Statistical Models of Inertial Sensors and Integral Error Bounds

Abstract

Inertial sensors such as gyroscopes and accelerometers are important components of inertial measurement units (IMUs). Sensor output signals are corrupted by additive noise plus a random drift component. This drift component, also called bias, is modeled using different types of random processes. This chapter considers the random components that are useful for modeling modern tactical-graded MEMS sensors. These components contribute to errors in the first and second integrals of the sensor output. The main contribution of this chapter is the derivation of a statistical bound on the magnitude of the error in the integral of a sensor signal due to noise and drift. This bound is a simple function of the Allan variance of a sensor.

Appendix

1.1 Program to Simulate Calibration Signal

% Script to compute calibration signal and compute % and plot Allan variance T=1/100/3600; % sampling interval (hours) D=48; % duration of calibration signal (hours) N=floor(D/T); % Define parameters for sensor Q=4.3047e-01; R=2.6518e-02; a1=3.0957e+01; Q1=2.1572e+01; a2=3.6367e+00; Q2=5.7928e+00; arw=randn(N,1)∗sqrt(R/T); rrw=randn(N,1)∗sqrt(Q1/T); rrw2=randn(N,1)∗sqrt(Q2/T); rrw3=randn(N,1)∗sqrt(Q∗T); A=[1 -exp(-a1∗T)]; B=(1-exp(-a1∗T))/a1; frrw=filter(B,A,rrw); A2=[1 -exp(-a2∗T)]; B2=(1-exp(-a2∗T))/a2; frrw2=filter(B2,A2,rrw2); y=arw+frrw+frrw2+cumsum(rrw3); % Compute Allan variance tau=logspace(-3,0.6,20)'; %smoothing lags in hours m=floor(tau/(T)); %smoothing lags in samples [avar]=compute_avar(y,m); loglog(tau,avar)

1.2 Program to Compute Allan Variance

function avar=compute_avar(y,m) % y is a calibration signal % m is a vector of smoothing interval lengths M=floor(length(y)./m); jj=length(m); avar=zeros(jj,1); for j=1:jj     mm=m(j);     MM=M(j);     D=zeros(MM,1);     for i=1:MM;         D(i)=sum(y((i-1)∗mm+1:i∗mm))/mm;     end     v=diff(D).ˆ2;     avar(j)=0.5∗(mean(v)); end end

1.3 Program to Estimate Sensor Parameters by Least Squares

function [R,Q,Q1,a1,Q2,a2]=LSfit(avar,tau,x0) % this function fits a model of additive noise, random walk, % and two GM components.  x0 is a vector of initial guesses %  for each parameter that the function returns % avar is a vector of Allan variance points computed % at each value of tau, which is a vector of smoothing intervals options=optimset('display','off','TolX',1e-6,    'TolFun',1e-6); x=fminsearch(@efun,x0,options,tau,avar); R=x(1); Q=x(2); Q1=x(3); a1=x(4); Q2=x(5); a2=x(6); function f=efun(x,tau,avar) R=x(1); Q=x(2); Q1=x(3); a1=x(4); Q2=x(5); a2=x(6); x1=R./tau; x2=Q∗tau/3; x3=Q1/a1ˆ2./tau.∗(1-1/2/a1./tau.∗(3-4∗exp(-a1∗tau) +exp(-2∗a1∗tau))); x4=Q2/a2ˆ2./tau.∗(1-1/2/a2./tau.∗(3-4∗exp(-a2∗tau) +exp(-2∗a2∗tau))); %err=(log10(avar)-log10(x1+x2+x3+x4)); err=((avar)-(x1+x2+x3+x4)); f=sum(err.∗err);