Is the velocity estimate generated by the PicoEncoder software accurate?

We will use frequency domain methods to see if the velocity estimate produced by the PicoEncoder software is accurate, because frequency domain methods are more robust to the noise on the velocity. We are going to use sinusoids data (sin9_3). To estimate the true velocity, we will differentiate the encoder position data, and low pass filter it to reduce the noise in the velocity. Here is the code to do that (the_data is the array of data):

full_revolution = (11*4*30*64);

% get position in radians for entire array
pos_left = 2*pi*( the_data(:,2) - mean( the_data(:,2) ) )/full_revolution;
pos_right = 2*pi*( the_data(:,5) - mean( the_data(:,5) ) )/full_revolution;
% filter the positions
[B,A] = butter(2,0.2)
pos_left = filtfilt(B,A,pos_left);
pos_right = filtfilt(B,A,pos_right);
% differentiate the positions
for i = 2:(length(pos_left) - 1)
  vel_filt_left( i ) = (pos_left( i+1 ) - pos_left( i-1 ))*500;
  vel_filt_right( i ) = (pos_right( i+1 ) - pos_right( i-1 ))*500;
end
% remove ends from arrays
% make the desired data length a power of 2 (makes FFT happier maybe)
log_samples = floor( log2( length( tim ) ) );
n_samples = power( 2, log_samples );
start_index = round( ( length( vel_filt_left ) - n_samples )/2 );
if start_index < 1
  start_index = 1;
end
end_index = n_samples + start_index - 1;
pos_left = pos_left( start_index:end_index );
pos_right = pos_right( start_index:end_index );
vel_filt_left = vel_filt_left( start_index:end_index );
vel_filt_right = vel_filt_right( start_index:end_index );

Here are the Bode plots for velocity to position:

This is what we expected for the magnitude plot. The position is the integral of the velocity, and an integrator generates a linear decrease in gain with frequency on a log-log plot. If the velocity is cos(w*t), the position is sin(w*t)/w.

This is what we expected for the phase plot. The position has a 90 degree (pi/2 radians) phase lag with respect to the velocity. (sine vs. cosine).

If the PicoEncoder velocity is the same as the true velocity, the gain should be a constant (1), and the phase lag should be 0. Here are the Bode plots for true velocity to PicoEncoder velocity:

The PicoEncoder velocity is accurate in terms of magnitude out to about 5 Hz, and then it becomes increasingly too small as the frequency increases.
The PicoEncoder introduces a phase lag starting at about 1Hz that increases with frequency.

The phase lag can be modeled as a 3msec. delay:

However, delays do not affect the magnitude, so a delay does not model the drop in magnitude.

The phase lag is modeled a little less well as a first order system with a time constant of 0.02s (corner frequency of 50Hz):

However, the drop in magnitude is modeled: