Application Note AN-01

                             Application Note AN-01
                                   Printable .pdf Format
       Extending the Frequency Response of the MHD Angular Rate Sensor

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’ rate transducers which demonstrates the large bandwidth capability of the MHD ARS. Exhibiting a wide rate bandwidth is deal for many 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. 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 is easily 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, 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 signal conditioning 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 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:

The low frequency (<10 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.

Figure 1. Typical Frequency Responses for Several Magnetohydrodynamic Angular Rate Sensors.

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 and is not shown in
the ARS-01 FRF. 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).

Where f3 and f4 are the new corners set lower than f1and 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 f

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 3 overlays the ARS-01 response H(s), the compensation filter
response C(s), and the compensated ARS-01 response HC(s). Figures 2 and Figure 3
are the typical normalized ARS-01 magnitude response 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 Magnitude 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).

The difference equation for C(z) is then,

Where x_k = uncompensated rate input
y_k = compensated rate output.

Implementation of the digital filter is relatively simple on a computer. Appendix A
provides a MATLAB™ 1 program to compensate the MHD ARS with the sensitivity or
scale factor, Kw, and the f1 and f2 corners currently set for an example ARS-01 as
described above. 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. The digital compensation algorithm provided in Appendix A is
typically used for post-processing although the compensation filter could also be
loaded into a Digital Signal Processor (DSP) for real time applications.

^1. MATLAB^TM is a software product from The Math Works, Inc., South Natick, MA.

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 4 overlays the input rate
reference, (differentiated optical encoder angle) with the non-compensated ARS-01
rate output, and the compensated ARS-01 using the same algorithm as provided in
Appendix A. Figure 4 shows the characteristic “droop” of the non-compensated
ARS-01 output where the rate error increases as the time increases. This is because
the ARS-01 cannot measure a constant angular rate and behaves analogous to a
gyro that is high passed with high pass corners at f1 and f2. The rate profile is
basically in one direction starting from rest and then increasing to a peak angular
rate of over 700 degrees/second 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 to restore the low frequency rate content of an MHD ARS
for rate measurements lasting up to two seconds.

Figure 3. Overlays of the Input Rate Reference (differentiated encoder angle), the
Non-Compensated ARS-01, and the Compensated ARS-01 Rate Results.

Angular Displacement Comparison:

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 5. Similar to the rate comparison, the non-compensated ARS-01 angular displacement result shows considerable error whereas versus 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 4. Overlays of the Optical Encoder Angle Reference, the Non-Compensated
ARS-01 Result, and the Compensated ARS-01 Angular Displacement Result.

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
f₃ and f₄ 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.

              Appendix A. MATLAB^TM Digital Compensation Filter Algorithm
                                                   Printable .pdf Format

% Frequency Compensation Filter Example
% Load rate table encoder position and raw ARS data
load ars_exam %includes time (time), angle (ang), and raw MHD sensor (ars) data
srate=1/(time(2)-time(1)); %sample rate
tau=1/srate;               %sample period
Kwr=.050;       %example ARS-01 scale factor
Kw=Kwr*pi/180; %converts scale factor into volts/(deg/s) from volts/(rad/s)
ars=(ars./Kw); %scale raw ARS-01 data via scale factor in volts/(deg/s)
%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); %total points
%Calculate and remove the pre-impact bias from the scale ARS-01 data
bias_ars=mean(ars(1:n_st));
ars=ars-bias_ars; %ars is the scaled ARS-01 data with pre-impact bias removed
%hpf pole descriptions
f1=.065; %hpf corner due to ARS-01 electronics
f2=.25;   %hpf corner due to ARS-01 sense channel
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);
%
rate=zeros(n,1);      %initialize output rate vector to zero
fact=c*d/a/b;                 %factor for unity gain at z = 0, s = inf
rate(n_st)=ars(n_st);       %initialize rate(1) to ars(1)
rate(n_st+1)=ars(n_st+1);   %initialize rate(2) to ars(2)
% 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
ang_comp =zeros(n,1);     %initialize output angle vector to zero
ang_noncomp =zeros(n,1); %initialize output angle vector to zero
for i=2:n;
ang_comp(i)=rate(i)+ang_comp(i-1);
ang_noncomp(i)=ars(i)+ang_noncomp(i-1);
end
ang_comp=ang_comp/srate;       %apply sample rate factor
ang_noncomp=ang_noncomp/srate; %apply sample rate factor
% 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