Method for estimating angular errors of angle coders in precision rotary devices, device
Method for estimating angular errors of coders in a looped system including a rotating actuator, an angle coder producing a position measurements, and a calculation device controlling the actuator, said device receiving a setpoint signal and the position measurements, for looping, and calculating in a corrector a control signal. In a test phase, a specific corrector C ω rj (q) is synthesised for a constant rotation speed ω rj , the corrector having zero gain at the corresponding frequency and harmonics, resulting in opening the loop, a matrix relation is established between position measurements and a function of parameters characterizing coder errors and torque ripples, the test phase repeats with different rotation speeds, the matrix relations are concatenated producing an identifiable global matrix relation, the parameters are estimated by resolution of the global matrix relation.
1 . A method for estimating angular errors of angle coders in a looped system including at least one actuator for rotating an axis and an angle coder providing a signal of position measurement of said axis, and including a calculation device for at least controlling said actuator, wherein the calculation device receives a setpoint signal at the system input, and the signal of position measurement for looping the system, the rotation actuator generating torque undulations d m (t), the coder producing coder errors d c (t) which, for a setpoint signal y c (t) that corresponds to a rotation of the axis at a speed of rotation ω r assumed to be constant, have a spectrum consisting of the fundamental rotation frequency of value ω r and its harmonics kω r , k∈ , the speed of rotation and the rotation frequency having a same unit, and wherein the calculating device calculates a command signal in a corrector, wherein,
in a test phase, for a speed of rotation indexed j, ω rj , assumed to be constant, a specific corrector C ω rj (q) of said constant speed of rotation ω ri is synthesized, said corrector C ω rj (q) being such that, for the frequency domain comprising the fundamental rotation frequency of value ω rj and a determined number of its first harmonics kω rj , k∈ , 1<k≤N l , N l being the maximum rank of the harmonics to be considered, the corrector has a zero gain at these frequencies ω ri , kω ri , which causes an opening of the loop at said frequencies, and a matrix relationship is established between the position measurements and a function of parameters characterizing the coder errors and the torque undulations, on the system with the specific corrector and at the constant speed of rotation ω rj ,
wherein the test phase is repeated a determined number n of times with different setpoints of rotation at speed of rotation ω ri each assumed to be constant, where, j∈ , 1<j≤n,
wherein the matrix relationships of the repeated test phases are concatenated in order to obtain an identifiable global matrix relationship, and
wherein the parameters characterizing the coder errors and the torque undulations are estimated by solving the global matrix relationship, the corrector being synthesized either by a mixed-sensitivity method on an H-infinity control, or by a robust pole placement method on RST corrector.
2 . The method according to claim 1 , wherein, moreover, once estimated the parameters characterizing the coder errors and the torque undulations, the calculation device is configured for the execution in real time of the calculation of a coder error compensation signal in order to compensate for the coder errors.
3 . The method according to claim 1 , wherein a range [ω low ,ω high ] of rotation frequency of the actuator in which an axis transfer function module has a decrease of 40 dB/decade is previously determined, and in the test phase, the different speeds of rotation ω rj are further chosen in such a way that each constant speed of rotation ω ri and its N l harmonics kω ri are in the range [ω low ,ω high ].
4 . The method according to claim 3 , wherein the range [ω low ,ω high ] of rotation frequency of the actuator in which the axis transfer function has a decrease of 40 dB/decade is determined by a method chosen among:
a graphical method using the curve of the gain
G
(
e
i
ω
)
=
❘
"\[LeftBracketingBar]"
B
(
e
i
ω
)
A
(
e
i
ω
)
❘
"\[RightBracketingBar]"
of the axis transfer function and determining a lower limit ω low and an upper limit ω high of frequencies surrounding an area of the curve having a decrease of 40 dB/decade, B being an output function and A being an input function, and
an identification method.
5 . The method according to claim 1 , wherein, in the test phase, the setpoint signal y c (t) at the system input is a ramp position setpoint signal or a constant speed of rotation setpoint signal.
