the corresponding positions ϕ k of which fulfill a predetermined criterion,
(e) forming a matrix Q according to
Q
=
(
q
->
1
⋮
q
->
m
)
,
(f) determining a new matrix M (i) from the matrix Q in such a way, that the transformation {right arrow over (p)}=M·{right arrow over (q)} is a linear mapping, which maps each of the m selected measurement vectors {right arrow over (q)} k on a 2-component vector {right arrow over (p)} k ,
until the matrix M (i) within a predetermined tolerance no longer differs from the matrix M (i−1) , whereby the tips of the vectors {right arrow over (p)} k with increasing index i substantially come to lie on a circle or on a circular arc; and
the measurement mode of operation configured to use the calibration matrix M to repeatedly determine a position ϕ of the position indicator by:
(A) detecting the sensor signals a 1 to a r of the magnetic field sensors and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ),
(B) forming a measurement vector {right arrow over (q)} according to the predetermined linear function g to {right arrow over (q)}=g({right arrow over (a)}), wherein the measurement vector q has a number of n components, with 2≤n≤r,
(C) calculating a vector {right arrow over (p)}=M·{right arrow over (q)}, wherein {right arrow over (p)} is a 2-component vector with the components p 1 and p 2 , and
(D) determining the position ϕ by means of a predetermined function ƒ({right arrow over (p)}) to ϕ=ƒ({right arrow over (p)}),
wherein the predetermined function ƒ({right arrow over (p)})=ƒ(p 1 ,p 2 ) is based on the equations p 1 =cos(ϕ) and p 2 =sin(ϕ).
2. The system of claim 1 , wherein the value r is at least 3.
3. The system of claim 1 , wherein the determining the calibration matrix M from the matrix Q comprises:
determining two matrices U and B, which fulfill the equation Q=U·B, wherein U is a m×2 matrix with U T ·U=I, and B is a symmetrical, positive definite 2×2 matrix, by
forming an auxiliary matrix D so that D=Q T ·Q,
calculating the matrix B so that B=D 1/2 ,
forming an auxiliary matrix H so that H=B −1 , and
forming the calibration matrix M so that M=N H, wherein
N
=
(
d
0
0
1
)
is an auxiliary matrix, whose parameter d has the value 1 or −1.
4. The system of claim 2 , wherein the determining the calibration matrix M from the matrix Q comprises:
determining two matrices U and B, which fulfill the equation Q=U·B, wherein U is a m×2 matrix with U T ·U=I, and B is a symmetrical, positive definite 2×2 matrix, by
forming an auxiliary matrix D so that D=Q T ·Q,
calculating the matrix B so that B=D 1/2 ,
forming an auxiliary matrix H so that H=B −1 , and
forming the calibration matrix M so that M=N H, wherein
N
=
(
d
0
0
1
)
is an auxiliary matrix, whose parameter d has the value 1 or −1.
5. The system of claim 1 , wherein the determining the calibration matrix M from the matrix Q comprises:
determining three matrices U, S and V T , which fulfill the equation Q=U·S·V T ,
wherein U is an orthogonal m×m matrix, V T is in orthogonal n×n matrix and S is a positive semidefinite m×n diagonal matrix, the diagonal elements of which are sorted according to decreasing size,
forming sub-matrices Ũ, {tilde over (S)} and {tilde over (V)} T in dependence of a predetermined parameter h,
which fulfills the condition 2≤h≤l, wherein l<min(m,n) designates the number of diagonal elements of S which are different from zero and wherein
Ũ is that m×h sub-matrix of the matrix U, which is formed by discarding the columns h+1 to m,
{tilde over (S)} is that h×h sub-matrix of the matrix S, which is formed by discarding the rows h+1 to m and discarding the columns h+1 to n, and
{tilde over (V)} T is that h×n sub-matrix of the matrix V T , which is formed by discarding the rows h+1 to n,
and if n>2, forming an auxiliary matrix H from elements of the sub-matrices Ũ, {tilde over (S)} and {tilde over (V)} T , or in the special case of n=2 alternatively forming the auxiliary matrix H to H={tilde over (S)} −1 {tilde over (V)} T ,
and forming the calibration matrix M so that M=N H, wherein
N
=
(
d
0
0
1
)
is an auxiliary matrix, whose parameter d has the value 1 or −1.
