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Recursive Nonlinear Set–Theoretic Estimation
Based on Pseudo Ellipsoids
Uwe D. Hanebeck
Institute of Automatic Control Engineering
Technische Universit¨at M¨unchen
80290 M¨unchen, Germany
Uwe.Hanebeck@ieee.org
Abstract
In this paper, the problem of estimating a vector
x
of unknown quantities based on a set of measure-
ments depending nonlinearly on x
is considered. The
measurements are assumed to be taken sequentially
and are corrupted by unknown but bounded uncer-
tainties. For this uncertainty model, a systematic de-
sign approach is introduced, which yields closed–form
expressions for the desired nonlinear estimates. The
estimates are recursively calculated and provide solu-
tion sets X containing the feasible sets, i.e., the sets
of all x
consistent with all the measurements available
and their associated bounds. The sets X are tight
upper bounds for the exact feasible sets and are in
general not convex and not connected. The proposed
design approach is versatile and the resulting nonlin-
ear filter algorithms are both easy to implement and
efficient.
1 Introduction
Estimating the state of a dynamic system based on
a sequence of uncertain measurements is a standard
problem in many applications. Usually, this problem
is approached in a stochastic setting. Alternatively,
set–theoretic methods can be used by assuming a pri-
ori bounds on the uncertainties. Estimation then con-
sists of constructing sets of possible states, which are
consistent with the a priori bounds and the measure-
ments. Good overviews about this topic can be found
in [1, 3, 19].
Most work has been done in set–theoretic state es-
timation for linear systems [17, 18, 21, 22]. Appli-
cations in the field of speech processing are found in
[4, 5]. Robotic applications are discussed in [7, 16]. A
comparison of stochastic and set–theoretic estimation
is given in [6].
In the case of nonlinear systems, the complex sets
resulting from the estimation procedure are either
approximated by simple–shaped sets, e.g. ellipsoids,
boxes [15, 12], polytopes [20], or by the union of simple
sets [11, 12].
In [20], an approach similar to the Extended
Kalman Filter (EKF) is pursued. As in the EKF,
the nonlinear state equations are linearized about the
current state estimate. Unlike the EKF, linearization
errors are not neglected, but rather considered as ad-
ditional exogeneous disturbances. Estimation is per-
formed recursively and provides polytopes as approx-
imation of the posterior feasible set.
The procedure in [15] works without linearization
of the nonlinear state equations and provides (recur-
sively) the smallest axis–aligned box enclosing the
posterior feasible set.
In [11], the posterior feasible set is characterized by
enclosing it between internal and external unions of
boxes on the basis of interval analysis. Recursive es-
timation has not been addressed. A more advanced
version of this approach with lower computational
complexity has been introduced in [12].
In this paper, a new nonlinear filtering algorithm
for nonlinear systems is proposed, that does not rely
in any way on linearization. In addition, the new ap-
proach is not based on a grid or on propagating par-
ticles, but provides a finite–dimensional closed–form
representation of the resulting complex–shaped sets.
This includes nonconvex sets or sets that are not even
connected. When applying the new filter recursively
to a sequential stream of measurements, the size of the
analytical representation of the resulting sets does not
grow with the number of measurements.
The key idea of the proposed filter is to trans-
form the original N –dimensional space S to an L–
dimensional hyperspace S
∗
with L>N.Thisre-
sults in an N–dimensional manifold U
∗
, called the
universal manifold, in the L–dimensional transformed
space S
∗
. Complex–shaped subsets of the original
N–dimensional space are then represented by N–
dimensional submanifolds of U
∗
in the space S
∗
.
These submanifolds are defined by the intersection
of L–dimensional simple–shaped sets, e.g. ellipsoids,
with the universal manifold U
∗
. Furthermore, the
nonlinear measurement equation is transformed to a
linear one in the hyperspace S
∗
. Hence, nonlinear fil-
tering can be performed by a linear filter operating in
the transformed space S
∗
.
Section 2 formulates the problem of nonlinear set–
theoretic estimation. In Sec. 3 the concept of modeling
complex–shaped sets is introduced. The nonlinear fil-
tering algorithm is given in Sec. 4. The performance of
the new nonlinear filtering algorithm is demonstrated
by estimating the parameters of a SCARA–type robot
manipulator in Sec. 5.
