# A bundle framework for observer design on smooth manifolds with symmetry

Anant A. Joshi1, D.H.S. Maithripala2, Ravi N. Banavar3 1Department of
Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai,
Maharashtra, 400069, India<EMAIL_ADDRESS>
2Sri Lanka Technological Campus, Padukka, KY20400, Sri Lanka.
<EMAIL_ADDRESS>
3Department of Systems and Control Engineering, Indian Institute of Technology
Bombay, Mumbai, Maharashtra, 400069, India<EMAIL_ADDRESS>
###### Abstract

The article presents a bundle framework for nonlinear observer design on a
manifold having a Lie group action. The group action on the manifold
decomposes the manifold to a quotient structure and an orbit space, and the
problem of observer design for the entire system gets decomposed to a design
over the orbit (the group space) and a design over the quotient space. The
emphasis throughout is on studying and exploiting the created geometric
structure. Gradient based observer design on a Lie group falls out as a
special case and is given explicit attention. The concepts developed are
illustrated by applying them on well known examples, which include SLAM and
rigid body observer design.

##  I Introduction

Observer design and estimator design have enjoyed a long history after the
appearance of seminal work [1, 2, 3]. The Kalman filter was developed for
linear systems but it has been modified and applied to various other systems
as well. Much of estimation theory in engineering has centred around the
Kalman filter. The setting of the problem is a vector space and the tools
involve linear systems theory. Nonlinear extensions to the Kalman filter have
not pushed the domain of theoretical ideas, but have largely been restricted
to ideas like linearization, such as the, extended Kalman filter (EKF) [4],
unscented Kalman filter [5, 6, 4], and the multiplicative extended Kalman
filter [7, 8, 9]. The work [10] presents more such techniques for attitude
estimation. The dynamics of a large class of mechanical and aerospace systems,
however, evolve in a nonlinear setting, and in particular, on smooth manifolds
or specifically on Lie groups. Control synthesis for such systems in an
intrinsic framework, that respects the geometry of the underlying manifold,
has been much studied in the past two decades [11, 12, 13, 14]. The parallel,
observer design or estimator design, has received less attention. Our work
focusses on the latter.

There is a large body of work in non-linear observer design, see [15, 16, 17,
18]. Many aero-mechanical systems are mathematically modelled as systems
evolving on Lie groups. Intrinsic observers directly on the Lie group are
designed to avoid the pitfalls of parametrization, like Euler angles (that
suffers from singularity at particualr configurations), or quaternions (that
suffer from over-parametrization of the rotation group). The concept of fusing
two measurements with different frequency characteristics to design an
observer for attitude estimation directly on $\mathop{\mathbb{SO}(3)}$ rather
than using quaternions or Euler angles was presented in [19]. This
complementary filtering was extended to the case when observations are made
from the inertial frame and relayed to the agent, and shows when observability
can be achieved with a single direction measurement [20]. Complementary
filtering on the Lie groups $\mathbb{SE}(3)$ and $\mathbb{SL}(3)$ has also
been done [21, 22]. The work of [23] contains an observer on $\mathbb{SE}(3)$,
that is similar in structure to what is presented in [19].

The work reported in [24] demonstrate a separation principle on Lie groups for
linearized controller and observer design. The invariant extended Kalman
filter in [25, 26] applies the ideas of the conventional extended Kalman
Filter to Lie groups. However, it also involves linearization of state error
to propagate covariance (just as in a conventional EKF). It has been studied
further in [27], in the setting of symmetry preserving observers presented in
[28]. Its stability properties have also been studied in [29, 30]. Extending
these notions to the discrete setting, an intrinsic discrete EKF for Lie
groups is presented in [31]. Motivated by the Kalman filter for linear
systems, an estimator design for discrete time systems evolving on Lie groups
perturbed by stochastic noise is also developed in [32].

Gradient based techniques for designing observers are very intuitive to
understand, since it can be visualised as a gradient descent kind of algorithm
moving to make the observer error zero. The work [33] introduces the idea of
gradient based observer design for kinematic systems evolving on Lie groups
assuming full state and input measurement while [34] gives a gradient based
observer for discrete time observer design on $\mathop{\mathbb{SO}(3)}$. These
ideas are also applied in [35] to the design of gradient based observer on
$\mathbb{SE}(3)$ assuming velocity measurements and measurement of position of
$n$ points whose inertial locations are known. Extending these results
gradient based observers for SLAM have been developed in [36, 37].

Similar to optimal control, optimal observer design has also received some
attention. Near optimal (deterministic) filters (filters on which we know how
far they are from optimality) have been developed for systems evolving on
$\mathop{\mathbb{S}^{1}},\mathbb{SO}(n)$ [38, 39]. The work presented in [40]
considers optimal attitude filtering considering only kinematics and [41]
extends that to dynamics as well (on the tangent bundle of the Lie group).

In this work we propose a unifying bundle framework for observer design for
systems that admit a symmetry. In particular, we focus on the case when there
is a Lie group $G$ acting on the configuration manifold $P$ of the system and
the system is also invariant under this action. It is also assumed that the
same Lie group acts on the output manifold so as to have a group action
equivariant output. When a Lie groups $G$ acts properly on a smooth manifold
$P$, it provides a stratification of the manifold where each stratum
corresponds to an orbit, $\mathcal{O}(\cdot)$, of the group action. The
collection of these orbits, the modulo space of the equivalent classes of
orbits $P/G$, is called the base space and has the structure of a smooth
manifold when the action is free. The triple $(P,P/G,\pi_{\phi})$, where
$\pi_{\phi}:P\to P/G$ is the canonical projection, has the structure of a
principle bundle with structure group $G$ when the action is free (for free
actions, a cross-section of this bundle allows one to associate with each
point, $p\in P$, on the manifold a unique pair $([p],g)$ where $[p]\in P/G$
and $g\in G$). An invariant vectorfield on $P$ induces a well defined unique
vectorfield on the base space $P/G$. Thus the flow of such a vectorfield will
carry orbits to orbits. Thus the flow an invariant vectorfield can be
projected on to the base space $P/G$ and the group $G$. The projection onto
$G$ depends on the cross section in a unique way. This splitting of the flow
induces a corresponding splitting of the system into one that is evolving on
the Lie group and one that is evolving on the base space.

The mathematical structure presented in the previous paragraph can naturally
be applied to observer design for systems evolving on the configuration
manifold. An observer can be designed for the two subsystems individually. For
the subsystem evolving on the Lie group, we define a group action on the
measurement as well. A constructive procedure for observer design is laid down
for this subsystem. In particular, we detail a gradient based observer design
technique stemming from a choice of suitable cost function, that makes the
error dynamics autonomous. However, for the system evolving on the base space,
the methodology is not uniform and is implemented on a case by case basis. We
do not examine this here.

The initial ideas behind this theory of symmetry-preserving observers appear
in [28]. We work in a similar setting as [28]. The same problem, if the
configuration manifold $P$ itself is a Lie group, is presented in [42]. They,
however, adopt a very different methodology from us. They use a method
inspired from observer design for linear systems, like the idea of the
Luenberger observer, in which they augment the vector field governing the
original system with a correction (gain and innovation) term to have desirable
characteristics of the error dynamics. Furthermore, eventually they rely on
linearization of the system to design the observer gains to obtain desirable
characteristics. Our work, however, makes no such approximations or
linearization assumptions and attempts to explot the intrinsic manifold
structure. Furthermore the natural decomposition of the system due to symmetry
is stated briefly and without proofs or much geometric insight in [28], they
don’t use it in their observer design methodology. Mahony et al [43, 44] work
is in the same setting as ours and [45] applies the methodology in [43] to
design an observer for the SLAM problem when the group action is transitive.
Relating this case in our setting, we note that the cross-section is a single
point on the configuration manifold, and thus there is exactly one orbit.
Hence given a base point on the manifold, the rest of the manifold can be
identified with the Lie group.(Consider the action of
$G=\mathop{\mathbb{SO}(3)}$ on $P=\mathbb{S}^{3}$. The action is transitive
i.e. given any element $p\in P$ we can obtain any other element $p^{\prime}\in
P$ via a suitable rotation i.e. via the action of a suitable element in
$\mathop{\mathbb{SO}(3)}$, see for example, Given’s rotations [46]. Hence,
just one element of $P$ suffices to describe the entire space $P$ based on the
action of $\mathop{\mathbb{SO}(3)}$.) It therefore essentially reduces to
observer design on a Lie group, a particular case of the methodology we
propose. However all group actions may not always be transitive. If the group
action is not transitive, there is the formation of quotient manifold and
orbit space which we highlight in this current work.

While we are aware and acknowledge the overlap of our work with existing
previous contributions to which our work is close [28], [43], [45] we believe
that the mathematical framework we present here is more general and intuitive
for the class of systems whose observer design we consider. Our work provides
a new geometric insight into the observer design problem, particularly through
our consideration of the geometric structure created on the configuration
manifold and measurement space (by the Lie group), which has not been
highlighted before.

The paper is organised as follows : Section II takes a brief look at the
decomposition of the manifold created by the Lie group. This is illustrated by
showing an example of the action of $\mathop{\mathbb{SO}(3)}$ on
$\mathbb{R}^{3}\setminus\\{0\\}$. It continues to present a description of how
to Lie group decomposes the output space. Section III addresses how
equivariant control systems decompose in the presence of the preceeding
geometric structure. Section IV details a method to design an observer on a
Lie group. Section V concludes by presenting examples highlighting some of the
developments in previous sections.

##  II Mathematical preliminaries

In this section we briefly introduce the reader to the mathematical tools that
we employ in the rest of the paper. Let $P$ be a smooth manifold of dimension
$n_{P}$, $G$ be a $n_{G}$ dimensional connected Lie group, with $I$ being the
identity element, and let $\mathcal{G}$ be its Lie algebra. In what follows we
will summarize several well known results that are crucial to this work such
as group actions, orbit spaces, infinitesimal generators and invariance under
the group action.

