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terms of continuous image dynamics with associated analytical stability analysis.
Recently, a nonlinear observer was developed in [10] to asymptotically identify the
structure given the camera motion (
i.e.
, the SfM problem) or to asymptotically iden-
tify the structure and the unknown time-varying angular velocities given the linear
velocities. In another recent result in [28], an IBO approach [20] was used to esti-
mate the structure and the constant angular velocity of the camera given the linear
velocities.
Two continuous nonlinear observers are developed in this chapter. The first ob-
server estimates the structure of the object provided the linear and angular camera
velocities are known with respect to a stationary object (
i.e.
, SfM). A Lyapunov-
based analysis is provided that illustrates global exponential stability of the ob-
server errors provided some observability conditions are satisfied. This observer is
extended to address the SaM problem where the structure, the time-varying angular
velocities and two of the three unknown time-varying linear velocities are estimated
(
i.e.
, one relative linear velocity is assumed to be known). A Lyapunov-based anal-
ysis is provided that indicates the SaM observer errors are globally asymptotically
regulated provided some additional (more restrictive) observability conditions are
satisfied.
The chapter is organized in the following manner. In Section 12.2 relationships
are developed between the 3D Euclidean coordinates and image space coordinates.
Section 12.3 describes the perspective camera motion model. Section 12.4 states the
objective of SfM and SaM, followed by Section 12.4.1 and Section 12.4.2 which
propose continuous nonlinear observers for SfM and SaM problems, respectively,
and the associated stability analyses.
12.2
Euclidean and Image Space Relationships
The development in this chapter is based on the scenario depicted in Figure 12.1
where a moving camera views four or more planar and noncollinear feature points
(denoted by
j
=
{
1
,
2
,....,
n
}∀
n
≥
4) lying fixed in a visible plane
π
r
attached to
an object in front of the camera. In Figure 12.1,
F
r
is a static coordinate frame
attached to the object. A static reference orthogonal coordinate frame
F
c
is attached
to the camera at the location corresponding to an initial point in time
t
0
where the
object is in the camera field of view (FOV). After the initial time, an orthogonal
coordinate frame
F
c
attached to the camera
3
undergoes some rotation
R
(
t
)
∈
SO
(3)
3
F
c
. The rotation between the camera frame
and translation
x
f
(
t
)
∈
R
away from
F
c
and the object frame
F
r
is denoted by
R
(
t
)
∈
SO
(3) and the constant rotation
F
c
and the object frame
F
r
is denoted by
R
∗
∈
between the camera frame
SO
(3).
3
,andthe
Likewise, the translation vector between
F
c
and
F
r
is given as
x
f
(
t
)
∈
R
F
c
and
F
r
is given as
x
∗
f
∈
R
3
. The constant normal
constant translation between
F
c
,isgivenby
n
∗
∈
R
3
,where
d
∗
∈
R
3
denotes the
vector to plane
π
r
, measured in
F
c
and
constant distance between
π
r
along the normal.
3
F
c
and
F
c
are collocated at
t
0
.
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