6. The system of claim 5 , wherein the forming an auxiliary matrix if from elements of the sub-matrices Ũ, {tilde over (S)} and {tilde over (V)} T comprises:
forming an auxiliary matrix W,
if the measurement vectors are present in an arbitrary order with the steps:
calculating values α i =a tan 2(U 12 U i1 −U 11 U i2 ,U 12 U i2 +U 11 U i1 ) for i=1 to m,
calculating the permutation k(i), which sorts the α i -values according to their size, so that α k(i) >α k(i−1) for i=2, . . . , m, and
forming the elements of the auxiliary matrix W so that
W
i
,
k
(
j
)
=
{
sin
(
2
π
(
j
-
1
)
m
)
,
i
=
1
cos
(
2
π
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator are distributed over approximately the entire range from 0 to 2π,
or
W
i
,
k
(
j
)
=
{
sin
(
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ],
or
W
i
,
k
(
j
)
=
{
sin
(
e
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
e
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ] and the position indicator has a multipole character,
wherein the parameter e characterizes the multipole character, or if the measurement vectors {right arrow over (q)} k with k=1 in are sorted in ascending or descending order according to their corresponding position ϕ k ,
forming the elements of the auxiliary matrix W with the auxiliary quantity ƒ=1 for ascending and ƒ=−1 for descending order so that
W
ij
=
{
sin
(
f
·
2
π
(
j
-
1
)
m
)
,
i
=
1
cos
(
f
·
2
π
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator are distributed over approximately the entire range from 0 to 2π,
or
W
ij
=
{
sin
(
f
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
f
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ],
or
W
ij
=
{
sin
(
fe
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
fe
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min , ϕ max ] and the position indicator has a multipole character, wherein the parameter e characterizes the multipole character,
forming an auxiliary matrix T so that T=W Ũ, and
forming the auxiliary matrix H so that H=T {tilde over (S)} −1 {tilde over (V)} T .
7. The system of claim 2 , wherein the determining the calibration matrix M from the matrix Q comprises:
determining three matrices U, S and V T , which fulfill the equation Q=U·S·V T , wherein U is an orthogonal in m×m matrix, V T is in orthogonal n×n matrix and S is a positive semidefinite m×n diagonal matrix, the diagonal elements of which are sorted according to decreasing size,
forming sub-matrices Ũ, {tilde over (S)} and {tilde over (V)} T in dependence of a predetermined parameter h, which fulfills the condition 2≤h≤l, wherein l≤min(m,n) designate the number of diagonal elements of S which are different from zero and wherein
Ũ is that m×h sub-matrix of the matrix U, which is formed by discarding the columns h+1 to m,
{tilde over (S)} is that h×h sub-matrix of the matrix S, which is formed by discarding the rows h+1 to m and discarding the columns h+1 to n, and
{tilde over (V)} T is that h×n sub-matrix of the matrix V T , which is formed by discarding the rows h+1 to n,
and, if n≥2, forming an auxiliary matrix H from elements of the sub-matrices Ũ, {tilde over (S)} and {tilde over (V)} T , or, in the special case of n=2, alternatively forming the auxiliary matrix H so that H={tilde over (S)} −1 {tilde over (V)} T ,
and forming the calibration matrix M so that M=N H, wherein
N
=
(
d
0
0
1
)
is an auxiliary matrix, whose parameter d has the value 1 or −1.
8. The system of claim 7 , wherein the forming an auxiliary matrix TI from elements of the sub-matrices Ũ, {tilde over (S)} and {tilde over (V)} T comprises:
forming an auxiliary matrix W,
if the measurement vectors are present in an arbitrary order with the steps:
calculating values α i =a tan 2(U 12 U i1 −U 11 U i2 , U 12 U i2 +U 11 U i1 ) for i=1 to m,
calculating the permutation k(i), which sorts the α i -values according to their size, so that α k(i) >α k(i−1) for i=2, . . . , m, and
forming the elements of the auxiliary matrix W so that
W
i
,
k
(
j
)
=
{
sin
(
2
π
(
j
-
1
)
m
)
,
i
=
1
cos
(
2
π
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator are distributed over approximately the entire range from 0 to 2π,
or
W
i
,
k
(
j
)
=
{
sin
(
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ],
or
W
i
,
k
(
j
)
=
{
sin
(
e
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
e
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ] and the position indicator has a multipole character, wherein the parameter e characterizes the multipole character,
or if the measurement vectors {right arrow over (q)} k with k=1, . . . , m are sorted in ascending or descending order according to their corresponding position ϕ k ,
forming the elements of the auxiliary matrix TV with the auxiliary quantity f=1 for ascending and f=−1 for descending order so that
W
ij
=
{
sin
(
f
·
2
π
(
j
-
1
)
m
)
,
i
=
1
cos
(
f
·
2
π
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m, if the m positions of the position indicator are distributed over approximately the entire range from 0 to 2π,
or
W
ij
=
{
sin
(
f
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
f
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=i to m if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ],
or
W
ij
=
{
sin
(
fe
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
1
cos
(
fe
·
(
ϕ
max
-
ϕ
min
)
(
j
-
1
)
m
)
,
i
=
2
,
for j=1 to m if the m positions of the position indicator occupy only a circular arc with ϕ∈[ϕ min ,ϕ max ] and the position indicator has a multipole character, wherein the parameter e characterizes the multipole character,
forming an auxiliary matrix T so that T=W Ũ, and
forming the auxiliary matrix H so that H=T {tilde over (S)} −1 {tilde over (V)} T .
9. The system of claim 2 , wherein ƒ(p 1 ,p 2 )=a tan 2(p 2 ,p 1 )−ϕ 0 , wherein ϕ 0 is a quantity which characterizes a zero position of the position measuring system.
10. The system of claim 7 , wherein ƒ(p 1 ,p 2 )=a tan 2(p 2 ,p 1 )−ϕ 0 , wherein ϕ 0 is a quantity which characterizes a zero position of the position measuring system.
11. The system of claim 8 , wherein ƒ(p 1 ,p 2 )=a tan 2(p 2 ,p 1 )−ϕ 0 , wherein ϕ 0 is a quantity which characterizes a zero position of the position measuring system.
12. The system of claim 2 , wherein the function g({right arrow over (a)}) is one of the following functions:
1) g({right arrow over (a)})={right arrow over (a)},
2) g({right arrow over (a)})={right arrow over (a)}−{right arrow over (a)} 0 ,
3) g({right arrow over (a)})=P·{right arrow over (a)}, or
4) g({right arrow over (a)})=P·({right arrow over (a)}−{right arrow over (a)} 0 ),
wherein {right arrow over (a)} 0 designates an offset vector and P a projection matrix.
13. The system of claim 7 , wherein the function g({right arrow over (a)}) is one of the following functions:
1) g({right arrow over (a)})={right arrow over (a)},
2) g({right arrow over (a)})={right arrow over (a)}−{right arrow over (a)} 0 ,
3) g({right arrow over (a)})=P·{right arrow over (a)}, or
4) g({right arrow over (a)})=P·({right arrow over (a)}−{right arrow over (a)} 0 ),
wherein {right arrow over (a)} 0 designates an offset vector and P a projection matrix.
14. The system of claim 13 , wherein g({right arrow over (a)})=P·{right arrow over (a)} or g({right arrow over (a)})=P·({right arrow over (a)}−{right arrow over (a)} 0 ) and wherein the projection matrix P is determined in the calibration mode of operation, either (i) in absence of the position indicator with the steps:
executing the following steps for a predetermined number of different external fields v=1 to b≥1, wherein the number h fulfills the condition b≤r−2:
applying the external field,
detecting the sensor signals a 1 to a r and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), and
forming an external field vector ƒ v ={right arrow over (a)} or, given the case, of an offset compensated external field vector {right arrow over (ƒ)} v ={right arrow over (a)}−{right arrow over (a)} 0 ,
or (ii) in presence of the position indicator with the steps:
moving the position indicator to a fixed position,
detecting the sensor signals a 1 to a r and storing as vector {right arrow over (a)} F =(a 1 , . . . , a r ),
executing the following steps for a predetermined number of different external fields v=1 to b≥1, wherein the number b fulfills the condition b≤r−2:
applying the external field,
detecting the sensor signals a 1 to a r and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), and
forming an external field vector {right arrow over (ƒ)} v ={right arrow over (a)}−{right arrow over (a)} F or, given the case, of an offset compensated external field vector {right arrow over (ƒ)} v ={right arrow over (a)}−{right arrow over (a)} F −{right arrow over (a)} 0 ,
and both in the presence and absence of the position indicator either with the further steps:
forming of c orthonormal vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c of the vector space spanned by the external field vectors {right arrow over (ƒ)} 1 , . . . {right arrow over (ƒ)} b , wherein c≤b,
complementing the c vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c with n=r−c further vectors {right arrow over (g)} c+1 , . . . , {right arrow over (g)} c+,n to a complete, orthonormal basis of the r-dimensional vector space of the signal vectors, and
forming the projection matrix P so that
P
=
(
g
->
c
+
1
⋮
g
->
c
+
n
)
.