###  II-A Group actions and orbit spaces

Let $\phi:G\times P\to P$ be a proper, constant rank, left or right action of
$G$ on $P$ and let $\mathcal{X}_{P}$ denote the set of smooth vector fields on
$P$. We will frequently use the notation $g\cdot p:=\phi(g,p)$. We distinguish
between two maps - $\phi^{p}(\cdot):G\rightarrow P$ and
$\phi_{g}(\cdot):P\rightarrow P$ that are associated with $\phi(\cdot,\cdot)$
as follows:

$\displaystyle\phi_{g}(p):=\phi(g,p)\quad\forall g\in
G\quad\mathrm{and}\quad\phi^{p}(g):=\phi(g,p)\quad\forall p\in P.$

The orbit of $\phi$ through $p$ is defined to be the set of points

$\displaystyle\mathcal{O}(p)$ $\displaystyle:=\\{\phi_{g}(p)\mid g\in G\\}.$

Since the orbits are equivalence classes we will also denote it compactly as
${}_{G}[p]$ 111Given a smooth manifold $P$ and a Lie group $G$, we will denote
the orbit of $p$ under $G$ (with left action) as ${}_{G}[p]$ and under right
action as $[p]_{G}$. In particular, the smooth manifold $P$ may itself be a
Lie group, and $G$ may be a Lie subgroup of $P$.. Let $P/G$ denote the space
of all orbits of $\phi$ with $\pi_{\phi}:P\to P/G$ denoting the canonical
projection map. That is let $\pi_{\phi}(p)=_{G}[p]$. We will distinguish
between $\mathcal{O}(p)$ and ${}_{G}[p]$ as follows : we will view
$\mathcal{O}(p)$ as a collection of points in $P$ i.e. a subset of $P$ and
${}_{G}[p]$ as an element of $P/G$.. _The infinitesimal generator_ of the
action is the vector field $\zeta_{P}\in\mathcal{X}_{P}$ that is explicitly
given by

$\displaystyle\zeta_{P}(p):=\dfrac{d}{dt}\left|{}_{t=0}\phi_{\exp{\zeta
t}}(p)=T_{e}\phi^{p}\cdot\zeta\quad\forall\quad p\in P\right.$

Note that the flow of $\zeta_{P}\in\mathcal{X}_{P}$ is thus
$\Phi_{\zeta_{P}}^{t}(p):=\phi_{\exp{\zeta t}}(p)$, where $\exp{\zeta t}\in
G\quad\forall t\in\mathbb{R}$. Since $T_{e}\phi^{p}$ is linear it follows that

$\displaystyle(\zeta+\eta)_{P}$ $\displaystyle=\zeta_{P}+\eta_{P},$
$\displaystyle(\alpha\,\zeta)_{P}$ $\displaystyle=\alpha\,\zeta_{P}.$

By definition we also have

$\displaystyle(\mathrm{Ad}_{g}\zeta)_{P}(p)=T_{e}\phi^{p}\cdot\mathrm{Ad}_{g}\zeta.$

Thus for left invariant actions

$\displaystyle(\mathrm{Ad}_{g}\zeta)_{P}(\phi_{g}(p))$
$\displaystyle=T_{e}\phi^{\phi_{g}(p)}\cdot\mathrm{Ad}_{g}\zeta$
$\displaystyle=\left.\dfrac{d}{ds}\right|_{s=0}\phi(g\exp{(\zeta
s)}g^{-1},\phi(g,p))=\left.\dfrac{d}{ds}\right|_{s=0}\phi(g\exp{(\zeta s)},p)$
$\displaystyle=\left.\dfrac{d}{ds}\right|_{s=0}\phi_{g}\circ\phi_{\exp{(\zeta
s)}}(p)=T_{p}\phi_{g}\cdot T_{e}\phi^{p}\cdot\zeta.$

This shows that for left actions

$\displaystyle{T_{p}\phi_{g}\cdot\zeta_{P}(p)=T_{p}\phi_{g}\cdot(T_{e}\phi^{p}\cdot\zeta)=(\mathrm{Ad}_{g}\zeta)_{P}(\phi_{g}(p))\neq\zeta_{P}(\phi_{g}(p)),}$
(1)

and hence that in general $\zeta_{P}$ is not a $\phi$ \- invariant vector
field. Differentiating this expression it also follows that for _left actions_

$\displaystyle-[\eta_{P},\zeta_{P}]=[\eta,\zeta]_{P},$

and thus for left actions the assignment $\zeta\to\zeta_{P}$ is a Lie algebra
antimorphism and the subspace of vectorfields
$\mathcal{X}_{G}:=\\{\zeta_{P}\,:\,\zeta\in\mathcal{G}\\}$ is a Lie-subalgebra
of the space of vectorfields $\mathcal{X}$ on $P$. Since this distribution is
involutive, it is integrable. Since it is tangent to the orbits at every point
of the orbit these integral manifolds are in fact the orbits, $\mathcal{O}$,
of the action. The space $T_{p}\mathcal{O}=\mathcal{X}_{G}(p)$ is called the
_vertical space_ of the tangent space $T_{p}P$ of $P$ at $p$.

For right actions (1) becomes

$\displaystyle{T_{p}\phi_{g}\cdot\zeta_{P}(p)=(\mathrm{Ad}_{g^{-1}}\zeta)_{P}(\phi_{g}(p))\neq\zeta_{P}(\phi_{g}(p)),}$
(2)

and hence $[\eta_{P},\zeta_{P}]=[\eta,\zeta]_{P}$ and $\zeta\to\zeta_{P}$ is a
Lie algebra homomorphism.

Let $G_{p}\subset G$ be the isotropy subgroup of $p$. That is let
$G_{p}:=\\{g\in G\mid g\cdot p=p\\}$. Then we see that $G_{g\cdot
p}=gG_{p}g^{-1}$ and that $\zeta_{P}(p)=0$ for all
$\zeta\in\mathcal{G}_{p}:=T_{e}G_{p}$. Hence we have that the
$\mathcal{X}_{G}$ is rank $(\mathrm{dim}(G)-\mathrm{dim}(G_{p}))$ at $p$. The
integral submanifolds of $\mathcal{X}_{G}$ coincide with the orbits of the
group action. When the action is proper these integral manifolds are
guaranteed to be embedded submanifols. The properness also implies that $P/G$
is Hausdorff and hence is a smooth manifold of dimension
$\mathrm{dim}(P)-(\mathrm{dim}(G)-\mathrm{dim}(G_{p}))$ with respect to the
usual quotient topology. We summarize these observations in the well known
theorem:

###### Theorem II.1

If the action $\phi:P\times G\to P$ is proper and constant rank then each
$\pi^{-1}_{\phi}(_{G}[p])$ is a closed embedded submanifold of $P$ of
dimension $(\mathrm{dim}(G)-\mathrm{dim}(G_{p}))$. Furthermore $P/G$ is a
smooth manifold of dimension
$r=\left(\mathrm{dim}(P)-(\mathrm{dim}(G)-\mathrm{dim}(G_{p}))\right)$.

###### Definition II.1

Define $\sigma_{P}:P/G\to P$ to be a map that assigns to every element
${}_{G}[p]\in P/G$, a point on the fiber $\mathcal{O}(p)$ in a smooth fashion.
That is

$\displaystyle\sigma_{P}(_{G}[p])\in P\;\;\;\text{is smooth and
}\;\;\;\;\pi_{\phi}\circ\sigma_{P}=id_{P/G}.$

Such a $\sigma_{P}$ is called a smooth cross section of $(P,P/G,\pi_{\phi})$.

Associated with the section $\sigma_{P}(\cdot)$ there exists a for every $p\in
P$ a $g\in G$ such that

$\displaystyle
p=\phi_{g}(\sigma_{P}(_{G}[p]))=\phi^{\sigma_{P}(_{G}[p])}(g):=g\cdot\sigma_{P}(_{G}[p])$
(3)

For a left action since
$p=\phi_{g}(\sigma_{P}(_{G}[p]))=\phi_{gh}(\sigma_{P}(_{G}[p]))$ for all $h\in
G_{\sigma_{P}(_{G}[p])}$ we see that the $g\in G$ that satisfies the above
relationship is unique only up to a right multiplication by an element of
$G_{\sigma_{P}(_{G}[p])}$. A similar result holds for right actions as well
where now the $g$ is only unique up to a left multiplication by an element of
$G_{\sigma_{P}(_{G}[p])}$. That is if $g_{p}\in G$ is such that
$p=\phi_{g_{p}}(\sigma_{P}(_{G}[p]))$ and $[g_{p}]_{G_{p}}:=\\{g_{p}h\mid h\in
G_{\sigma_{P}(_{G}[p])}\\}$ then $[g_{p}]_{G_{\sigma_{P}([p])}}\in
G/G_{\sigma_{P}([p])}$ can be uniquely identified with $p\in\mathcal{O}(p)$.
Thus we see that there exists a unique $[g_{p}]_{G_{\sigma_{P}(_{G}[p])}}\in
G/G_{\sigma_{P}(_{G}[p])}$ such that (3) holds for all
$g\in[g_{p}]_{G_{\sigma_{P}(_{G}[p])}}$ and hence that the cross section
$\sigma_{P}$ allows us to identify points in ${}_{G}[p]=\mathcal{O}(p)$ with
points in $G/G_{\sigma_{P}([p])}$ in a unique way.

###### Definition II.2

Based on the above fact, we define
$\gamma_{\sigma_{P}}:P\to\bigcup\limits_{z\in\sigma_{P}(P/G)}G/G_{z}$ such
that $P\ni p\mapsto[g]_{{G_{\sigma_{P}(_{G}[p])}}}\in
G/G_{\sigma_{P}(_{G}[p])}$ such that (3) holds for all
$g\in[g]_{{G_{\sigma_{P}(_{G}[p])}}}$. In other words,
$\gamma_{\sigma_{P}}(p):=\\{g\in G\mid p=g\cdot\sigma_{P}(_{G}[p])\\}$.

Observe that $\sigma_{P}(_{G}[\phi_{h}(p)])=\sigma_{P}(_{G}[p])$ for all $h\in
G$. Thus using the expression (3) we see that
$\gamma_{\sigma_{P}}(\phi_{h}(p))=\bar{L}_{h}\gamma_{\sigma_{P}}(p)$, where
$G/G_{z}\ni[g]_{G_{z}}\mapsto\bar{L}_{h}([g]_{G_{z}}):=\\{h\cdot
g^{\prime}\mid g^{\prime}\in[g]_{G_{z}}\\}=[hg]_{G_{z}}$. That is the
following commutative diagram holds.