6 . The method according to claim 1 , wherein the matrix relationship between the position measurements and a function of parameters characterizing the coder errors and the torque undulations is expressed by:
Y
j
=
ϕ
j
θ
+
D
j
with
ϕ
j
=
[
φ
(
0
)
φ
(
1
)
⋮
φ
(
t
f
)
]
Y
j
=
[
y
m
′
(
0
)
y
m
′
(
1
)
⋮
y
m
′
(
t
f
)
]
where Y j is a vector of centered position measurement of coefficients
y
m
′
(
t
)
,
and
φ
(
t
)
=
[
cos
(
y
m
(
t
)
)
sin
(
y
m
(
t
)
)
cos
(
2
y
m
(
t
)
)
sin
(
2
y
m
(
t
)
)
⋯
⋯
1
(
ω
r
)
2
cos
(
y
m
(
t
)
)
1
(
ω
r
)
2
sin
(
y
m
(
t
)
)
1
(
2
ω
r
)
2
(
2
y
m
(
t
)
)
1
(
2
ω
r
)
2
sin
(
2
y
m
(
t
)
)
⋯
]
and
θ
T
=
[
a
1
b
1
a
2
b
2
⋯
α
1
′
β
1
′
α
2
′
β
2
′
⋯
]
a transposed vector of the parameters characterizing the coder errors and the torque undulations, y m (t) is the signal of position measurement provided by the angle coder, and
D j a vector of centered disturbance terms.
7 . The method according to claim 6 , wherein the components φ(t) of φ i and the components
y
m
′
(
t
)
of Y j are filtered, wherein the filtering can be high-pass or low-pass or band-pass.
8 . The method according to claim 7 , wherein the identifiable global matrix relationship
Y=φθ+D
with:
Y
=
[
Y
1
Y
2
⋮
Y
n
]
ϕ
=
[
ϕ
1
ϕ
2
⋮
ϕ
n
]
is used to produce an estimation {circumflex over (θ)} of the parameters characterizing the coder errors and the torque undulations by linear regression and application of a least squares relationship according to {circumflex over (θ)}=(φ T φ) −1 φ T Y.
9 . The method according to claim 7 , wherein the identifiable global matrix relationship
Y=φθ+D
with:
Y
=
[
Y
1
Y
2
⋮
Y
n
]
ϕ
=
[
ϕ
1
ϕ
2
⋮
ϕ
n
]
is used to produce an estimation {circumflex over (θ)} of the parameters characterizing the coder errors and the torque undulations and with a regularization method using the following relationship:
{circumflex over (θ)}=(φ T φ+H ) −1 φ T Y
where H is a square matrix defined as positive and of the size of φ T φ.
10 . The method according to claim 7 , wherein an interferometric optical measurement method is implemented for measuring the coder errors, said measurements being made on a finite number u p of positions of the axis, and wherein the estimation {circumflex over (θ)} of the parameters characterizing the coder errors and the torque undulations is obtained by minimizing a quadratic criterion j=(Y−φθ) T (Y−φθ) under the equality constraint E c =Cθ, where C is a matrix such that
C
=
[
cos
(
y
1
*
)
sin
(
y
1
*
)
⋯
cos
(
Ny
1
*
)
sin
(
Ny
1
*
)
0
1
,
N
l
⋮
⋮
⋮
⋮
⋮
⋮
cos
(
y
n
p
*
)
sin
(
y
n
p
*
)
⋯
cos
(
Ny
n
p
*
)
cos
(
Ny
n
p
*
)
0
1
,
N
l
]
y
i
*
corresponding to the n p measurement positions on the axis, i∈[1, . . . n p ],
and N being the maximum harmonic rank of a Fourier series decomposition of the coder errors.
11 . The method according to claim 6 , wherein the identifiable global matrix relationship
Y=φθ+D
with:
Y
=
[
Y
1
Y
2
⋮
Y
n
]
ϕ
=
[
ϕ
1
ϕ
2
⋮
ϕ
n
]
is used produce an estimation {circumflex over (θ)} of the parameters characterizing the coder errors and the torque undulations by linear regression and application of a least squares relationship according to {circumflex over (θ)}=(φ T φ) −1 φ T Y.