or with the further steps:
forming an orthonormal basis of c vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c of the vector space spanned by the external field vectors {right arrow over (ƒ)} 1 , . . . , {right arrow over (ƒ)} b , and
forming the projection matrix P so that
P
=
𝕀
-
∑
i
=
1
c
g
->
i
⊗
g
->
i
,
wherein {right arrow over (g)} 1 ⊗{right arrow over (g)} i are the tensor products of the vectors {right arrow over (g)} i .
15. The system of claim 8 , wherein the function g(d) is one of the following functions:
1) g({right arrow over (a)})={right arrow over (a)},
2) g({right arrow over (a)})={right arrow over (a)}−{right arrow over (a)} 0 ,
3) g({right arrow over (a)})=P·{right arrow over (a)}, or
4) g({right arrow over (a)})=P·({right arrow over (a)}−{right arrow over (a)} 0 ),
wherein {right arrow over (a)} 0 designates an offset vector and P a projection matrix.
16. The system of claim 15 , wherein g({right arrow over (a)})=P·{right arrow over (a)} or g({right arrow over (a)})=P·({right arrow over (a)}−{right arrow over (a)} 0 ) and wherein the projection matrix P is determined in the calibration phase, either (i) in absence of the position indicator with the steps:
executing the following steps for a predetermined number of different external fields v=1 to b≥1, wherein the number b fulfills the condition b≤r−2:
applying the external field,
detecting the sensor signals {right arrow over (a)} 1 to {right arrow over (a)} r and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), and
forming an external field vector {right arrow over (ƒ)} v ={right arrow over (a)} or, given the case, of an offset compensated external field vector {right arrow over (ƒ)} v ={right arrow over (a)}−{right arrow over (a)} 0 ,
or (ii) in presence of the position indicator with the steps:
moving the position indicator to a fixed position,
detecting the sensor signals a 1 to a r and storing them as vector {right arrow over (a)} F =(a 1 , . . . , a r ),
executing the following steps for a predetermined number of different external fields v=1 to b≥1, wherein the number h fulfills the condition b≤r−2:
applying the external field,
detecting the sensor signals a 1 to a r and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), and
forming an external field vector {right arrow over (ƒ)} v ={right arrow over (a)}−{right arrow over (a)} F or, given the case, of an offset compensated external field vector {right arrow over (ƒ)} v ={right arrow over (a)}−{right arrow over (a)} F −{right arrow over (a)} 0 ,
and both in the presence and absence of the position indicator either with the further steps:
forming c orthonormal vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c of the vector space spanned by the
external field vectors {right arrow over (ƒ)} 1 , . . . , {right arrow over (ƒ)} b , wherein c≤b,
complementing the c vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c with n=r−c further vectors {right arrow over (g)} c+1 , . . . , {right arrow over (g)} c+n to a complete, orthonormal basis of the r-dimensional vector space of the signal vectors, and
forming the projection matrix P so that
P
=
(
g
->
c
+
1
⋮
g
->
c
+
n
)
.
or with the further steps:
forming an orthonormal basis of c vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c of the vector space spanned by the external field vectors {right arrow over (ƒ)} 1 , . . . , {right arrow over (ƒ)} b , and
forming the projection matrix P so that
P
=
𝕀
-
∑
i
=
1
c
g
->
i
⊗
g
->
i
,
wherein {right arrow over (g)} 1 ⊗{right arrow over (g)} i are the tensor products of the vectors {right arrow over (g)} i .
