Can the Kalman filter handle variations in sampling interval?

[bolly]

Dear Community,

I want to use a Kalman- filter approach to calculate inclination angles (pitch and roll) obtained from accelerometer- and gyroscope- raw data. However the problem is that my sampling interval dt is not constant – it has a significant jitter instead. To investigate whether the kalman filter can handle such data I’ve experimented using the following MATLAB code copied from here (http://web.csulb.edu/~hill/ee400d/Project%20Folder/Kalman%20Filter%20Research.pdf):

stacksize = 100;dt_stack  = rand(stacksize,1)/2;
t_stack = zeros(stacksize,1);

t_stack(1) = 0;
for count=2:stacksize
t_stack(count) = t_stack(count-1) + dt_stack(count);
end

x =     sin(t_stack); % angle
x_dot = cos(t_stack); % angular velocity

figure;
subplot(3,1,1);
plot(t_stack);
subplot(3,1,2);
plot(t_stack,x);
hold on;
plot(t_stack,x_dot,    'Color','red');
P = [1 0; 0 1];R_angle = 0.9;
Q_angle = 0.05;
Q_gyro = 0.5;

Q = [Q_angle 0; 0 Q_gyro];
A = [0 -1; 0 0];
q_bias = 0; % Initialize gyro bias
angle = 0; % Initialize gyro angle
q_m = 0;
X = [0; 0];
x1 = zeros(stacksize);
x2 = zeros(stacksize);

 H = [1 0];

 x_calc  = 0;
 x_dot_corr = 0;
 gyro_bias = 0;

 angle_err = 0;

 for i=1:100 

  x_dot_corr = x_dot(i) - gyro_bias; % /* Pitch gyro measurement */ 
  x_calc = x_calc + x_dot_corr*dt_stack(i);
  x_err = x(i)- x_calc;
 
  Pdot = A*P + P*A' + Q; 
  P = P + Pdot*dt_stack(i);
  E = H*P*H' + R_angle;

  K = P*H'*inv(E);
  P = P - K*H*P;

  X = X + K * x_err; 
  x1(i) = X(1);
  x2(i) = X(2);
  x_calc= x1(i);
  gyro_bias = x2(i); 
end

 x = x(1:stacksize);
 x1 = x1(1:stacksize);

 length(x)
 length(x1)
 length(t_stack)% Plot the result using kalman filter
subplot(3,1,3);
plot(t_stack,x);
hold on
plot(t_stack,x1,'Color','green');

xlabel('time(s)');
ylabel('x(t)');
legend('actual','kalman');

I’ve found that with increasing bandwidth of dt the filter refuses to work. As far I’ve understood the kalman approach can be applied to zero the linear integration-error obtained by integrating the angular velocity measured using a gyroscope (if the this error propagtes linear).
However can it also be applied to zero the sampling error caused by a jitter of dt ?

I would be very happy if someone could give a hint why the kalman approach refuses to work with increasing bandwidth of dt.

 

[bolly]

Dear Community,

after investigating the Kalman filter in more detail I've found that it can handle dt jitter if the sampling rate is set high compared with the settling time of the filter since the filter only yields good results after a settling time of approx. 20 times higher than the raw data sampling interval dt. Shorting the settling time by changing the kalman variances causes noisy results.

 

[Dr. Courtney]

https://arxiv.org/pdf/1507.01832.pdf

Computing Fourier transforms with explicit integration rather than FFT methods is better able to handle variations in step sizes. The speed of most computers no longer requires FFT methods for many data analysis tasks.