2 Problem Formulation
Consider a nonlinear discrete–time dynamic system
with system state x
k
(not directly observable) at time
step k.Measurementsˆz
k
of the system output are
taken at time instants k =1, 2,... according to the
nonlinear measurement equation
ˆz
k
= h
k
(x
k
)+v
k
(1)
with measurement uncertainty v
k
, which represents
exogenous noise sources or model parameter uncer-
tainties.
The uncertainties v
k
, k =1, 2,..., are assumed to
be bounded by a known set V
k
according to v
k
∈V
k
.
The set can be of complicated shape, i.e., can be
nonconvex or not connected.
The goal is to estimate at each time instant k the
state x
k
based on all available measurements ˆz
l
for
l =1, 2,... ,k. Of course, a recursive estimation pro-
cedure is preferred, which calculates a state estimate
based on the estimate at the previous time step and
the current measurement. Therefore, it is not required
to store and reprocess all measurements. Further-
more, instead of trying to construct point estimates,
we prefer to calculate at each time instant k all states
that are compatible with the measurements and their
corresponding uncertainties.
On a theoretical level, the problem can easily be
solved: Let X
s
k−1
denote the set of all states com-
patible with all the measurements up to time step
k − 1 and their respective uncertainties. Furthermore,
X
m
k
denotes the set of states defined solely by the
measurement at time k according to
X
m
k
= {x
k
:ˆz
k
− h
k
(x
k
) ∈V
k
} .
Then the estimate X
s
k
is given by the intersection
X
s
k
= X
s
k−1
∩X
m
k
.
However, representing these sets in practical applica-
tions at least approximately by a finite set of parame-
ters is not a trivial task. On one hand, the parameter
set should not be too large, even more, the approx-
imation should degrade gracefully with a decreasing
number of parameters. On the other hand, the num-
ber of parameters should not be permanently growing
with an increasing number of incoming measurements.
Hence, the remainder of this paper is concerned with
a new parametric representation of complex–shaped
sets and an efficient procedure for calculating the cor-
responding parameters.
3 Pseudo Ellipsoids
The key idea of this paper is to represent an un-
certainty X
k
with a complicated shape in the N –
dimensional original space S by a simpler shaped un-
certainty X
∗
k
in an L–dimensional hyperspace S
∗
with
L>N.Pointsx
k
in S are related to points x
∗
k
in S
∗
via a nonlinear transformation T (.) according to
x
∗
k
= T (x
k
)=
T
1
(x),T
2
(x), ... , T
L
(x)
T
.
Hence, T
(.) defines an N –dimensional manifold U
∗
in
an L–dimensional space.
In addition, L–dimensional sets X
∗
k
of simple shape
are defined in the transformed space S
∗
. Here, ellip-
soidal sets according to
X
∗
k
=
x
∗
k
:(x
∗
k
− ˆx
∗
k
)
T
(X
∗
k
)
−1
(x
∗
k
− ˆx
∗
k
) ≤ 1
are used, where ˆx
∗
k
is the ellipsoid midpoint and X
∗
k
is
a symmetric positive definite matrix.
The intersection of an ellipsoid X
∗
k
with the mani-
fold U
∗
defines a submanifold of U
∗
, which, in turn,
defines a complicated set in the original space S.
Remark 3.1 A complex–shaped set in the original
space S is defined by both the transformation T
(.) and
the pseudo ellipsoid X
∗
k
.
In many cases, the type of transformation T
(.)re-
sults directly from the nonlinearities considered. For
example, when considering polynomial nonlinearities,
a polynomial transformation is used. For trigonomet-
ric nonlinearities, a trigonometric transformation can
be used.
However, to simplify application of the new filtering
approach, a generic transformation is helpful. For that
purpose, Bernstein polynomials are used, since their
approximation capabilities are sufficient for a large
class of nonlinearities. In addition, they lead to better
conditioned calculations than standard polynomials.
