Instrumentation & Measurement Magazine 25-4 - 6

the posterior |ˆpk k
fp z conditional pdf. Finally, the pdf fm
(zk
) assumes
the role of a normalization factor.
Notice that, due to the Markovian property inherited by the
dynamic system of a robot expressed as in (1) or (2), the prior
ˆ
pk
fp zpkˆ 11
| k
and εk
Let us consider an example of a robot moving along a curvilinear
path on a plane. Let us assume that p0
state of the robot, where x0
=[x0,v0]T
fp is generated by the posterior  at the previous
time instant. The nature of the uncertainties ηk
rules
the choice and the attainable performance of the estimator:
◗ If they are generated by stochastic processes that are
white and zero-mean, one solution from the Kalman
Filter (KF) family turns to be a quite effective solution to
implement a Bayesian filter [5];
◗ If, additionally, the system is described by the linear
equations (2) and (4), the KF turns to be the Best Linear
Unbiased Estimator (BLUE);
◗ If the uncertainties are also Gaussian (as assumed previously),
the KF is optimal in the Mean Squared Error sense,
i.e., it reaches the Cramer-Rao lower bound [6]. As a final
remark, when the dynamic model is not explicitly considered
in the estimator design, no prior is given and, hence,
only the likelihood pdf ˆ|lk k
fz p
can be used. In such a
case, popular approaches such the Maximum Likelihood
(ML) or the Least Squares (LS) can be effectively used [6].
Positioning and Localization:
Two Different Problems
A localization problem deals with the estimates ˆk
where uk
is the
is the initial position on the curvilinear
path (expressed in W), while v0 is the initial velocity.
We assume a (time invariant) linear dynamic as in (2), here reported
explicitly:
   u Ap Bu ,
  
   xxT
2
kks
    
01
vv
kk


1
1
s
T
represents the acceleration of the robot. Let us assume
that the vehicle is equipped with odometers and uses an external
reference system to collect a position measurement xk
It then follows that with odometers readings and knowing the
sampling time Ts
, it is possible to define an indirect measurement
of the velocity vk. Therefore, the output equation (4) turns to:
z 
kk
k


 

10
01
xk
v
Cp
(7)
This is a very unusual situation since observability is ensured
with just the first set of measurements, since z0
p representing
the pose of the AS in a global and fixed reference frame W. In
such a case, the exteroceptive sensors are strictly needed. The
measurement results should be expressed directly in W (such
as wireless anchors in known positions or GPS) or local quantities
that are matched against an available map (i.e., a metric
representation of measurable quantities whose coordinates
in W are given). Notice that localization is not positioning, the
latter having radically different characteristics in terms of feasible
estimators, number of sensors and structural properties,
as will be evident in the next section. Localization is probably
the most important problem to solve to ensure autonomy for
mobile robots, since with no knowledge of the pose, it is not
possible to solve any task.
Observability: A Necessary Condition
The design of an estimator for ˆk
observability analysis to prove that the initial state p0
ments z0
p always requires a preliminary
of the robot
can be reconstructed, assuming the knowledge of both the
sequence of input values u0
,...,zk
,...,uk and the sequence of measure.
Since the observability is a structural property of
the system [7], it just depends on the nominal dynamic and
output functions, i.e., neglecting the presence of the uncertainties
ηk
and εk. It is then evident why observability refers to
the initial state po: if the process uncertainties are neglected,
by knowing u0,...,uk it is possible to reconstruct all the state sequence
p1,...,pk using (1) or (2). Furthermore, if the system is
unobservable, it is not possible to design any estimator able to
estimate
p ˆk
timation error covariance Pk
subspace) grows unbounded.
6
, i.e., at least a subset of the eigenvalues of the es(associated
to the unobservable
IEEE Instrumentation & Measurement Magazine
can be
1
k kk
 2
Ts
(6)
in W.
=p0. In other
words, observability is ensured even without the need for the
knowledge of the system dynamic (6), thus resulting in a static
observability condition. Instead, if only the position xk
measured, i.e., if C=[1,0]T
(i.e., with k=0 and k=1), we have z0
1 00

=Cp0
=x0
and:
in (7), static observability does not
hold. However, after two consecutive measurements at time 0
and Ts
z Cp C Ap Bu ,
1   
(8)
in which we have substituted explicitly the dynamic (6). Rearranging
the terms, we can define the following linear system:


z0
 10 CA 00
z CBu 
  CA   

   
   

 
CC zz
pp
  
z CBu 10 1z CBu0 
 O1
1
  00 (9)
The state p0 can then be computed if and only if the observability
matrix O in (9) is invertible, which is the case for the example
at hand. It is now clear why the linear system observability is
a structural property: it is based on the structure of the system
expressed by the matrices C and A. Of course, this analysis can
be extended to a linear system having a state variable 0
p  ,
n
where the observability matrix should be computed up to order
n (Isidori, 2013), i.e.,:
O CA
CA 
2








C
CA
(10)


n 1
and then verifying that O is a of full column rank.
June 2022
Instrumentation & Measurement Magazine 25-4

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