${P}$${P}$${G/G_{\sigma_{P}(_{G}[p])}}$${G/G_{\sigma_{P}(_{G}[p])}}$$\scriptstyle{\phi_{h}}$$\scriptstyle{\gamma_{\sigma_{P}}}$$\scriptstyle{\gamma_{\sigma_{P}}}$$\scriptstyle{\bar{L}_{h}}$

Also observer that $\gamma_{\sigma_{P}}(\sigma_{P}(_{G}[p]))=[I]$. Similarly,
for a right group action, $\gamma_{\sigma_{P}}(\phi_{h}(p))=_{G_{z}}[gh]$.

###### Remark II.1

Notice that the map $\gamma_{\sigma_{P}}$ depends on the cross section
$\sigma_{P}:P/G\to P$. In the special case where the action is transitive
picking the cross section simply amounts to identifying a particular point
$p_{\sigma}\in P$ and then we see that $P\simeq G/G_{p_{\sigma}}$.

Let $\mathcal{U}_{{}_{G}[p_{0}]}$ be an open neighborhood of ${}_{G}[p_{0}]$
with respect to the quotient topology of $P/G$. Then
$\mathcal{U}_{p_{0}}:=\pi^{-1}_{\phi}(\mathcal{U}_{{}_{G}[p_{0}]})$ is an open
neighborhood of $p_{0}$ (a tubular open region of $p_{0}$). Define the map
$\psi_{\sigma_{P}}:\mathcal{U}_{p_{0}}\to\mathcal{U}_{{}_{G}[p_{0}]}\times
G/G_{\sigma_{P}(_{G}[p])}$ by the relationship
$\psi_{\sigma_{P}}(p):=\left(\pi_{\phi}(p),\gamma_{\sigma_{P}}(p)\right)=\left({}_{G}[p],g(p)\right)\in\mathcal{U}_{{}_{G}[p_{0}]}\times
G/G_{\sigma_{P}(_{G}[p])}$. From the previous results $\psi_{\sigma_{P}}$ is
an isomorphism (that is every point $p$ in a neighborhood of $p_{0}$ belongs
to some unique orbit in the neighborhood of the orbit ${}_{G}[p_{0}]$ and
there exists a unique $g\in G/G_{\sigma_{P}(_{G}[p])}$ such that
$p=\phi_{g}(\sigma_{P}(p))$). Thus we have the following lemma.

###### Lemma II.1

For every $p_{0}\in P$ there exists a neighborhood $\mathcal{U}_{p_{0}}$ of
$p_{0}$ that is isomorphic to $\pi_{\phi}(\mathcal{U}_{p_{0}})\times
G/G_{\sigma_{P}(_{G}[p])}$.

The coordinate ${}_{G}[p]$ is called the base coordinate of $p$ while $g(p)$
is called the fibre coordinate of $p$. In general $P\neq(P/G)\times
G/G_{\sigma_{P}(_{G}[p])}$ implying the local (and not global) nature of this
decomposition. If there exists a globally defined cross section then one sees
that $P=(P/G)\times G/G_{\sigma_{P}([p])}$.

Consider a left action (a similar argument holds for the right action as
well). Let $\sigma_{P}^{U}$ and $\sigma_{P}^{V}$ be two cross sections and let
$\gamma_{\sigma_{P}}^{U}$ and $\gamma_{\sigma_{P}}^{U}$ be the corresponding
fibre coordinates induced by the two cross sections. Let
$g^{u}:=\gamma_{\sigma_{P}}^{U}(p)$ and $g^{V}:=\gamma_{\sigma_{P}}^{V}(p)$
and $g_{UV}(p)\in G/G_{\sigma_{P}^{V}}$ be such that
$\sigma_{P}^{U}(p)=g_{UV}(p)\sigma_{P}^{V}(p)$. Thus we have that
$g^{V}\sigma_{P}^{V}(p)=g^{U}\sigma_{P}^{U}(p)=g^{U}g_{UV}\sigma_{P}^{V}(p)$
and hence that $g^{V}=g^{U}g_{UV}$ with $g_{UV}(p)\in G/G_{\sigma_{P}^{V}}$.
When the action is free $G_{\sigma_{P}^{V}}=\\{e\\}$ and hence we have the
following fact structure.

###### Remark II.2

If the action $\phi:P\times G\to G$ is free then the bundle $(P,G,\pi_{\phi})$
is a principle fibre bundle [47].

###### Comment II.1

The group action leads to the decomposition of the tangent space. At any point
$p\in P$, $T_{p}P$ decomposes into two complementary vector spaces - the
vertical space and horizontal space. (We have extended nomenclature defined
for the case when the group action is free [47]). The horizontal space
($\operatorname{\text{Hor}}_{p}(P)$), is non-unique and can be chosen by the
user to satisfy
$\operatorname{\text{Hor}}_{p}(P)\bigoplus\operatorname{\text{Ver}}_{p}(P)=T_{p}P$
and $\operatorname{\text{Hor}}_{g\cdot
p}(P)=T_{p}\phi_{g}\operatorname{\text{Hor}}_{p}(P)$. The horizontal space is
useful and interesting because it gives us a direct pathway to analyse
dynamics on the quotient manifold, since $\operatorname{\text{Hor}}_{p}(P)$ is
isomorphic to $T_{{}_{G}[p]}(P/{G})$. The vertical
space($\operatorname{\text{Ver}}_{p}(P)$) at each point is tangent to the
orbit passing through that point and is also the kernel of $T_{p}\pi_{\phi}$.
In case of the free action, it is isomorphic to the Lie algebra. In case of
non-free action, it is isomorphic to $T_{[g]_{G_{p}}}(G/G_{p})$.
($T_{[g]_{G_{p}}}(G/G_{p})$ is in turn the horizontal space when $G_{p}$ acts
on $G$.)

The group action leads to the decomposition of the tangent space. At any point
$p\in P$, $T_{p}P$ decomposes into two complementary vector spaces - the
vertical space and horizontal space. (We have extended nomenclature defined
for the case when the group action is free [47]). The vertical
space($\operatorname{\text{Ver}}_{p}(P)$) at each point is tangent to the
orbit passing through that point and is also the kernel of $T_{p}\pi_{\phi}$.

Let us here delve into the special case of free group action. This is
necessary in order to understand the case of non-free action with more
clarity. In this case, the vertical space at each point is isomorphic to the
Lie algebra. Choosing a basis for the Lie algebra yields a basis for the
vertical space via $T_{I}\phi_{g}$. The horizontal space
($\operatorname{\text{Hor}}_{p}(P)$), is non-unique and can be chosen by the
user to satisfy
$\operatorname{\text{Hor}}_{p}(P)\bigoplus\operatorname{\text{Ver}}_{p}(P)=T_{p}P$
and $\operatorname{\text{Hor}}_{g\cdot
p}(P)=T_{p}\phi_{g}\operatorname{\text{Hor}}_{p}(P)$.

In case of the non-free action, the situation is a little more complicated.
Let a basis for the Lie algebra be $\mathcal{B}_{1}$, and let the basis for
the Lie algebra of $G_{p}$ be $\mathcal{B}_{2}$. Then the vertical space is
$\text{span}(\mathcal{B}_{1}\setminus\mathcal{B}_{2})$. In other words, it is
isomorphic to $T_{[g]_{G_{p}}}(G/G_{p})$. From the free action case we can see
that $T_{[g]_{G_{p}}}(G/G_{p})$ is in turn the horizontal space
$\operatorname{\text{Hor}}_{g}(G)$ when $G_{p}$ acts as a Lie group on the
smooth manifold $G$. The horizontal space in the non-free case has the same
definition as the case of free action. It is useful and interesting because it
gives us a direct pathway to analyse dynamics on the quotient manifold, since
$\operatorname{\text{Hor}}_{p}(P)$ is isomorphic to $T_{{}_{G}[p]}(P/{G})$.

We illustrate some of these ideas using a well known example.

###### Example II.1

Consider the action of the rotation group $\mathop{\mathbb{SO}(3)}$ on the
elements of $\mathbb{R}^{3}\setminus\\{0\\}$. Here,
$P=\mathbb{R}^{3}\setminus\\{0\\}$ and $G=\mathop{\mathbb{SO}(3)}$ and the
left action $\phi\,:\,SO(3)\times P\to P$ that maps $q\in\mathbb{R}^{3}\to
Rq\in\mathbb{R}^{3}$ for each $R\in SO(3)$. That is for any
$q\in\mathbb{R}^{3}$ we let $\phi_{R}(q)=Rq$. Let $p=(1,0,0)^{T}$. The
isotropy subgroup $G_{p}$ of $G$ that leaves $p$ fixed is explicitly given by

$\displaystyle G_{p}$ $\displaystyle=\\{\begin{pmatrix}1&0\\\
0&S(\theta)\end{pmatrix}\mid
S(\theta)\in\mathop{\mathbb{SO}(2)},\theta\in[0,2\pi)\\}=:\mathbb{S}.$

Then we see that the orbit of the action through $p$ is

$\displaystyle\mathcal{O}(p)=\\{Rp\mid R\in SO(3)\\}=\mathbb{S}^{2}.$

It is well known that $SO(3)/G_{p}=SO(3)/\mathbb{S}=\mathbb{S}^{2}$. Thus the
orbit $\mathcal{O}(p)$ can be identified with elements of $SO(3)/\mathbb{S}$.
In particular if $q\in\mathcal{O}(p)=\mathbb{S}^{2}$, then there exists an
$R_{qp}\in SO(3)$ such that $q=R_{qp}p$ and thus the equivalence class
$[R_{qp}]=\\{R_{qp}R_{p}\mid R_{p}\in G_{p}\\}$ can be uniquely identified
with $q$. This identification depends on the choice of $p$. Below we will show
that such a choice corresponds to a choice of cross section for the bundle
$\pi:\mathbb{R}^{3}\to(\mathbb{R}^{3}\setminus\\{0\\})/SO(3)$.