Multidimensional Bernstein polynomials are defined
on the basis of one–dimensional Bernstein polynomi-
als, which on the interval [l, r]aregivenby
H
n
i
(x)=
n
i
l − x
l − r
i
r − x
r − l
n−i
for i =0,... ,n.With
x
k
=
x
k,1
x
k,2
... x
k,N
T
,
the above transformation is defined by
T
i
(x
k
)=
N
j=1
H
L
j
−1
i
j
(x
k,j
) ,
for i
j
=0,... ,L
j
− 1, j =1,... ,N, L =
N
j=1
L
j
,
and i =
N
j=1
i
j
.
4 Filtering
Based on the concept of pseudo ellipsoids, which
represent complex–shaped sets in the original space S
by pseudo ellipsoids in the hyperspace S
∗
, the nonlin-
ear filter step can now be performed by a linear filter
in the hyperspace S
∗
. For that purpose, a pseudo–
linear expansion of the nonlinear measurement equa-
tion h
k
(.) is performed according to
h
k
(x
k
)=H
∗
k
x
∗
k
+ e
h
k
≈ H
∗
k
x
∗
k
,
where e
h
k
represents the approximation error defined
by e
h
k
= h
k
(x
k
) − H
∗
k
x
∗
k
.
In general, the expansion can be enhanced by an
additional nonlinear transformation g
(.)ofthemea-
surements according to
g
(ˆz
k
− v
k
)=g(h
k
(x
k
)) .
The left hand side can be approximated by
g
(ˆz
k
− v
k
)=ˆz
∗
k
− G
∗
k
v
∗
k
− e
v,∗
k
≈ ˆz
∗
k
− G
∗
k
v
∗
k
,
where ˆz
∗
k
and G
∗
k
are nonlinear functions of ˆz
k
and
v
∗
k
is a nonlinear function of v
k
. e
v,∗
k
accounts for the
approximation error.
The right hand side is again approximated accord-
ing to
g
(h
k
(x
k
)) = H
∗
k
x
∗
k
+ e
h,∗
k
≈ H
∗
k
x
∗
k
with approximation error e
h,∗
k
.Asaresult,themea-
surement equation in the hyperspace is obtained ac-
cording to
z
∗
k
= H
∗
k
x
∗
k
+ e
h,∗
k
+ G
∗
k
v
∗
k
+ e
v,∗
k
w
∗
k
with w
∗
k
representing the total uncertainty.
Let the set of all predicted states be given by the
set X
p
k
, which is defined in the transformed space S
∗
by
X
p,∗
k
=
x
∗
k
:(x
∗
k
− ˆx
p,∗
k
)
T
(E
p,∗
k
)
−1
(x
∗
k
− ˆx
p,∗
k
) ≤ 1
.
Furthermore, let w
∗
k
be bounded by the set
W
∗
k
=
w
∗
k
:(w
∗
k
)
T
(W
∗
k
)
−1
w
∗
k
≤ 1
.
Then, the set defined by the measurement is given by
X
m,∗
k
=
x
∗
k
:(ˆz
∗
k
− H
∗
k
x
∗
k
)
T
(W
∗
k
)
−1
(ˆz
∗
k
− H
∗
k
x
∗
k
) ≤ 1
.
The fusion result is given by a set X
s,∗
k
(again an el-
lipsoid in the transformed space, but a set of compli-
cated shape in the original space!) that contains the
intersection of the ellipsoids X
p,∗
k
and X
m,∗
k
. Hence,
X
s,∗
k
is obtained by a linear set–theoretic filter in the
hyperspace S
∗
[19]
X
s,∗
k
=
x
∗
k
:(x
∗
k
− ˆx
s,∗
k
)
T
(E
s,∗
k
)
−1
(x
∗
k
− ˆx
s,∗
k
) ≤ 1
with
ˆx
s,∗
k
=ˆx
p,∗
k
+ λ
∗
k
E
p,∗
k
(H
∗
k
)
T
W
∗
k
+ λ
∗
k
H
∗
k
E
p,∗
k
(H
∗
k
)
T
−1
ˆz
∗
k
− H
∗
k
ˆx
p,∗
k
,
and
E
s,∗
k
= d
∗
k
P
s,∗
k
P
s,∗
k
= E
p,∗
k
− λ
∗
k
E
p,∗
k
(H
∗
k
)
T
W
∗
k
+ λ
∗
k
H
∗
k
E
p,∗
k
(H
∗
k
)
T
−1
H
∗
k
E
p,∗
k
,
where
d
∗
k
=1 + λ
∗
k
− λ
∗
k
ˆz
∗
k
− H
∗
k
ˆx
p,∗
k
T
W
∗
k
+ λ
∗
k
H
∗
k
E
p,∗
k
(H
∗
k
)
T
−1
ˆz
∗
k
− H
∗
k
ˆx
p,∗
k
.
