Instrumentation & Measurement Magazine 25-4 - 8

In practice, since the measurement results are expressed as in
(4), where pk is the ideal true value of the measured quantity, the
estimates are only related to the likelihood function obtained
from a sampled distribution (in the frequentist sense) that is obtained
by repeated measurements (Type A analysis [2]).
Notice how the WLS, written here in its recursive form,
and the KF basically differ in the prior (e.g., process-based
prediction) part. Furthermore, while the KF is optimal if the
uncertainties are Gaussian, the WLS, which is solely based on
the likelihood function, turns out to be a disguised version of
the ML for Gaussian uncertainties and optimal in its turn [6].
In the case of nonlinear dynamics (1) and/or of nonlinear output
function (3), extension of both algorithms can be easily
derived, such as the Extended KF, the Unscented KF or the
Nonlinear WLS [5].
The Dead-reckoning Phenomenon
We have seen that a localization problem involves both the dynamic
and the output function of a system, while a positioning
problem relies only on the output functions (3) or (4). However,
it is also possible to use just the model (1) or (2) to estimate
the position of a robot. In this case, there is a lack of observability
(being the output matrix C=0); hence, the initial position p0
in the fixed reference frame W should be known a-priori, while
the use of only proprioceptive sensors (e.g., odometers) or exteroceptive
sensors measuring quantities in the mobile frame
Mk
only (e.g., visual odometry) is adopted. In such a case, only
the prediction part of the KF in Fig. 2 can be carried out. Since
the covariance matrices Pk
and Qk
involved in the prediction
step are positive semi-definite and combined through a sum
of quadratic forms, the eigenvalues of the covariance matrix
Pk
cannot asymptotically converge towards 0, even if an infinite
number of measurements is collected, i.e., for k  
. Moreover, a typical mobile agent moving on a plane is usually
referred to as a driftless system, having an equilibrium (i.e.,
pk+1
=pk) when uk
=0 in (1) or (2). Hence, for a vehicle moving on
a plane without output functions (3) or (4), the eigenvalues of
the covariance matrix Pk
grow unbounded when k  : the
dead-reckoning phenomenon. This is somehow trivial since the
system is unobservable.
Notice that this phenomenon is implicit in Simultaneous
Localization and Mapping (SLAM) problems, where
the mobile agent is supposed to estimate a map of an unknown
environment while simultaneously localizing in the
estimated map. Indeed, the measurement results can only be
expressed in the moving reference frame Mk
. Nevertheless,
with a loop closure, the uncertainty of a SLAM problem can
be reduced [1].
Positioning and Localization: Final Comments
The previously introduced concepts can be applied to any estimation
problem for any AS. For example, the same ideas
underlying the design of estimators for localization or positioning
problems can be applied to human tracking, provided
that a model, e.g., [11], is given. Moreover, it is becoming popular
nowadays the idea of active localization, for which the
8
agent follows desired paths to control the uncertainty growth.
Finally, it is worth mentioning that a maximum target uncertainty
can be always managed, either by instrumenting the
environment with a suitable set of sensors [12] or by accessing
on-demand to the available localization infrastructure [13].
Distributed Localization: The Role of Covariance
Consider a simple example: two robots X and Y are moving
on the same curvilinear abscissa (e.g., a corridor), whose coordinates
are xk
and yk
, respectively. Assuming known constant
velocities vx,k and vy,k, respectively, we have the following scalar
linear systems derived from (2):
1
x x bv fky y b v f
k k y y k
k
x xk x xk, and
,
,,
xk
where~ 0,
x k
2 yk ~ 0,

2
y k
  1 ,,  y y k , (11)
 and ,, are the two
process uncertainties. Suppose that exteroceptive sensors
measuring the actual locations in W are available and modelled
from (4) as:
zx z yk   y k
xk k xk and
yk ~ 0, yk . Since all of the uncertainties are white and
,,

x and ˆk
y . However, let us assume that
. Hence, there is
k
, ,, ,
y k
with measurement uncertainties given byxk ~ 0, xk
2
,,
 and
2
mutually uncorrelated, each robot can implement a KF individually
to estimate ˆk
robot X can measure the relative distance to robot Y (e.g., using
a laser scanner) in the moving frame Mx,k
k
 xy xy k
~ (0,,,xyk
xyk ): if the robot Y sends its own estimated posian
additional measure ,,xy k   with uncertainty
2
tion to X, we have an indirect measurement of the position xk
in W, i.e.:
z
where
xyk xyk ky xyk
,, ,xyk k
y yy 

k kk
     
ˆ
xy k, ,
(13)
ˆ is the estimation error of the KF executed on
the robot Y with variance given by Py,k (Fig. 2). Assuming that
εxy,k
variance Py,k, the overall measurement uncertainty of zxy,k
available to the robot X and equals 2
 P .
xy,,k
y k
If this measurement is additionally used in the KF of robot
X, the estimation error variance Px,k certainly decreases [6].
Hence, it would be beneficial to do the same for robot Y if endowed
with a similar relative sensor. Unfortunately, this is not
working so straightforwardly: indeed, once X first uses zxy,k
estimate ˆk
x becomes correlated with ˆk
z
yxk yxk kx yxk
yxk k
    
ˆ
fore, if the process is now repeated for Y, the measurements:
,, ,
yxk, ,
are now correlated with ˆk
(14)
y , which violates the KF assumption
of model and measurement uncertainties to be uncorrelated.
The problem can be circumvented by exchanging the mutual
covariance quantity between X and Y. Indeed, by rewriting
the problem as a single KF with state pk
=[xk,yk]T
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and applying
June 2022
, the
y by means of (13). Thereis,
again, generated by a white stochastic process, and assuming
that the robot Ysends the estimate ˆk
y along with the
is
,
(12)
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