Extending the Frequency Response of the MHD Angular
Rate Sensor
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The
Magnetohydrodynamic Angular Rate Sensor (MHD ARS) has a
very wide bandwidth when compared to other types of rate
sensors. The MHD ARS is designed to measure angular
motion over a 1 Hz to 1,000 Hz frequency range. Figure 1
provides the magnitude responses for several of ATA
Sensors' transducers and demonstrates the large
bandwidth capability of the MHD ARS. Exhibiting a wide
rate bandwidth is ideal formany applications, including
crash dummy testing, active control of pointing systems,
and measuring the rotational vibration of complex
structures. However, engineers interested in other
applications, such as biodynamic, ergonomic, vehicle
roll-over, and short duration inertial navigation are
interested in the lower frequency end of the ARS
measurement spectrum, typically below 10 Hz.
Figure 1. Typical Frequency Responses for Several
Magnetohydrodynamic Angular Rate Sensors.
If attitude or
angular position knowledge is required for longer
periods of time than a few hundred milliseconds, then it
is desirable to extend the effective low frequency
corner of the MHD ARS such that the rate or displacement
error is minimized. A recursive digital filter can be
implemented to perform the frequency extension. The
advantage of using a digital filter as opposed to an
analog filter is that the digital filter can be
conveniently reconfigured, even updated adaptively
during operation whereas an analog filter requires
additional board space and power and cannot be easily
reconfigured for varying applications.
Extending the
bandwidth at the lower frequencies is viable for time
histories of only a few seconds in most cases. This is
because the sensitivity of the MHD ARS will fall off at
least 20 dB per decade as the frequency approaches zero
Hz (steady state or constant rate). In other words, the
MHD ARS cannot measure a steady state rate like a gyro
and will have a zero output with a constant rate input.
However, on the other end of the spectrum, there are
very few rate sensors that can measure angular rate
above 1 kHz. Most gyros have upper –3 dB points below
100 Hz whereas the MHD sensors can have –3dB points
above 1.5 kHz, set via the low pass filter in the
internal electronics.
The first high pass
filter corner in the MHD ARS is actually dominated by
the physics of the sense channel. The back-EMF produced
by the MHD effect coupled with the viscosity will cause
the fluid in the sensor to ‘catch up’ with the sensor
case when subjected to low frequency angular rotation.
In addition to the sense element corner, a second high
pass corner is placed in the internal electronics of
most MHD ARS models to remove the offset bias after the
first stage of amplification. As mentioned before, the
upper –3dB LPF corner is set in the integral electronics
and is not limited by the sense element below 5 kHz.
Low Frequency Compensation for
MHD ARS Models with 2 HPF Poles:
The low frequency (<100 Hz)
response of the ARS-01, ARS-03, ARS-04, ARS-09, or
Dynacube™ can be represented as:
where: K = angular rate scale
factor
f1 = sense element corner
f2 = electronics high pass filter (HPF)
corner
We will use the
typical ARS-01 response as an example to illustrate how
the compensation filter C(s) can be used to restore the
low frequency content of the ARS-01 rate output. The
first frequency corner f1 for a typical
ARS-01 is the physical corner of the sense element at
about 0.25 Hz. The second lower corner, f2,
is the high pass filter set to 0.065 Hz within the
internal signal conditioning electronics enclosed within
the header of the ARS-01.
An upper low pass
filter corner fH, typically 1650 Hz, is also
set within the integral electronics of the ARS-01 but
has negligible effect below 100 Hz. Using pole
cancellation, a compensation filter was designed to
effectively reduce the low frequency corner for the
ARS-01. The continuous compensation filter C(s) can be
used to effectively move the HPF poles down in frequency
to improve the low frequency response (FRF).
The
compensation filter can be represented as:
where f3
and f4 are the new corners set lower than f1
and f2. The compensated FRF HC(s)
for the sensor using the compensation filter C(s)
becomes:
where f3
and f4 are the new corner frequencies. The
compensated response Hc(s) will behave as though it has
lower frequency poles at f3 and f4
(f3 = f4 =0.002 Hz for this
example) which are lower than the original ARS-01 poles
at f1 and f2 (f1 =0.25
Hz, f2 =0.065 Hz). Figure 2 is the typical
normalized ARS-01 magnitude and phase response overlaid
with the compensation filter response, C(s), and the
extended (compensated) ARS-01 response, Hc(s)=H(s)C(s).