Given any $q=(q_{1},q_{2},q_{3})\in\mathbb{R}^{3}\setminus\\{0\\}$ the orbit
of $q$ is the sphere of radius $||q||_{2}$. The collection of all such orbits
in $\mathbb{R}^{3}\setminus\\{0\\}$ is the quotient space
$(\mathbb{R}^{3}\setminus\\{0\\})/\mathop{\mathbb{SO}(3)}$. Since all such
orbits are spheres of radius $||q||_{2}$ we can identify all points in
$(\mathbb{R}^{3}\setminus\\{0\\})/\mathop{\mathbb{SO}(3)}$ with
$\mathbb{R}_{>0}$. This helps define he section as follows. Define
${\sigma_{P}}:(\mathbb{R}^{3}\setminus\\{0\\})/\mathop{\mathbb{SO}(3)}\to(\mathbb{R}^{3}\setminus\\{0\\})$
as ${\sigma_{P}}([q]):=||q||_{2}(1,0,0)^{T}$. This corresponds to a smooth
section of the bundle
$\pi:(\mathbb{R}^{3}\setminus\\{0\\})\to(\mathbb{R}^{3}\setminus\\{0\\})/SO(3)$.
Given any $q\in\mathbb{R}^{3}\setminus\\{0\\}$ there exist Given’s rotations
$R_{1},R_{2}\in\mathop{\mathbb{SO}(3)}$ such that $R_{2}q=(*,*,0)^{T}$ and
$R_{1}R_{2}q=(||q||_{2},0,0)^{T}$ [46, Chapter 3]. Hence
$\gamma_{\sigma_{P}}(q)=[R_{2}^{T}R_{1}^{T}]$ where
$[R_{2}^{T}R_{1}^{T}]=\\{R_{2}^{T}R_{1}^{T}R_{{\sigma_{P}}([q])}\mid
R_{{\sigma_{P}}([q])}\in G_{{\sigma_{P}}([q])}\\}$. Thus
$q=(||q||_{2},R_{2}^{T}R_{1}^{T})$ and $||q||_{2}$ is called the base
coordinate and $[R_{2}^{T}R_{1}^{T}]$ is called the fibre coordinate of $q$.
The decomposition of the tangent space is shown later in a continuation of
this example.

###  II-B Equivariant Maps

The previous content introduces much of the existing mathematical framework
that is needed for our approach. In this section we introduce a few additional
structure on the measurement space in order to cast the observer problem in an
invariant setting.

Let $\mathcal{Y}$ be a smooth $n_{Y}$ dimensional manifold and
$\varphi:G\times\mathcal{Y}\to\mathcal{Y}$ be a proper and constant rank
action of $G$ on $\mathcal{Y}$. Let
$\pi_{\varphi}:\mathcal{Y}\to\mathcal{Y}/G$ be the respective canonical
projection. We will restrict our attention to the case where the fibers of the
two actions, $\varphi$ on $\mathcal{Y}$ and $\phi$ on $P$, are isomorphic to
each other.

###### Definition II.3

A smooth onto map $H:P\to\mathcal{Y}$ is said to be $G$ \- equivariant if
$\varphi_{g}\left(H(p)\right)=H\left(\phi_{g}(p)\right)$ for all $p\in P$ and
$g\in G$. That is if the top part of the following commutative diagram holds.

${P}$${\mathcal{Y}}$${P}$${\mathcal{Y}}$${P/G}$${\mathcal{Y}/G}$$\scriptstyle{H}$$\scriptstyle{\phi_{g}}$$\scriptstyle{\varphi_{g}}$$\scriptstyle{H}$$\scriptstyle{\pi_{\phi}}$$\scriptstyle{\pi_{\varphi}}$$\scriptstyle{\bar{H}}$

In essence what this says is that $H$ maps fibers of $P$ to fibers of
$\mathcal{Y}$. We will assume that $H$ restricted to the fibers is a one-to-
one map.

Let us define a smooth map $\bar{H}:P/G\to\mathcal{Y}/G$ such that the bottom
half of the preceding commutative diagram holds. We define it as follows :
given any ${}_{G}[p]\in P/G$, $\bar{H}(_{G}[p]):=\pi_{\varphi}(H(p))$.
$\bar{H}$ thus evaluated is well defined, irrespective of the point
$p\in\mathcal{O}(p)$ that is chosen. This is shown as follows : assume that
$\phi_{g}(p)$ is another point on $\mathcal{O}(p)$ for some $g\in G$. If we
evaluate $\bar{H}$ using this point, we have
$\bar{H}(_{G}[p])=\pi_{\varphi}(H(\phi_{g}(p)))=\pi_{\varphi}(\varphi_{g}(h((p)))=\pi_{\varphi}(H(p))$
(where the last equality follows from the fact that
$\pi_{\varphi}\circ\varphi=\pi_{\varphi}$).

Let $\sigma_{Y}$ be a cross section of the bundle
$(\mathcal{Y},G,\pi_{\varphi})$ and $\gamma_{\sigma_{Y}}:\mathcal{Y}\to
G/G_{\sigma_{Y}(_{G}[y])}$ be such that
$y=\varphi_{\gamma_{\sigma_{Y}}(y)}(\sigma_{Y}(_{G}[y]))$ holds. Then we see
that $y=(_{G}[y],\gamma_{\sigma_{Y}}(y))$. In the case where the output map
$H$ is one-to-one when restricted to fibers we see that $[p_{i}]\neq[p_{j}]$
for any $p_{i},p_{j}\in H^{-1}\left(\sigma_{Y}(_{G}[y])\right)$. This allows
us to define, in a unique way, the associated cross section $\sigma_{PY}$ of
$(P,G,\pi_{\phi})$ by the relationship

$\displaystyle\sigma_{PY}(_{G}[p])$ $\displaystyle=\mathcal{O}(p)\cap
H^{-1}\left(\sigma_{Y}(_{G}[H(p)])\right).$ (4)

Note that $H(\sigma_{PY}(_{G}[p]))=\sigma_{Y}(_{G}[H(p)])$. Let
$p\neq\sigma_{PY}(_{G}[p])$ and $\gamma_{\sigma_{y}}(H(p))\in[g]_{G_{H(p)}}$
such that
$H(p)=\varphi_{\gamma_{\sigma_{y}}(H(p))}\left(\sigma_{Y}([H(p)])\right)$.
Thus we have

$\displaystyle
H(p)=\varphi_{\gamma_{\sigma_{y}}(H(p))}\left(H(\sigma_{PY}(_{G}[p]))\right)=H\left(\phi_{\gamma_{\sigma_{y}}(H(p))}\sigma_{PY}(_{G}[p])\right).$

Now since $H$ is one-to-one when restricted to fibers we have that

$\displaystyle p=\phi_{\gamma_{\sigma_{y}}(H(p))}\sigma_{PY}(_{G}[p]),$

and hence that $\gamma_{\sigma_{PY}}(p)=\gamma_{\sigma_{y}}(H(p))$.

We summarize this result in the lemma below.

###### Lemma II.2

Let $H:P\to\mathcal{Y}$ be a G - equivariant onto map that is also one-to-one
when restricted to the fibers. Let $\sigma_{y}:\mathcal{Y}/G\to\mathcal{Y}$ be
a cross section of the bundle $(\mathcal{Y},G,\pi_{\varphi})$ then the map
$\sigma_{PY}:P/G\to P$ defined by (4) is a cross section of $(P,G,\pi_{\phi})$
that we will call the cross section that is associated with the cross section
$\sigma_{Y}$. Furthermore we also have that
$\gamma_{\sigma_{PY}}(p)=\gamma_{\sigma_{y}}(H(p))$ for all $p\in P$.

###### Remark II.3

Constant rank, proper group actions allows us to identify $y$ with
$(_{G}[y],\gamma_{\sigma_{y}}(y))$ and $p$ with
$(_{G}[p],\gamma_{\sigma_{PY}}(p))$. This identification depends on the cross
section $\sigma_{Y}$. Thus from Lemma II.2 we have that the $G$ \-
equivariance of $H$ implies that the fibre coordinates of $p$ and $h(p)$ can
be identified with each other if the map $H$ is additionally one-to-one when
restricted to the fibers. Thus from an observer point of view when $H$
represents the output of a system with state $p$ then the equivariance of the
output along with the one-to-one nature when restricted to the fibers ensure
that the measurement contains direct information of the fiber coordinate of
the state $p$.

##  III Equivariant Control Systems

Let $P$ (state-space), $\mathcal{U}$ (control input), and $\mathcal{Y}$
(measurement) be $n_{P},n_{\mathcal{U}},n_{\mathcal{Y}}$ \- dimensional smooth
manifolds respectively and let $X:{P}\times\mathcal{U}\to T{P}$ be a smooth
map such that $X(p,u)\in T_{p}{P}$ for all $p\in{P}$ and $u\in\mathcal{U}$ and
$H:P\to\mathcal{Y}$ is smooth and onto. Based on these structures, the
equations

$\displaystyle\dot{p}$ $\displaystyle=X(p,u),$ (5) $\displaystyle y$
$\displaystyle=H(p),$ (6)

define a control system with state evolving on $P$ with control taking values
in $\mathcal{U}$ and the output taking values in $\mathcal{Y}$. The 5-tuple
$(P,\mathcal{U},\mathcal{Y},X,H)$ will be referred to as a _Control System_ on
$P$. Since the notion of symmetries plays an important role in our evolution
of ideas, we define group actions on each of the mathematical objects.

Let $\phi:G\times P\to P$, $\psi:G\times\mathcal{U}\to\mathcal{U}$ be proper
and constant rank left actions of $G$ on $P$ and $\mathcal{U}$ respectively,
and let $\varphi:G\times\mathcal{Y}\to\mathcal{Y}$ be a proper left or right
action of $G$. Define the proper constant rank left or right action
$\rho:G\times(P\times\mathcal{U})\to P\times\mathcal{U}$ by
$\rho_{g}(p,u)=(\phi_{g}(p),\psi_{g}(u))=(\phi_{g}\times\psi_{g})(p,u)$. We
now define projection maps on each of the spaces to impart a bundle structure
to each. Let $\pi_{\phi}:P\to P/G$, $\pi_{\psi}:\mathcal{U}\to\mathcal{U}/G$,
$\pi_{\varphi}:\mathcal{Y}\to\mathcal{Y}/G$ and
$\pi_{\rho}:P\times\mathcal{U}\to(P\times\mathcal{U})/G$ be the respective
canonical projections. Denote by ${}_{G}[p](\in P/G)=\pi_{\phi}(p)$ the orbit
of the $\phi$ action of $G$ through $p$, ${}_{G}[u]=\pi_{\psi}(u)$ the orbit
of the $\psi$ action of $G$ through $u$ ($\in\mathcal{U}/G$),
${}_{G}[y]=\pi_{\varphi}(y)$ ($\in\mathcal{Y}/G$) the orbit of the $\varphi$
action of $G$ through $y$, and ${}_{G}[p,u]=\pi_{\rho}(p,u)$ the orbit of the
$\rho$ action of $G$ through $(p,u)$. Note that, in general,
${}_{G}[p,u]\neq(_{G}[p],_{G}[u])$ unless $\psi_{g}=id$ in which case
${}_{G}[p,u]=(_{G}[p],u)$.

