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.
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Appendix
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);
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Vaccaro, R.J., Zaki, A.S. (2018). Statistical Models of Inertial Sensors and Integral Error Bounds. In: Ruffa, A., Toni, B. (eds) Advanced Research in Naval Engineering. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-95117-1_9
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