Figure 2.
Overlay of the Normalized Response of the ARS-01, H(s),
the compensation filter C(s), and the extended
(compensated) ARS-01 Response, Hc(s)=H(s)C(s).
A z-domain digital
filter can be used to compensate the rate signal time
history to restore low frequency rate information, which
can then be integrated to yield angle. The digital
filter C(z) is found using zero-pole mapping technique.
where
t is the sample
period or 1/sampling frequency.
The difference equation for C(z) is then,
where xk =
uncompensated rate input
yk
= compensated rate output.
t = the time
step between samples
Low
Frequency Compensation for MHD ARS Models with 1 HPF
Pole (ARS-06):
The low
frequency (<100 Hz) response of the ARS-06 can be
represented as:
where: K = angular rate scale
factor
f1
= sense element corner
An upper low pass filter corner fH, typically
1650 Hz, is also set within the integral electronics of
the ARS-06 but has negligible effect below 100 Hz. The
compensation filter for the single pole HPF MHD can be
represented as:
where f2
is the new corner set lower than f1. The
compensated FRF HC(s) for the sensor using
the compensation filter C(s) becomes:
where f2
is the new corner frequency. The compensated response
Hc(s) will behave as though it has a lower frequency
pole at f2 rather than at f1.
The
difference equation for C(z) is then,
where
xk = uncompensated rate input
yk
= compensated rate output.
t = the time
step between samples
Implementation of
either digital filter is relatively simple on a
computer. Appendix A provides an example of the digital
filter implemented in MATLABTM that was used
to compensate an ARS-01 with the sensitivity or scale
factor, Kw=50mV/rad/sec,
and f1 and f2 corners of 0.25 Hz
and 0.065 Hz respectively. This program could be
implemented for any MHD ARS using the measured scale
factor Kw, and
corner frequencies f1 and f2 that
are supplied with the test data for each MHD ARS model.
The extended corners f3 and f4 are
set within the program. This particular program can also
be implemented in any spreadsheet program if MATLABTM
is not available. The digital compensation algorithm
provided is typically used for post-processing although
the compensation filter could also be loaded into a
Digital Signal Processor (DSP) for real time
applications.
Summary
A digital
compensation filter can be used to extend the low
frequency response of the MHD ARS to yield accurate rate
and angular measurements for time events lasting a few
seconds, with best results for time histories of 1
second or less. Each application requires different rate
or displacement accuracy based on time history length.
Experimentation using the digital compensation filter by
varying the extended corners is recommended to get an
understanding of how the filter behaves. Several factors
can effect the performance of the digital compensation
filter, i.e. the accuracy of the A/D converters, the
stability of the sample rate, and the methods used to
remove pre-event biases and trends from the ARS-01 raw
measurement data that cause errors when using the
compensation filter.
MATLABTM
is a software product from The Math Works, Inc., South
Natick, MA.
Appendix A: ARS-01 Compensation
Example:
An actual example
using real rate data measured with an ARS-01 is useful
to illustrate how the compensation filter can be used to
restore low frequency content. An optical encoder was
used as the reference to measure the input angular
displacement. The rate reference was calculated from the
encoder displacement by differentiating the encoder
reference angle data. Figure 3 overlays the input rate
reference, (differentiated optical encoder angle) with
the non-compensated ARS-01 rate output, and the
compensated ARS-01 using the algorithm provided in
Appendix A. Figure 4 shows the same overlays in
displacement. The rate profile is basically in one
direction starting from rest and then increasing to a
peak angular rate of over 600 º/s and then back to zero
rate again. The compensated ARS-01 shows very close
agreement to the true input rate as compared to the
non-compensated ARS-01 result. This example illustrates
how effective the compensation filter can be in
restoring the low frequency rate content of an MHD ARS
for rate measurements lasting up to two seconds.