###### Definition III.1

The control system $(P,\mathcal{U},\mathcal{Y},X,h)$ will be called a $G$ \-
equivariant control system if the maps $X:{P}\times\mathcal{U}\to T{P}$ and
$H:{P}\to\mathcal{Y}$ are G - equivariant. That is, if the following two
commutative diagram holds.

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We will demonstrate below the known result that the flow of $G$ \- equivariant
control systems take orbits to orbits. Let
$p(t):=\Psi^{X}_{t}(p_{0},u([0,t]))$ be the solution of of (5) for a control
history $u([0,t])$. Consider the curve $\phi_{g}(p(t))$ for some $g\in G$.
Then from $G$ \- invariance we have

$\displaystyle\dfrac{d}{dt}\phi_{g}\left(p(t)\right)$
$\displaystyle=T_{p(t)}\phi_{g}\cdot
X\left(p(t),u(t)\right)=X\left(\rho_{g}(p(t),u(t))\right).$

Thus we have that $\phi_{g}(p(t))$ is the solution of $X$ that originates at
$\phi_{g}(p_{0})$ with control history $\psi_{g}(u(\cdot))$. Thus it follows
that for any $p_{0}\in P$

$\displaystyle\Psi^{X}_{t}\left(\rho_{g}(p_{0},u([0,t]))\right)=\phi_{g}\left(\Psi^{X}_{t}(p_{0},u([0,t]))\right)\quad\forall\quad
t\geq 0,\>\&\>g\in G.$

and hence

$\displaystyle\pi_{\phi}\left(\Psi^{X}_{t}(p_{0},u([0,t]))\right)=\pi_{\phi}\left(\phi_{g}\left(\Psi^{X}_{t}(p_{0},u([0,t]))\right)\right)=\pi_{\phi}\left(\Psi^{X}_{t}\left(\rho_{g}(p_{0},u([0,t]))\right)\right).$

That is
$\pi_{\phi}\circ\Psi^{X}_{t}(p^{\prime},u^{\prime}([0,t]))=\pi_{\phi}\circ\Psi^{X}_{t}(p_{0},u([0,t]))$
for all $p^{\prime}=\phi_{g}(p_{0})$ and
$u^{\prime}(\cdot)=\psi_{g}(u(\cdot))$ for all $g\in G$ and hence that the
flow of $G$ \- equivariant control systems take orbits to orbits.

Let us define a smooth map
$\bar{X}:({P}\times\mathcal{U})/G{}\rightarrow{}T(P/G)$ such that the
following commutative diagram holds:

${{P}\times\mathcal{U}}$${({P}\times\mathcal{U})/G}$${TP}$${T(P/G)}$$\scriptstyle{\pi_{\rho}}$$\scriptstyle{X}$$\scriptstyle{\bar{X}}$$\scriptstyle{T\pi_{\phi}}$

Therefore, $\bar{X}\circ\pi_{\rho}=T\pi_{\phi}\circ X$. Hence given any
${}_{G}[p,u]\in({P}\times\mathcal{U})/G$,
$\bar{X}(_{G}[p,u])=T_{p}\pi_{\phi}\cdot X(p,u)$. $\bar{X}$ thus evaluated is
a well defined map, and yields the same result irrespective of the particular
point $(p,u)\in{P}\times\mathcal{U}$ on the orbit ${}_{G}[p,u]$ which is
chosen at which to evaluate $X$ and $T_{p}\pi_{\phi}$. This is shown as
follows: assume that $(\phi_{g}(p),\psi_{g}(u))$ is another point on
${}_{G}[p,u]$ for some $g\in G$. Then if we evaluate $\bar{X}$ using this
point we have $\bar{X}(_{G}[p,u])=T_{\phi_{g}(p)}\pi_{\phi}\cdot
X(\phi_{g}(p),\psi_{g}(u))=T_{\phi_{g}(p)}\pi_{\phi}(T_{p}\phi_{g}\cdot
X(p,u))=T_{p}(\pi_{\phi}\circ\phi_{g})\cdot X(p,u)$. Since
$\pi_{\phi}\circ\phi_{g}=\pi_{\phi}$ we have that
$\bar{X}(_{G}[p,u])=T_{p}\pi_{\phi}\cdot X(p,u)$. Hence $\bar{X}$ is a well
defined map.

Since ${}_{G}[p](t)=\pi_{\phi}(p(t))$, and ${}_{G}[y](t)=\pi_{\varphi}(y(t))$
we have that

$\displaystyle\dfrac{d}{dt}_{G}[p]$ $\displaystyle=T_{p}\pi_{\phi}\cdot
X(p,u)=\bar{X}(_{G}[p,u]),$

does not depend on $g\in G$.

Recall the map $\gamma_{\sigma_{P}}:P\to G/G_{\sigma_{P}(_{G}[p])}$ takes
$p\to G/G_{\sigma_{P}(_{G}[p])}$ such that the relationship (3) holds. _Note
that this map depends on the cross section $\sigma_{P}$._ Also recall that
$\gamma_{\sigma_{P}}\circ\phi_{h}={L}_{h}\circ\gamma_{\sigma_{P}}$ for all
$h\in G$.

###### Lemma III.1

The maps $\sigma_{P}(\cdot)$ and $\gamma_{\sigma_{P}}(\cdot)$ determine the
dynamics of $[g]_{G_{p}}$ as

$\dfrac{d}{dt}{[g]_{G_{p}}}=T_{[I]_{G_{p}}}{L}_{g}(T_{\sigma_{P}(_{G}[p])}\gamma_{\sigma_{P}}\cdot
X\left(\sigma_{P}(_{G}[p]),\psi_{g^{-1}}(u)\right))$

where $g\in[g]_{G_{p}}$. If $\psi$ is identity222This is the case when the
controls are open loop. In observer design, the control $u$ are indeed open
loop since they are given to the user and known., then the dynamics are
independent of the particular $g\in[g]_{G_{p}}$ that is chosen.

Proof Using the expression (3) we see that
$\gamma_{\sigma_{P}}\left(\phi_{h}(p)\right)=\bar{L}_{h}\circ\gamma_{\sigma_{P}}(p)$.
Fix $g$ such that $g\cdot\sigma_{P}(_{G}[p])=p$. Choose a particular element
$h\in G_{p}$ then $gh\in[g]_{G_{p}}$. Note that $g\in[g]_{G_{p}}$ also (if
$h=I$). Let $g_{1}:=gh$ Then we have,

$\displaystyle\dfrac{d}{dt}{[g]_{G_{p}}}$
$\displaystyle=T_{p}\gamma_{\sigma_{P}}\cdot X(p,u),$
$\displaystyle=T_{p}\gamma_{\sigma_{P}}\cdot T_{g_{1}^{-1}\cdot
p}\phi_{g}\cdot X\left(g_{1}^{-1}\cdot(p,u)\right),$
$\displaystyle=T_{p}\gamma_{\sigma_{P}}\cdot
T_{\sigma_{P}(_{G}[p])}\phi_{g_{1}}\cdot X(g_{1}^{-1}\cdot(p,u)),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\gamma_{\sigma_{P}}\circ\phi_{g_{1}}\right)\cdot
X\left(\phi_{g_{1}^{-1}}(p),\psi_{g_{1}^{-1}}(u)\right),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\gamma_{\sigma_{P}}\circ\phi_{g_{1}}\right)\cdot
X\left(\sigma_{P}(_{G}[p]),\psi_{g_{1}^{-1}}(u)\right),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\bar{L}_{g_{1}}\circ\gamma_{\sigma_{P}}\right)\cdot
X\left(\sigma_{P}(_{G}[p]),\psi_{g_{1}^{-1}}(u)\right),$
$\displaystyle=T_{[I]}\bar{L}_{g_{1}}(T_{\sigma_{P}(_{G}[p])}\gamma_{\sigma_{P}}\cdot
X\left(\sigma_{P}(_{G}[p]),\psi_{g_{1}^{-1}}(u)\right))$

Choosing $h=I$ yields to the first part of the proposition. Now assume that
$\psi$ is identity. Then

$\displaystyle\dfrac{d}{dt}{[g]_{G_{p}}}$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\gamma_{\sigma_{P}}\circ\phi_{g_{1}}\right)\cdot
X\left(\phi_{g_{1}^{-1}}(p),u\right),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\gamma_{\sigma_{P}}\circ\phi_{g_{1}}\right)\cdot
X\left(\phi_{g^{-1}}\circ\phi_{h^{-1}}(p),u\right),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\gamma_{\sigma_{P}}\circ\phi_{g_{1}}\right)\cdot
X\left(\phi_{g^{-1}}(p),u\right),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\gamma_{\sigma_{P}}\circ\phi_{g}\right)\cdot
X\left(\sigma_{P}(_{G}[p]),u)\right),$
$\displaystyle=T_{\sigma_{P}(_{G}[p])}\left(\bar{L}_{g}\circ\gamma_{\sigma_{P}}\right)\cdot
X\left(\sigma_{P}(_{G}[p]),u\right),$
$\displaystyle=T_{[I]}\bar{L}_{g}(T_{\sigma_{P}(_{G}[p])}\gamma_{\sigma_{P}}\cdot
X\left(\sigma_{P}(_{G}[p]),u\right))$

Since the dynamics are independent of the particular $h$ chosen, they are
independent of the particular representative element in $[g]_{G_{p}}$ chosen.