17. The system of claim 5 , wherein a matrix C is formed in the calibration mode of operation so that
C=I−{tilde over (V)}·{tilde over (V)} T ,
and wherein in the measurement mode of operation a check value v is formed with the steps
forming a vector {right arrow over (d)}=C {right arrow over (q)},
calculating the check value v from the vector {right arrow over (d)}, and
triggering a predetermined action, if the check value v exceeds a predetermined value.
18. The system of claim 6 , wherein a matrix C is formed in the calibration phase so that
C=I−{tilde over (V)}·{tilde over (V)} T ,
and wherein in the measurement mode of operation a check value v is formed with the steps
forming a vector {right arrow over (d)}=C {right arrow over (q)},
calculating the check value v from the vector {right arrow over (d)}, and
triggering a predetermined action, if the check value v exceeds a predetermined value.
19. The system of claim 1 , further comprising:
periodically, while engaged in the measurement mode of operation, capturing and buffering a plurality of measurement vectors (and of the corresponding positions c which have been calculated according to the steps C and D of the measurement mode of operation, and
performing a recalibration by means of the steps:
selecting a number of buffered measurement vectors {right arrow over (q)}, the corresponding positions ϕ of which fulfill a predetermined criterion,
rebuilding the matrix Q from the selected measurement vectors {right arrow over (q)}, and
rebuilding the calibration matrix M.
20. A system comprising:
a sensor chip comprising a plurality, r, of magnetic field sensors,
a rotationally moveable or linearly displaceable position indicator having a magnet configured to produce a periodically changing magnetic field at the locations of the magnetic field sensors when the magnet is moved or displaced, respectively, and
a position measuring system operated in two modes, a first mode and a second mode, the first mode being a calibration mode and the second mode being a measurement mode, the position measuring system operated at first in the calibration mode and then in the measurement mode;
the calibration mode of operation configured to determine a projection matrix P either (i) in absence of the position indicator with the steps:
executing the following steps for a predetermined number of different external fields v=1 to b=r−2:
applying the external field,
detecting the sensor signals a 1 to a r and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), and
forming an external field vector {right arrow over (ƒ)}v={right arrow over (a)} or, given the case, of an offset compensated external field vector {right arrow over (ƒ)}v={right arrow over (a)}−{right arrow over (a)} 0 ,
or (ii) in presence of the position indicator with the steps:
moving the position indicator in a fixed position,
detecting the sensor signals a 1 to a r and storing as vector {right arrow over (a)} F =(a 1 , . . . , a r ),
executing the following steps for a predetermined number of different external fields v=1 to b≥1, wherein the number b fulfills the condition b=r−2:
applying the external field,
detecting the sensor signals a 1 to a r of the sensors and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ), and
forming an external field vector {right arrow over (ƒ)}v={right arrow over (a)}−{right arrow over (a)} F or, given the case, of an offset compensated external field vector {right arrow over (ƒ)}v={right arrow over (a)} F −a 0 ,
and both in the presence and absence of the position indicator with the further steps:
forming c orthonormal vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c of the vector space spanned by the external field vectors {right arrow over (ƒ)} 1 , . . . , {right arrow over (ƒ)} b , wherein c=b,
complementing the c vectors {right arrow over (g)} 1 , . . . , {right arrow over (g)} c with two further vectors {right arrow over (g)} c+1 , {right arrow over (g)} c+2 to a complete, orthonormal basis of the r-dimensional vector space of the signal vectors, and
forming the projection matrix P so that
P
=
(
g
->
c
+
1
g
->
c
+
2
)
,
and
the measurement mode of operation configured to use the projection matrix P to repeatedly determine a position ϕ of the position indicator by:
(A) detecting the sensor signals a 1 to a r of the sensors and forming based thereon a signal vector {right arrow over (a)}=(a 1 . . . , a r ),
(B) forming a measurement vector {right arrow over (q)} according to a predetermined linear function g to {right arrow over (q)}=g({right arrow over (a)}), wherein the measurement vector {right arrow over (q)} has a number of n=2 components, wherein the function g({right arrow over (a)}) is one of the following functions:
1) g({right arrow over (a)})=P·{right arrow over (a)}, or
2) g({right arrow over (a)})=P·({right arrow over (a)}−{right arrow over (a)} 0 ),
wherein {right arrow over (a)} 0 designates an offset vector, and P the projection matrix,
(C) determining the position ϕ of the position indicator by means of a predetermined function ƒ({right arrow over (q)}) so that ϕ=ƒ({right arrow over (q)}).