In many
applications precision attitude or angular position
versus time is needed. The compensated and
non-compensated ARS-01 angular rates were integrated
versus time to yield angular displacement and
subsequently overlaid with the encoder reference angle
for direct comparison as shown in Figure 4. Similar to
the rate comparison, the non-compensated ARS-01 angular
displacement result shows considerable error whereas the
compensated ARS-01 result was in close agreement with
the encoder reference angle. This example clearly
illustrates the importance of using the compensation
filter to compensate the angular rate prior to time
integration of the MHD ARS to yield angular
displacement. The compensation filter should be used for
applications where angular position must be precisely
known for up to a few seconds.
Figure 3. Overlays of the Input Rate Reference
(differentiated encoder angle), the Non-Compensated
ARS-01, and the Compensated ARS-01 Rate Results.
Figure 4. Overlays of the Optical Encoder Angle
Reference, the Non-Compensated ARS-01 Result, and
the Compensated ARS-01 Angular Displacement Result.
MATLABTM Digital
Compensation Filter Algorithm
% Frequency Compensation Filter
Example
% Load rate table encoder position and raw ARS data
%file ars_exam includes time
(time), angle (ang), and raw MHD sensor (ars) data
load ars_exam
%calculate sample rate, assumes
a fixed sample rate
tau=time(2)-time(1);
srate=1/tau;
%example ARS-01 scale factor in
V/rad/s
Kwr=.050;
%converts scale factor into
V/º/s
Kw=Kwr*pi/180;
%scale raw ARS-01 data by scale
factor in V/º/s
ars=ars/Kw;
%Loop finds start of the impact
using the encoder position data
i=1;
while ang(i) == 0,i=i+1;end
n_st=i-1
n=length(time);
%Calculate and remove the
pre-impact bias from the scaled ARS-01 data
bias_ars=mean(ars(1:n_st));
%ars becomes the scaled ARS-01
data with pre-impact bias removed
ars=ars-bias_ars;
%hpf pole descriptions
f1=.25; %hpf corner due to ARS-01 sense channel
f2=.065; %hpf corner due to ARS-01 electronics
f3=.002; %new compensated hpf corner 1
f4=.002; %new compensated hpf corner 2
%Calculate digital filter
coefficients based on hpf poles & sample period tau
a=exp((-1)*2*pi*f1*tau);
b=exp((-1)*2*pi*f2*tau);
c=exp((-1)*2*pi*f3*tau);
d=exp((-1)*2*pi*f4*tau);
%initialize output rate vector
to zero
rate=zeros(n,1);
%factor for unity gain at z = 0,
s = inf
fact=c*d/(a*b);
%initialize the 1st 2 samples of
rate after the motion starts
rate(n_st)=ars(n_st);
rate(n_st+1)=ars(n_st+1);
% Apply digital compensation
filter, rate(i) is the frequency compensated ARS-01 rate
for i=n_st+2:n;
rate(i)=fact*(ars(i)-(a+b)*ars(i-1)+a*b*ars(i-2))+(c+d)*rate(i-1)-c*d*rate(i-2);
end;
%Overlay compensated vs
uncompensated ARS-01 rates
plot(time,rate,'-',time,ars,'--');
title('Compensated vs Uncompensated ARS-01 Angular
Rates'),
xlabel('Seconds'),
ylabel('Degrees/Second');
grid
pause
% Integrate compensated and
uncompensated rates to obtain angular displacement
%initialize compensated output angle vector to zero
ang_comp =zeros(n,1);
%initialize uncompensated output
angle vector to zero
ang_noncomp =zeros(n,1);
%sum the rate and ars samples
for i=2:n;
ang_comp(i)=rate(i)+ang_comp(i-1);
ang_noncomp(i)=ars(i)+ang_noncomp(i-1);
end
%divide by the sample rate to
obtain the integrated angle
ang_comp=ang_comp/srate;
ang_noncomp=ang_noncomp/srate;
% Overlay Encoder Angle,
Compensated and Uncompensated ARS-01 results
plot(time,ang,'-',time,ang_comp,'--',time,ang_noncomp,'-');
title('Compensated vs Uncompensated ARS-01 Angular
Displacement'),
xlabel('Seconds'),
ylabel('Degrees');
grid
break |
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