Thus we finally have the following reduction theorem:

###### Theorem III.1

Let the control system (5) – (6) be $G$ \- equivariant and the output map $H$
be one-to-one when restricted to the fibers. If
$\sigma_{Y}:\mathcal{Y}/G\to\mathcal{Y}$ is a cross section of the bundle
$(\mathcal{Y},\mathcal{Y}/G,\pi_{\varphi})$ and $\sigma_{PY}:P/G\to P$ is the
cross section of the bundle $(P,P/G,\pi_{\phi})$ that is associated with
$\sigma_{Y}$ and is given by (4) then (5) – (6) is equivalent to the system

$\displaystyle\dfrac{d}{dt}{[g]_{G_{p}}}$
$\displaystyle=T_{[I]}\bar{L}_{g}(T_{\sigma_{P}(_{G}[p])}\gamma_{\sigma_{P}}\cdot
X\left(\sigma_{P}(_{G}[p]),\psi_{g^{-1}}(u)\right))$ (7)
$\displaystyle\gamma_{\sigma_{Y}}(y)$ $\displaystyle=[g]_{G_{p}}$ (8)
$\displaystyle\dfrac{d}{dt}_{G}[p]$ $\displaystyle=\bar{X}(_{G}[p,u])$ (9)
${}_{G}[y]$ $\displaystyle=\pi_{\varphi}\circ
h\circ\pi_{\phi}^{-1}(_{G}[p]):=\bar{H}(_{G}[p])$ (10)

###### Remark III.1

From (8) we see that the fibre coordinate
$\gamma_{\sigma_{PY}}(p)=[g]_{G_{p}}$ of $p=(_{G}[p],[g]_{G_{p}})$ is directly
measured and hence that an observer problem for (5) – (6) reduces to that of
designing an observer on the quotient space and filter on the fibers. That is
designing an observer for (9) – (10) and a filter for (7) – (8).

###### Remark III.2

This decomposition is brielfy (and without proofs) mentioned in [28]. However,
they do not use it in their observer design procedure. They linearize their
observer around the desired equilibrium to design the gain and establish local
results. This decomposition howveer yields significant geometric insight.

###### Remark III.3

In addition when the group action is transitive on both the state space and
the output space the choice of a particular point in output space reduces the
problem to that of designing an observer on the fibres (i.e. the Lie group).
That is designing a filter for (7) – (8). This physically corresponds to the
case of the unicycle and rigid body attitude observation using IMUs considered
in [43] and SLAM with known spatial markers that was considered in [45].

##  IV Gradient Based Observers for Kinematic Systems on Lie Groups

In this section we present observer design when the system evolves on a Lie
group. This is a restatement of results in [43] in a much more lucid manner.
This is relevant if there is no geometric structure assumed on the output
space $\mathcal{Y}$ i.e. the special case of designing an observer for (7) –
(8). This physically corresponds to the situations dealt with in [43],[45].

Let $\Phi:{G}\to G$ be a general constant rank proper left or right action and
let $\varphi:G\times\mathcal{Y}\to\mathcal{Y}$ be some proper and free $\Phi$
group action of $G$ on $\mathcal{Y}$. That is
$\varphi(h,\varphi(g,y))=\varphi\left(\Phi_{h}(g),y\right)$. When $\Phi$
corresponds to left multiplication (ie. $\Phi_{g}=L_{g}$) or right
multiplication (ie. $\Phi_{g}=R_{g}$).

We consider a system that evolves according to

$\displaystyle\dot{g}$ $\displaystyle=T_{I}\Phi_{g}\cdot\zeta(t),$ (11)
$\displaystyle y$ $\displaystyle=\varphi_{g^{-1}}(y_{0}),$ (12)

where $\zeta(t)\in\mathcal{G}$ is known and $y_{0}\in\mathcal{Y}$ is a
constant. This corresponds to (7) – (8) where we have treated
$\sigma_{P}([p](t))$ as an input.

###### Definition IV.1

We will say that the system (11) – (12) is a _left observed system_ if both
$\Phi$ and $\varphi$ are left actions and a _right observed system_ if both
$\Phi$ and $\varphi$ are right actions.

The problem we consider is that of estimating $g$ from the measurement of $y$
given the information of $\Omega$.

We consider the pre observer

$\displaystyle\dot{\widetilde{g}}$
$\displaystyle=T_{I}\Phi_{\tilde{g}}\cdot(\zeta-\Delta(\widetilde{g},{y})),$
(13) $\displaystyle\widetilde{y}$
$\displaystyle=\varphi_{\widetilde{g}^{-1}}(y_{0}).$ (14)

Here the innovation term $\Delta$ will be decided in the next section. Note
that the innovation is a Lie-algebraic valued function of the estimate and the
measurement. The term - $(\zeta-\Delta(\widetilde{g},{y}))$ \- is like a
residual

Let $V^{y}(\cdot,\cdot):\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}$ be some
suitable distance function on $\mathcal{Y}$ that is $\varphi$ invariant. Or in
other words
$V^{y}(\varphi_{g}({y_{1}}),\varphi_{g}({y_{2}}))=V^{y}(\varphi_{\Phi_{h}(g)}({y_{1}}),\varphi_{\Phi_{h}(g)}({y_{2}}))$
for all $g\in G$ and ${y_{1}},{y_{2}}\in\mathcal{Y}$. In particular we assume
$V^{y}(\cdot\,,\,\cdot)$ to be symmetric.

Consider the estimation error and the output error given respectively by

$\displaystyle e_{g}$ $\displaystyle:=\Phi_{g}(\widetilde{g}^{-1}),$ (15)
$\displaystyle e_{y}$
$\displaystyle:=V^{y}(\varphi_{g^{-1}}(y_{0}),\varphi_{\tilde{g}^{-1}}(y_{0}))=V^{y}(\varphi_{e_{g}}(y_{0}),y_{0})=:V^{e}(e_{g}).$
(16)

###### Lemma IV.1

If $\left\langle\left\langle\zeta_{e},\cdot\right\rangle\right\rangle$ is a
$\Phi_{g}$ invariant metric, then $\zeta_{e}$ defined by

$\displaystyle\left\langle(\Phi_{e_{g}})^{*}d_{e_{g}}V^{e}\>,\>\cdot\right\rangle=\left\langle\left\langle\zeta_{e}\>,\>\cdot\right\rangle\right\rangle$

is independent of $g$.

Proof When $\varphi$ is left invariant:

$\displaystyle V^{y}(\varphi_{g^{-1}}(y_{0}),\varphi_{\tilde{g}^{-1}}(y_{0}))$
$\displaystyle=V^{y}(y_{0},\varphi_{e_{g}}(y_{0}))=V^{y}(\varphi_{e_{g}}(y_{0}),y_{0}):=V^{e}(e_{g}).$

Consider the curve $c(s)={e_{g}}\exp{(\xi s)}$ that passes through $e_{g}$ at
$s=0$ with tangent vector $T_{I}L_{e_{g}}\cdot\xi$. Then

$\displaystyle\langle(TL_{e_{g}})^{*}d_{e_{g}}V^{e}\,,\,\xi\rangle$
$\displaystyle=\langle
d_{e_{g}}V^{e}\,,\,e_{g}\cdot\xi\rangle=\dfrac{d}{ds}|_{s=0}V^{e}({e_{g}}\exp{(\xi
s)})=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{e_{g}\exp{(\xi s)}}(y_{0}),y_{0}),$
$\displaystyle=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{g\widetilde{g}^{-1}\exp{(\xi
s)}}(y_{0}),y_{0})=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{\widetilde{g}^{-1}\exp{(\xi
s)}}(y_{0}),y).$

The right hand side depends only on $y$ and $\tilde{y}$ thus since
$\langle\langle\cdot\,,\,\cdot\rangle\rangle$ is left invariant $\xi$ depends
only on $y$ and $\tilde{y}$.

When $\varphi$ is right invariant

$\displaystyle V^{y}(\varphi_{g^{-1}}(y_{0}),\varphi_{\tilde{g}^{-1}}(y_{0}))$
$\displaystyle=V^{y}(y_{0},\varphi_{\widetilde{g}^{-1}g}(y_{0}))=V^{y}(\varphi_{e_{g}}({y_{0}}),y_{0}):=V^{e}(e_{g})$

Consider the curve $c(s)=\exp{(\xi s)e_{g}}$ that passes through $e_{g}$ at
$s=0$ with tangent vector $T_{I}R_{e_{g}}\cdot\xi$. Then

$\displaystyle\langle(TR_{e_{g}})^{*}d_{e_{g}}V^{e}\,,\,\xi\rangle$
$\displaystyle=\langle
d_{e_{g}}V^{e}\,,\,e_{g}\cdot\xi\rangle=\dfrac{d}{ds}|_{s=0}V^{e}(\exp{(\xi
s)}{e_{g}})=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{\exp{(\xi
s)}e_{g}}(y_{0}),y_{0}),$
$\displaystyle=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{\exp{(\xi
s)}}(y),\varphi_{e_{g}^{-1}}(y_{0}))=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{\exp{(\xi
s)}}(y),\varphi_{e_{g}^{-1}}(y_{0}))$
$\displaystyle=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{\exp{(\xi
s)}}(y),\varphi_{g^{-1}\widetilde{g}}(y_{0}))=\dfrac{d}{ds}|_{s=0}V^{y}(\varphi_{\exp{(\xi
s)}}(y),\varphi_{\widetilde{g}}(y))$

The right hand side depends only on $y$ and $\tilde{y}$ thus since
$\langle\langle\cdot\,,\,\cdot\rangle\rangle$ is left invariant $\xi$ depends
only on $y$ and $\tilde{y}$.

From the definition of $V^{e}$ it is clear that $V^{e}(I)=0$. Differentiating
$e_{g}$ and $V^{e}$ we have

$\displaystyle\dot{e}_{g}$
$\displaystyle=\Phi^{\tilde{g}}_{e_{g}}\Delta(\widetilde{g},y)$
$\displaystyle\dot{V}^{e}$ $\displaystyle=\left\langle
d_{e_{g}}V^{e}\,,\,\Phi^{e}_{e_{g}}\cdot\Delta(\tilde{g},y)\right\rangle,$

where

$\displaystyle\Phi^{\tilde{g}}_{e_{g}}$
$\displaystyle:=\left\\{\begin{array}[]{lr}\Phi_{e_{g}}\mathrm{Ad}_{\widetilde{g}}&\quad\mathrm{for\quad
left\quad observed\quad systems}\\\
\Phi_{e_{g}}\mathrm{Ad}_{\widetilde{g}^{-1}}&\quad\mathrm{for\quad right\quad
observed\quad systems}\end{array}\right.,$ (19)

It is now easy to see that the innovation term

$\displaystyle\Delta(\widetilde{g},y)$
$\displaystyle:=\left\\{\begin{array}[]{lr}-k\mathrm{Ad}_{\tilde{g}^{-1}}\zeta_{e}&\quad\mathrm{for\quad
left\quad observed\quad systems}\\\
-k\mathrm{Ad}_{\tilde{g}}\zeta_{e}&\quad\mathrm{for\quad right\quad
observed\quad systems}\end{array}\right.,$ (22)

will yield the time invariant error dynamics

$\displaystyle\dot{e}_{g}$
$\displaystyle=\Phi_{e_{g}}\cdot\left(-k\zeta_{e}\right),$ (23)
$\displaystyle\dot{V}^{e}$
$\displaystyle=-k\,\langle\langle\zeta_{e}\,,\,\zeta_{e}\rangle\rangle.$ (24)

Further assume that $I$ is a non-degenerate critical point for $V^{e}$ (this
will need to be kept in consideration while picking $V$ when designing an
observer). Then $I$ is an isolated critical point. Thus there exists a
nighbourhood $\mathcal{B}$ of $I$ such that if $e_{g}\in\mathcal{B}$ then
$\dot{V}^{e}\leq 0$ ensures that $e_{g}$ converges to $I$.

##  V Examples

In this section we present two examples to illustrate some of the concepts
developed so far. We continue example II.1 to show the decomposition of the
tangent space. In a second example we illustrate observer design on a Lie
group. We do not show any examples based on geometry of the output space.

###  V-A Motion on a Sphere

We continue example II.1. Table I provides a summary of the structure in the
problem.

$P=\mathbb{R}^{3}\setminus\\{0\\}$ | $G=\mathop{\mathbb{SO}(3)}$ | $\phi(g,p)=gp$
---|---|---
TABLE I: Summary of Structure

Let us now look at how the kinematics split as per the base and fibre co-
ordinate structure. Define $r:=||q||_{2}$. Let $R\in\mathop{\mathbb{SO}(3)}$
be such that $q=rRe_{1}$ (using Given’s rotations [46]).

$\displaystyle\dot{p}=$ $\displaystyle v$ $\displaystyle=$
$\displaystyle\dot{r}ge_{1}+r\dot{R}e_{1}$ $\displaystyle=$
$\displaystyle\dot{r}ge_{1}+rR{\Omega}_{\times}e_{1}$

where the last equality assumes that $\dot{R}=R{\Omega}_{\times}$ for some
$\Omega\in\mathbb{R}^{3}$ and
$(\cdot)_{\times}:\mathbb{R}^{3}{}\rightarrow{}\mathfrak{so(3)}$ is the
canonical isomorphism between the two. Although we express $p=rRe_{1}$, and
the base co-ordinate is $r$, note that $R\mathbb{S}$ is the fibre co-ordinate
and not $R$, because of the non-trivial isotropy subgroup. Hence the
kinematics on $P$ will split across
$T_{{}_{G}[p]}(\mathbb{R}^{3}/\mathop{\mathbb{SO}(3)})$ which will be the
horizontal component $\operatorname{\text{hor}}_{P}(v)$ and
$T_{(g\mathbb{S})}(\mathop{\mathbb{SO}(3)}/S)$ which will be the vertical
component $\operatorname{\text{ver}}_{P}(v)$.

$\pi_{\phi}(p)$ is the sphere of radius $r$ in $\mathbb{R}^{3}$. Hence a
smooth curve with $r$ constant looks like $r_{0}R(t)e_{1}$ with tangent vector
$r_{0}R{\Omega}_{\times}e_{1}$. Hence $rR{\Omega}_{\times}e_{1}$ will be
annihilated by $T\pi_{\phi}$. Thus
$\operatorname{\text{hor}}_{P}(v)=e_{1}^{T}R^{T}v$.

$\operatorname{\text{ver}}_{P}(v)$ is not simply $rR{\Omega}_{\times}e_{1}$
because of the non-trivial isotropy subgroup. It is the projection of
$rR{\Omega}_{\times}e_{1}\in T_{R}\mathop{\mathbb{SO}(3)}$ onto
$\operatorname{\text{Hor}}_{R}(\mathop{\mathbb{SO}(3)})$. Let us analyse what
$\operatorname{\text{Hor}}_{R}(\mathop{\mathbb{SO}(3)})$ looks like. Consider
a smooth curve $\alpha(t)\in\mathbb{S}$ with $\alpha(0)=I$. Then
$\dot{\alpha}(0)=\begin{pmatrix}0&0\\\ 0&\xi\end{pmatrix}$ for some
$\xi\in\mathfrak{se(2)}$. Hence given any
$({\Omega}_{0})_{\times}=\begin{pmatrix}0&a&b\\\ -a&0&c\\\
-b&-c&0\end{pmatrix}$,
$\operatorname{\text{ver}}_{g}((\Omega_{0})_{\times})=\begin{pmatrix}0&0&0\\\
0&0&c\\\ 0&-c&0\end{pmatrix}$
$\operatorname{\text{hor}}_{R}(({\Omega}_{0})_{\times})=\begin{pmatrix}0&a&b\\\
-a&0&0\\\ -b&0&0\end{pmatrix}=(\Omega_{\times}e_{1})_{\times}$. Thus the
vertical component of $v$ is
$\dot{g}_{p}={i}(g({{\Omega}_{\times}e_{1}})_{\times})$ where ${i}$ is the
isomorphism between $\operatorname{\text{Hor}}_{R}(\mathop{\mathbb{SO}(3)})$
and $T_{R\mathbb{S}}(\mathop{\mathbb{SO}(3)}/S)$.

Notice that the horizontal component can change if $R$ takes different values
in $R\mathbb{S}$ the horizontal component changes. This is not shocking, since
the horizontal space is not unique, it depends on the user.

###  V-B Observer Design for Attitude Kinematics

Consider the case of attitude kinematics with $G=\mathop{\mathbb{SO}(3)}$
measured body angular velocities, $\hat{\Omega}\in\mathfrak{so(3)}$, where the
kinematic equations are

$\displaystyle\dot{R}$ $\displaystyle=R\widehat{\Omega}$

The measured outputs are the inertial directions $e_{3}$ and $e_{2}$ in the
body frame.
$\mathcal{Y}=\mathbb{S}^{2}\times\mathbb{S}^{2}\subset\mathbb{R}^{3}\times\mathbb{R}^{3}$
and
$y=(y_{2},y_{3})=(R^{T}e_{2},R^{T}e_{3})=(\varphi_{R^{T}}(e_{2}),\varphi_{R^{T}}(e_{3}))$
where $\varphi\,:\,SO(3)\times\mathbb{S}^{2}\to\mathbb{S}^{2}$ is a left
action that is simply given by multiplication by $R$. Table II contains a
summary of the structure.

$P=\mathbb{R}^{3}\setminus\\{0\\}$ | $G=\mathop{\mathbb{SO}(3)}$ | $\mathcal{Y}=\mathbb{S}^{2}\times\mathbb{S}^{2}$
---|---|---
TABLE II: Summary of Structure

Let

$\displaystyle V^{y}(y,\tilde{y})$
$\displaystyle:=||R^{T}\,e_{3}-\tilde{R}^{T}\,e_{3}||^{2}+||R^{T}\,e_{2}-\tilde{R}^{T}\,e_{2}||^{2}=||e_{3}-R\tilde{R}^{T}\,e_{3}||^{2}+||e_{2}-R\tilde{R}^{T}\,e_{2}||^{2},$
$\displaystyle=||e_{3}-E\,e_{3}||^{2}+||e_{2}-E\,e_{2}||^{2}:=V^{e}(E)$

Let $c(s)=E\exp{(\widehat{\xi}s)}$

$\displaystyle\langle(TL_{E})^{*}d_{E}V^{e}\,,\,\xi\rangle$
$\displaystyle=\langle
d_{E}V^{e}\,,\,E\widehat{\xi}\,\,\rangle=\dfrac{d}{ds}|_{s=0}V^{e}({E}\exp{(\widehat{\xi}s)})$
$\displaystyle=\sum_{k=2}^{3}\dfrac{d}{ds}|_{s=0}||e_{k}-{E\exp{(\widehat{\xi}s)}}e_{k}||^{2},$
$\displaystyle=\sum_{k=2}^{3}\dfrac{d}{ds}|_{s=0}||R^{T}e_{k}-{\tilde{R}^{T}\exp{(\widehat{\xi}s)}}e_{k}||^{2}=\sum_{k=2}^{3}\dfrac{d}{ds}|_{s=0}||y_{k}-{\tilde{R}^{T}\exp{(\widehat{\xi}s)}}e_{k}||^{2}$
$\displaystyle=\sum_{k=2}^{3}\dfrac{d}{ds}|_{s=0}\left(y_{k}^{T}y_{k}-2y_{k}^{T}{\tilde{R}^{T}\exp{(\widehat{\xi}s)}}e_{k}+e_{k}^{T}e_{k}\right)=\sum_{k=2}^{3}-\left(2y_{k}^{T}{\tilde{R}^{T}\widehat{\xi}}e_{k}\right)$
$\displaystyle=\sum_{k=2}^{3}2y_{k}^{T}\tilde{R}^{T}\left(e_{k}\times{\xi}\right)=\sum_{k=2}^{3}\left\langle\left(-2\widehat{e_{k}}\tilde{R}y_{k}\right)\,,\,\xi\right\rangle$

Thus

$\displaystyle(TL_{E})^{*}d_{E}V^{e}=\sum_{k=2}^{3}\left(-2\widehat{e_{k}}\tilde{R}y_{k}\right)$

only depends on $\tilde{R},y$ and not on $R$. If one uses a bi-invarant metric
then

$\displaystyle\zeta_{e}=\sum_{k=2}^{3}\left(-2\widehat{e}_{k}\tilde{R}y_{k}\right)$

###  V-C SLAM

Assume that there is a fixed inertial frame represented by $\mathcal{I}$ and
the body frame is represented by $\mathcal{B}$. Assume that there are $N$
fixed but unknown landmarks represented by $\mathcal{L}_{i}\in\mathbb{R}^{3}$
for $i=1,2,\ldots N$ in $\mathcal{I}$. $\mathcal{L}_{i}$ are assumed to be
unknown to the user. The $\mathcal{L}_{i}$ are measured in $\mathcal{B}$ as
$l_{i}\in\mathbb{R}^{3}$. Assume that $S=\begin{pmatrix}R&r\\\
0&1\end{pmatrix}\in\mathop{\mathbb{SE}(3)}$ represents the euclidean
transformation between $\mathcal{B}$ and $\mathcal{I}$. Then $Rl_{i}+r=L_{i}$.
Our goal is to observe $S$ and all $L_{i}$. However it is well known that the
SLAM problem is not fully observable; it can only be observed upto a euclidean
transformation [36, 45, 48].

The state space is
$P=\mathop{\mathbb{SE}(3)}\times\mathbb{E}^{3}\times\mathbb{E}^{3}\ldots\mathbb{E}^{3}$
where $\mathbb{E}^{3}$ consists of elements of $\mathbb{R}^{3}$ appended with
a 1 below (for ease of depicting the action of $\mathop{\mathbb{SE}(3)}$ on
$\mathbb{R}^{3}$). Define $\bar{\mathcal{L}}_{i}:=\begin{pmatrix}L_{1}\\\
1\end{pmatrix}$ and $\bar{l}_{i}:=\begin{pmatrix}l_{1}\\\ 1\end{pmatrix}$
Hence
$p=\begin{pmatrix}S&\bar{\mathcal{L}}_{1}&\bar{\mathcal{L}}_{1}\ldots\bar{\mathcal{L}}_{N}\end{pmatrix}\in
P$. The group is $G=\mathop{\mathbb{SE}(3)}$ and the group action $\phi_{g}$
(a right action) is
$\phi_{g}(p):=\begin{pmatrix}g^{-1}S&g^{-1}\bar{\mathcal{L}}_{1}&g^{-1}\bar{\mathcal{L}}_{2}&\ldots&g^{-1}\bar{\mathcal{L}}_{N}\end{pmatrix}$.
It is a free action. The measurement is
$y=\begin{pmatrix}\bar{l}_{1}&\bar{l}_{2}&\ldots&\bar{l}_{N}\end{pmatrix}=\begin{pmatrix}S^{-1}\bar{\mathcal{L}}_{1}&S^{-1}\bar{\mathcal{L}}_{2}&\ldots&S^{-1}\bar{\mathcal{L}}_{N}\end{pmatrix}$.

$P=\mathop{\mathbb{SE}(3)}\times\mathbb{E}^{3}\times\mathbb{E}^{3}\ldots\mathbb{E}^{3}$ | $G=\mathop{\mathbb{SE}(3)}$ | $\mathbb{E}^{3}\times\mathbb{E}^{3}\ldots\mathbb{E}^{3}$
---|---|---
TABLE III: Summary of Structure

The orbit of $p$ is
$\mathcal{O}(p)=\\{\begin{pmatrix}g^{-1}S&g^{-1}\bar{\mathcal{L}}_{1}&g^{-1}\bar{\mathcal{L}}_{2}&\ldots&g^{-1}\bar{\mathcal{L}}_{N}\end{pmatrix}\mid
g\in\mathop{\mathbb{SE}(3)}\\}$. The quotient space space thus formed is
$P/G=\\{_{G}[p]\mid p\in P\\}$.

Choose a $W\in\mathfrak{se}(3)$ then $\chi^{W}(p):=T_{I}\phi^{p}\cdot
W=(-WS,-W\bar{\mathcal{L}}_{1},\ldots,-W\bar{\mathcal{L}}_{n})$. Thus
$T_{p}\mathcal{O}(p)=\\{\chi^{W}(p)\mid W\in\mathfrak{se}(3)\\}$. Consider a
smooth curve $\gamma(t):=\phi_{p}(g(t))$ where $g(\cdot)$ is a smooth curve on
$\mathop{\mathbb{SE}(3)}$ with $g(0)=I$, $\dot{g}(0)=W$. Observe that
$\dot{\gamma}(0)=(-WS,-W\bar{\mathcal{L}}_{1},\ldots,-W\bar{\mathcal{L}}_{N})$
Since $\pi(\gamma(t))=_{G}[p]$ then $\dot{\gamma}(0)\in\ker T_{p}\pi$. Hence
$\operatorname{\text{Ver}}(p)=\\{(-WS,-W\bar{\mathcal{L}}_{1},\ldots,-W\bar{\mathcal{L}}_{N})\mid
W\in\mathfrak{se}(3)\\}$. It can be seen that
$\operatorname{\text{Ver}}(p)=T_{p}\mathcal{O}(p)$.

One choice for horizontal space is
$\operatorname{\text{Hor}}(p)=\\{(0,{u}^{\sharp}_{1},\ldots,{u}^{\sharp}_{n})\mid{u}^{\sharp}_{1},\ldots,{u}^{\sharp}_{n}\in\bar{\mathbb{R}}^{3})\\}$
($\bar{\mathbb{R}}^{3}$ is $\mathbb{R}^{3}$ with a zero appended below it).
$\operatorname{\text{Hor}}(g\cdot p)=\operatorname{\text{Hor}}(p)$ since
$\operatorname{\text{Hor}}(p)$ is the same set for all $p$. It is easy to see
that $\operatorname{\text{Hor}}(p)\cap\operatorname{\text{Ver}}(p)=\\{0\\}$.
Hence
$\operatorname{\text{Ver}}(p)\bigoplus\operatorname{\text{Hor}}(p)=T_{p}P$ for
this choice of $\operatorname{\text{Hor}}(p)$.

Consider a choice of cross-section
$\mathcal{K}=\\{I\\}\times\mathbb{E}^{3}\times\ldots\mathbb{E}^{3}$. It is
evident that $T_{z}\mathcal{K}=\operatorname{\text{Hor}}(z)$ for all
$z\in\mathcal{K}$. Hence $\mathcal{K}$ intersects each orbit transversally.

###### Comment V.1

$\pi:\mathcal{K}{}\rightarrow{}P/G$ is a bijection. To see this, assume
$\pi(z_{1})=\pi(z_{2})\implies z_{1}=z_{2}\cdot g$ but this possible only if
$g=I$ since the first element of both $z_{1}$ and $z_{2}$ is $I$. Thus
$z_{1}=z_{2}$. Consider arbitrary ${}_{G}[p]\in P/G$. Let
$z=(I,p_{0}^{-1}\bar{p}_{1},\ldots,p_{0}^{-1}\bar{p}_{n})$. Then
$\pi(z)=_{G}[p]$. This shows that under the said restriction $\pi$ is
bijective.

Consider a chart on $\mathcal{K}\ni(I,\bar{p}_{1},\ldots,\bar{p}_{n})\mapsto
c(I,\bar{p}_{1},\ldots,\bar{p}_{n}):=(\bar{p}_{1},\ldots,\bar{p}_{n})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\ldots\times\mathbb{R}^{3}$
which is actually an atlas. Using $\pi$ we can transfer this atlas onto $P/G$
and have $\mathcal{K}$ globally diffeomorphic to $P/G$.

$P/G\ni_{G}[p]\mapsto\sigma_{P}(_{G}[p]):=(I,S^{-1}\bar{\mathcal{L}}_{1},\ldots,S^{-1}\bar{\mathcal{L}}_{n})$
is a section and it is easy to observe that $\pi\circ\sigma=id_{P/G}$. $P\ni
p\mapsto\gamma_{\sigma_{P}}(p):=S\in\mathop{\mathbb{SE}(3)}$.
$p=\phi(\gamma_{\sigma_{P}}(p),\sigma_{P}(_{G}[p]))$.

For the SLAM problem, $\dot{p}=X(p,v)$ with
$X(p,v):=(SV,S{v}^{\sharp}_{1},...,S{v}^{\sharp}_{n})$ and
$v=(V,{v}^{\sharp}_{1},\ldots,{v}^{\sharp}_{n})$. Let $z:=\sigma_{P}(p)$.
Then,

$\displaystyle\dot{z}=$ $\displaystyle T_{z}\sigma\cdot f(z,v)$
$\displaystyle=$
$\displaystyle(0,-VS^{-1}\bar{\mathcal{L}}_{1}+{v}^{\sharp}_{1},\ldots,-VS^{-1}\bar{\mathcal{L}}_{n}+{v}^{\sharp}_{n})$
$\displaystyle=$ $\displaystyle T_{p}\sigma\cdot f(p,v)$

and ${{d\vbox to1.29167pt{}}\over{\vbox to0.90417pt{}dt}}{(S)^{-1}}=-VS^{-1}$
is the splitting of the vector field in the cross-section and group variables.

Notice that under the given section, the measurement directly gives us the
point on the cross-section. Hence we directly determine the fibre. To
determine the exact group element, more measurements are needed.

In discrete time, there’s another perspective to this. We will slightly modify
notation here. Assume that $\mathcal{B}_{k}$ is the body frame at time instant
$k$ for $k=1,2,\ldots$, $l_{i}^{k}$ the measurement of $L_{i}$ from
$\mathcal{B}_{k}$ and $S^{k}$ is the euclidean transformation between
$\mathcal{B}_{k+1}$ and $\mathcal{B}_{k}$, i.e. $S^{k}l_{i}^{k+1}=l_{i}^{k}$.
$S^{0}$ is the transformation between $\mathcal{B}_{1}$ and $\mathcal{I}$ and
$\bar{l}_{i}^{0}=L_{i}$. Define
$M_{k}:=\begin{pmatrix}\begin{pmatrix}l_{1}^{k}\\\
1\end{pmatrix}&\begin{pmatrix}l_{2}^{k}\\\
1\end{pmatrix}&\ldots&\begin{pmatrix}l_{N}^{k}\\\
1\end{pmatrix}\end{pmatrix}$. Then $S^{k}M_{k+1}=M_{k}$. Assuming
$M_{k}M_{k}^{T}$ to be invertible for each $k$ (which is easily satisfied in
real applications), we see that
$S^{k}=M_{k}M_{k+1}^{T}(M_{k+1}M_{k+1}^{T})^{-1}$. If we therefore can observe
$S^{0}$ and $M_{0}$ we will be able to determine $\mathcal{B}^{k}$ at any $k$.
Hence $\mathcal{B}_{k}$ gets determined modulo a euclidean transformation. The
choice of $S^{0}$ and $M_{0}$ determines the fibre on which the system is
evolving, and the remainder of the sequence of $S^{k}$ takes us along that
orbit. Hence if we are able to determine $S^{0}$ and $M_{0}$, we can determine
$\mathcal{B}_{k}$ exactly.

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