model, which is built up here by the application of autoregres- 
sive integrated (ARI-) processes. The time-characteristic of 
each orientation parameter will be modelled by a specific 
representation of an ARI-process. 
In chapter 2 the theoretical background of the time-series 
model will be outlined. Chapter 3 will present some applica- 
tions of the theory : 1. modelling the position parameters of 
an aircraft flight measured by the Global Positioning System 
GPS, and 2. modelling the attitude parameters of an aircraft 
and of a space shuttle measured by an Inertial Navigation 
System INS. 
2. Theoretical background 
2.1 Autoregressive integrated processes 
Autoregressive integrated (ARI-) processes are a widely applied 
class of stochastic processes for various kinds of time-series 
(Haykin 1979, Kay and Marple 1981). 
An autoregressive (AR-) process of order p describes a statio- 
nary time-series x: by 
P 
Xt = — I aı-Xt-ı + et V(et) = ce? (1) 
1=1 
The time-series considered in our applications are generally 
not stationary. These time-series have to be transformed into 
stationary time-series by the elimination of trends. A usual 
way is to take derivatives of the time-series. The autoregres- 
sive integrated process of order (p,d) is achieved if the d-th 
derivative of the original time-series can be described by a 
stationary AR(p) process. Formally, the derivation can be ex- 
pressed by an additional number d of process parameters ai in 
eq.(1). Then the ARI(p,d) process fully describes the dynamic 
behaviour of any orientation parameter by a number of (p+d) 
process parameters ai and by the variance ce 2 of the prediction 
errors. 
2.2 The filtering algorithm 
Two kinds of equations build up a Gauss-Markov model to esti- 
mate the filtered time-series x: from the observed series y: 
E(Ys) ext V(yt) = On 2 (2) 
ai *Xt-1 Viet) Oe 2 (3) 
T 
o 
1 
" 
M 
E(et ) 
Eq.(2) expresses the observation process with on 2 being the 
variance of the observation noise. Eq.(3) presents the ARI 
model to which the unknown time-series xt has to correspond. 
The estimation of the filtered time-series Xt presumes the 
knowledge of the stochastic part of the Gauss-Markov model, 
i.e. the variances on 2 and oe 2. As the variances in general are 
a priori unknown, they must be estimated by applying the 
variance component estimation (VCE) technique. The formulation 
in the frequency domain published by Fórstner (1984) is recom- 
mended for our application. 
a3