Using this form of bounding ellipsoid for the exact
intersection of X
p,∗
k
, X
m,∗
k
in the transformed space S
∗
offers the advantage that the resulting set X
s
k
(λ
k
) in
the original space S possesses the following property,
which is desirable in applications in the sense, that no
new uncertainty is introduced:
l
1
l
2
x
EE
x
Figure 1: Schematical top view of the considered
type of 2D SCARA robot manipulator with segment
lengths l
1
, l
2
and joint angles φ
1
, φ
2
.
Lemma 4.1 If the two sets X
p
k
und X
m
k
overlap, the
filtering result X
s
k
(λ
k
) contains the exact intersection
X
p
k
∩X
m
k
and is itself contained in their union X
p
k
∪
X
m
k
. Hence, it holds
(X
p
k
∩X
m
k
) ⊂X
s
k
(λ
k
) ⊂ (X
p
k
∪X
m
k
)
for all λ
k
∈ [0, ∞].
The fusion parameter λ
k
is selected in such a way,
that a certain measure of the size of the set X
s
k
is min-
imized. (How to obtain the minimum volume ellipsoid
in a linear setting is discussed in [2].)
5 Simulation Example
We consider a SCARA–type robot manipulator
with two degrees of freedom according to Fig. 1. The
segment lengths l
1
, l
2
are assumed to be unknown and
are not amenable to direct measurements. Only uncer-
tain measurements of the distance r of the end–effector
from the origin are available for different angles φ
1
, φ
2
.
The position of the end–effector with respect to the
origin is given by
x
EE
=
l
1
cos(φ
1
)+l
2
cos(φ
1
+ φ
2
)
l
1
sin(φ
1
)+l
2
sin(φ
1
+ φ
2
)
.
Hence, the distance r of the end–effector from the
origin is given by
r =
l
2
1
+ l
2
2
+2l
1
l
2
cos(φ
2
) .
Given this nonlinear relation, the segment lengths l
1
,
l
2
are estimated (N = 2) based on measured distances
ˆr
k
, k =1, 2,... for different angles φ
2,k
.Themea-
surement uncertainties are assumed to be bounded
according to
|r
k
− ˆr
k
|≤R,
which gives
ˆr
k
=
l
2
1
+ l
2
2
+2l
1
l
2
cos(φ
2,k
)+v
k
with v
2
k
≤ R
2
.
The parameters used during the simulation are
l
1
= 500 mm ,
l
2
= 350 mm ,
R =40mm .
The filtering procedure starts with an axis–aligned
uncertainty box
X
p
1
=[0, 1000] × [0, 1000] mm
2
.
Sequentially, measurements ˆr
k
, k =1,... ,4, for
φ
2,1
=90
o
, φ
2,2
= 120
o
, φ
2,3
= 160
o
, φ
2,4
= 170
o
are used to update the initial estimate.
The function g
(.) has been selected to g(x)=
x
2
x
4
x
6
T
, T
i
(x), i =1,... ,L, are chosen as
multidimensional Bernstein polynomials. The results
are visualized
1
for n = 6 in Fig. 2, where n is the
order of the one–dimensional Bernstein polynomials,
which gives L = 49. Obviously, the result of the pro-
posed new filter is a tight approximation of the exact
estimation result at every time step.
6 Conclusions
In this paper, an efficient algorithm for the recur-
sive calculation of tight closed–form approximations of
the feasible sets in nonlinear set–theoretic estimation
problems has been presented. The resulting sets are
a much better approximation compared to simple sets
like hyper–rectangles or ellipsoids and are in general
of complex shape, i.e., nonconvex and not connected.
The same methodology has been applied to stochas-
tic nonlinear systems [10]. For the case of mixed
stochastic and set–theoretic uncertainties, the filter-
ing algorithms presented in [8, 9] will be generalized
to nonlinear systems.
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The result of updating the estimate based on measurement
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