12 . The method according to claim 11 , wherein the calculation device is implemented, which is configured for the execution in real time, in a compensation calculation block, of the calculation of a coder error compensation signal in order to compensate for the coder errors, and the calculation device is configured to take into account in real time the torque undulations, and wherein the estimation {circumflex over (θ)} of the vector θ of the parameters characterizing the coder errors and the torque undulations is calculated by iteration: after a first repetition of test phases and a first calculation of the parameters characterizing the coder errors and the torque undulations by solving the global matrix relationship, a first estimated vector {circumflex over (θ)} 1 is estimated, the coder error compensation signal being then calculated and the torque undulations being then taken into account on the basis of said first estimated vector {circumflex over (θ)} 1 , a new repetition of test phases and a new calculation of the parameters characterizing the coder errors and the torque undulations by solving the global matrix relationship are performed on the system so-compensated on the basis of the first estimated vector {circumflex over (θ)} 1 , a new estimated vector {circumflex over (θ)} 2 being estimated, the coder error compensation signal being then calculated and the torque undulations being then taken into account on the basis of the sum of the previous estimated vector {circumflex over (θ)} 1 and the new estimated vector {circumflex over (θ)} 2 , and subsequent iterations are performed on the compensated system on the basis of the sum of the previous estimated vectors {circumflex over (θ)} 1 +{circumflex over (θ)} 2 +{circumflex over (θ)} 3 . . . .
13 . The method according to claim 6 , wherein the identifiable global matrix relationship
Y=φθ+D
with:
Y
=
[
Y
1
Y
2
⋮
Y
n
]
ϕ
=
[
ϕ
1
ϕ
2
⋮
ϕ
n
]
is used to produce an estimation {circumflex over (θ)} of the parameters characterizing the coder errors and the torque undulations and with a regularization method using the following relationship:
{circumflex over (θ)}=(φ T φ+H ) −1 φ T Y
where H is a square matrix defined as positive and of the size of φ T φ.
14 . The method according to claim 13 , wherein the calculation device is implemented, which is configured for the execution in real time, in a compensation calculation block, of the calculation of a coder error compensation signal in order to compensate for the coder errors, and the calculation device is configured to take into account in real time the torque undulations, and wherein the estimation {circumflex over (θ)} of the vector θ of the parameters characterizing the coder errors and the torque undulations is calculated by iteration: after a first repetition of test phases and a first calculation of the parameters characterizing the coder errors and the torque undulations by solving the global matrix relationship, a first estimated vector {circumflex over (θ)} 1 is estimated, the coder error compensation signal being then calculated and the torque undulations being then taken into account on the basis of said first estimated vector {circumflex over (θ)} 1 , a new repetition of test phases and a new calculation of the parameters characterizing the coder errors and the torque undulations by solving the global matrix relationship are performed on the system so-compensated on the basis of the first estimated vector {circumflex over (θ)} 1 , a new estimated vector {circumflex over (θ)} 2 being estimated, the coder error compensation signal being then calculated and the torque undulations being then taken into account on the basis of the sum of the previous estimated vector {circumflex over (θ)} 1 and the new estimated vector {circumflex over (θ)} 2 , and subsequent iterations are performed on the compensated system on the basis of the sum of the previous estimated vectors {circumflex over (θ)} 1 +{circumflex over (θ)} 2 +{circumflex over (θ)} 3 . . . .
15 . The method according to claim 6 , wherein an interferometric optical measurement method is implemented for measuring the coder errors, said measurements being made on a finite number n p of positions of the axis, and wherein the estimation {circumflex over (θ)} of the parameters characterizing the coder errors and the torque undulations is obtained by minimizing a quadratic criterion I=(Y−φθ) T (Y−φθ) under the equality constraint E c =Cθ, where C is a matrix such that
C
=
[
cos
(
y
1
*
)
sin
(
y
1
*
)
⋯
cos
(
Ny
1
*
)
sin
(
Ny
1
*
)
0
1
,
N
l
⋮
⋮
⋮
⋮
⋮
⋮
cos
(
y
n
p
*
)
sin
(
y
n
p
*
)
⋯
cos
(
Ny
n
p
*
)
cos
(
Ny
n
p
*
)
0
1
,
N
l
]
y
i
*
corresponding to the n p measurement positions on the axis, i∈[1, . . . n p ],
and N being the maximum harmonic rank of a Fourier series decomposition of the coder errors.
16 . The method according to claim 15 , wherein the calculation device is implemented, which is configured for the execution in real time, in a compensation calculation block, of the calculation of a coder error compensation signal in order to compensate for the coder errors, and the calculation device is configured to take into account in real time the torque undulations, and wherein the estimation {circumflex over (θ)} of the vector θ of the parameters characterizing the coder errors and the torque undulations is calculated by iteration: after a first repetition of test phases and a first calculation of the parameters characterizing the coder errors and the torque undulations by solving the global matrix relationship, a first estimated vector {circumflex over (θ)} 1 is estimated, the coder error compensation signal being then calculated and the torque undulations being then taken into account on the basis of said first estimated vector {circumflex over (θ)} 1 , a new repetition of test phases and a new calculation of the parameters characterizing the coder errors and the torque undulations by solving the global matrix relationship are performed on the system so-compensated on the basis of the first estimated vector {circumflex over (θ)} 1 , a new estimated vector {circumflex over (θ)} 2 being estimated, the coder error compensation signal being then calculated and the torque undulations being then taken into account on the basis of the sum of the previous estimated vector {circumflex over (θ)} 1 and the new estimated vector {circumflex over (θ)} 2 , and subsequent iterations are performed on the compensated system on the basis of the sum of the previous estimated vectors {circumflex over (θ)} 1 +{circumflex over (θ)} 2 +{circumflex over (θ)} 3 . . . .
17 . The method according to claim 1 , wherein the number N i of harmonics kω ri is identical for the n tests with rotation setpoints at different speeds of rotation ω ri .
18 . The method according to claim 1 , wherein certain harmonics kω ri are omitted for the estimation of the parameters characterizing the coder errors and the torque undulations.
19 . A device having at least one rotary axis, said device being a motion simulator or a centrifuge and including the looped system configured to execute the method according to claim 1 and including the at least one actuator for rotating the axis and the angle coder producing the signal of position measurement of said axis, and including the calculation device for at least controlling said actuator, the looped system is configured in that the calculation device receives the setpoint signal at the system input, and the signal of position measurement for looping the system, the rotation actuator generating torque undulations d m (t), the coder producing coder errors d c (t) which, for the setpoint signal y c (t) that corresponds to the rotation of the axis at a speed of rotation ω r assumed to be constant, have the spectrum consisted of the fundamental rotation frequency of value ω r and its harmonics kω r , k∈ , the speed of rotation and the rotation frequency having a same unit, the calculation device of the looped system being configured to calculate the command signal in the corrector,
wherein the device is configured for:
in a test phase, for a speed of rotation indexed j, ω rj , assumed to be constant, a specific corrector C ωrj (q) of said constant speed of rotation ω rj is synthesized, said corrector C ωrj (g) being such that, for the frequency domain comprising the fundamental rotation frequency of value ω rj and a determined number of its first harmonics kω rj , k∈ , 1<k≤N l , N l being the maximum rank of the harmonics to be considered, the corrector having a zero gain at these frequencies ω rj , kω rj , which causes an opening of the loop at said frequencies, and a matrix relationship is established between the position measurements and a function of parameters characterizing the coder errors and the torque undulations, on the system with the specific corrector and at the constant speed of rotation ω rj ,
the test phase is repeated a determined number n of times with different setpoints of rotation at speed of rotation ω rj assumed to be constant, where, j∈ , 1<j≤n, the matrix relationships are concatenated in order to obtain an identifiable global matrix relationship,
the parameters characterizing the coder errors and the torque undulations are estimated by solving the global matrix relationship.