21. A system comprising:
a rotationally moveable position indicator having a geometrically structured, electrically conductive disk,
at least three coils, wherein at least one of the at least three coils is used as a transmitting coil fed with an alternating current to produce a magnetic field and the others of the at least three coils are used as receiving coils, the receiving coils being a number, r, of magnetic field sensors, the magnetic field generated by the transmitting coil(s) inducing in each receiving coil a voltage having an amplitude varying harmonically when the disk is rotated,
a position measuring system operated in two modes, a first mode and a second mode, the first mode being a calibration mode and the second mode being a measurement mode, the position measuring system operated at first in the calibration mode and then in the measurement mode;
the calibration mode of operation configured to determine a calibration matrix M, the calibration matrix M being a 2×n matrix, using one of:
A) (1) detecting the sensor signals a k1 to a kr of the magnetic field sensors for a predetermined number of k=1 to m≥3 positions of the position indicator,
(2) forming m signal vectors {right arrow over (a)} k =(a k1 , . . . a kr ) from the sensor signals with k=1 to m,
(3) forming m measurement vectors {right arrow over (q)} k with k=1 to m according to a predetermined function g to {right arrow over (q)} k =g({right arrow over (a)} k ),
(4) forming a matrix Q according to
Q
=
(
q
→
1
⋮
q
→
m
)
,
and
(5) determining the matrix M from the matrix Q in such a way, that the transformation {right arrow over (p)}=M·{right arrow over (q)} is a linear mapping, which maps each of the m measurement vectors {right arrow over (q)} k on a 2-component vector {right arrow over (p)} k , wherein the tips of the vectors {right arrow over (p)} k substantially lie on a circle or on a circular arc,
or:
B) (1) setting an index i=0 and configuring a predetermined starting matrix M (0) , the matrix M (0) being a 2×n matrix,
(2) detecting and storing a plurality of measurement vectors q according to the steps A and B of the measurement mode of operation,
(3) repeatedly executing the following steps:
(a) setting the calibration matrix M to M=M (i)
(b) increasing the index i by 1,
(c) calculating the positions ϕ of the stored measurement vectors {right arrow over (q)} according to the steps C and D of the measurement mode of operation,
(d) selecting a number of k=1 to m of the stored measurement vectors {right arrow over (q)}k i the corresponding positions ϕ k of which fulfill a predetermined criterion,
(e) forming a matrix Q according to
Q
=
(
q
→
1
⋮
q
→
m
)
,
(f) determining a new matrix M (i) from the matrix Q in such a way, that the transformation {right arrow over (p)}=M·{right arrow over (q)} is a linear mapping, which maps each of the m
selected measurement vectors {right arrow over (q)} k on a 2-component vector {right arrow over (p)} k ,
until the matrix M (i) within a predetermined tolerance no longer differs from the matrix M (i-1) , whereby the tips of the vectors {right arrow over (p)} k with increasing index i substantially come to lie on a circle or on a circular arc; and
the measurement mode of operation configured to use the calibration matrix M to repeatedly determine a position ϕ of the position indicator by:
(A) detecting the sensor signals a 1 to a r of the magnetic field sensors and forming a signal vector {right arrow over (a)}=(a 1 , . . . , a r ),
(B) forming a measurement vector {right arrow over (q)} according to the predetermined linear function g to {right arrow over (q)}=g({right arrow over (a)}), wherein the measurement vector {right arrow over (q)} has a number of n components, with 2≤n≤r,
(C) calculating a vector {right arrow over (p)}=M·{right arrow over (q)}, wherein {right arrow over (p)} is a 2-component vector with the components p 1 and p 2 , and
(D) determining the position ϕ by means of a predetermined function ƒ({right arrow over (p)}) to ϕ=ƒ({right arrow over (p)}),
wherein the predetermined function ƒ({right arrow over (p)})=ƒ(p 1 ,p 2 ) is based on the equations p 1 =cos(ϕ) and p 2 =sin(ϕ).