degenerate into integrable systems related to the Hitchin system, leaving open
the description of these conjectural systems·

In this section we provide an explicit description of the degenerations of
Section 4.3 in the cases studied in Section 2.4, focusing on the cases
specified in [MT, ASF, Mn, SS2, Sh], which are constructed using primitive
first Chern classes. We also review the degeneration given by Sawon–Shen [SS1]
into the $\mathrm{Sp}(2m,\mathbb{C})$-Higgs moduli space, studying as well the
degeneration of the associated natural Lagrangian. It is worth noticing that
we find that this natural Lagrangian degenerates into the fixed locus of an
involution associated to $\mathrm{U}(m,m)$-Higgs bundles, whose
Nadler–Langlads group is $\mathrm{Sp}(2m,\mathbb{C})$.

Our work provides an answer to the question posed by Sawon in [Sw] (which,
strictly speaking, refers only to the case of primitive first Chern classes)
and to the reformulation of Sawon’s question on the non-primitve case.

####  4.4.1. The general case

Consider a K3 surface $X$ equipped with an antisymplectic involution
$\zeta_{X}^{-}:X\to X$, whose quotient $\mathrm{q}_{X}:X\to
Y:=X/\zeta_{X}^{-}$ has ramification divisor $\Delta_{X}$. Let $D\subset Y$ be
a smooth curve and $C$ as in (2.18), again smooth. We denoted by
$\zeta_{C}:C\to C$ and $\mathrm{q}_{C}:C\to D=C/\zeta_{C}$ the restriction of
$\zeta_{X}^{-}$ and $\mathrm{q}_{X}$ to $C$. Note that the ramification
divisor of $\zeta_{C}$ is $\Delta_{C}=C\cap\Delta_{X}$. Since $K_{X}$ is
trivial, the normal bundle of $C$ is

$L=K_{C},$

giving the ruled surface $\mathbb{K}_{C}$, and the normal bundle of $D$ is

(4.24) $W=K_{D}K_{Y}^{-1}|_{D},$

associated to the ruled surface $\mathbb{W}$. Observe that $K_{Y}^{-1}|_{D}$
is a square-root of the branching divisor of $\mathrm{q}_{C}$, and its pull-
back is the ramification divisor.

Recall from Proposition 4.5, that
$d\zeta_{X}^{-}:(\zeta_{X}^{-})^{*}\mathcal{T}X\to\mathcal{T}X$ gives the
isomorphism
$-\partial_{C}:\zeta_{C}^{*}K_{C}\stackrel{{\scriptstyle\cong}}{{\to}}K_{C}$
giving the associated involution $\zeta_{\mathbb{K}}$ described in (3.6). The
associated natural antisymplectic involution
$\widehat{\zeta}_{\mathbb{K}}^{-}$ is sent to $\check{\zeta}_{K}^{-}$ under
the spectral correspondence as described in (3.15). As a consequence of
Proposition 4.5, Theorems 4.4 and 4.12 and Corollary 4.13 one has the
following.

######  Corollary 4.15.

There exists a non-linear degeneration
$\mathbf{N}_{\mathbf{X},\mathbf{H}}(\mathbf{v}_{a},n\mathbf{D})$ of the
natural Lagrangian subvariety
$\mathrm{N}^{\mathrm{H}}_{X}(\mathrm{v}_{a},nD)\subset\mathrm{M}_{X,\mathrm{H}}(\mathrm{v}_{a},nC)$
obtained from the fixed locus of $\zeta_{X}^{-}$, into the Lagangian
subvariety of the Higgs moduli space
$\mathcal{N}_{C}(n,d)\subset\mathcal{M}_{C}(n,d)$, with $d=a-n(n-1)(g_{C}-1)$,
given by the fixed locus of the involution

(4.25)
$\begin{array}[]{cccc}\check{\zeta}_{K}^{-}:&\mathcal{M}_{C}(n,d)&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathcal{M}_{C}(n,d)\\\
&(E,\varphi)&\longmapsto&(\zeta_{C}^{*}E,-\zeta_{C}^{*}\varphi).\end{array}$

For $a=2b$ and a $\zeta_{X}$-invariant polarization
$\widetilde{\mathrm{I}}=\mathrm{q}_{X}^{*}\mathrm{I}$, the morphism

$\widehat{\mathrm{q}}_{\mathbf{X}}:\mathbf{M}_{\mathbf{Y}/\mathbb{A}^{1}}^{\mathbf{I}}(\mathbf{w}_{b},n\mathbf{D})\longrightarrow\mathbf{N}_{\mathbf{X}/\mathbb{A}^{1}}^{\widetilde{\mathbf{I}}}(\mathbf{v}_{2b},n\mathbf{D})\subset\mathbf{M}_{\mathbf{X}/\mathbb{A}^{1}}^{\widetilde{\mathbf{I}}}(\mathbf{v}_{2b},n\mathbf{C}),$

given in (4.22), is generically 1:1 when the ramification locus of
$\mathrm{q}_{S}$ intersects $C$ non-trivially, and generically $2:1$
otherwise, for instance whenever $\zeta_{S}$ is unramified. The generic fibres
of
$\mathbf{M}_{\mathbf{Y}/\mathbb{A}^{1}}^{\mathbf{I}}(\mathbf{w}_{b},n\mathbf{D})$
are the moduli spaces $\mathrm{M}_{Y,\mathrm{I}}(\mathrm{w}_{b},nD)$ and the
central fibre is $\mathcal{M}_{D}^{W}(n,d^{\prime})$, for $W$ described in
(4.24) and $d^{\prime}=b-n(n-1)(g_{D}-1)$.

As a consequence of Proposition 4.5 and Theorems 4.4 and 4.14 one has the
following.

######  Corollary 4.16.

Suppose that (2.11) holds for the choice of a line bundle $J$ and a Mukai
vector $\mathrm{v}_{a}$ as in (2.1) for some integer $a$ and let $d$ be
$a-\delta=a-n(n-1)(g_{C}-1)$.

There exists a non-linear degeneration
$\mathbf{P}_{\mathbf{X},\mathbf{H}}(\mathbf{v}_{a},\mathbf{J},n\mathbf{D})$ of
the Prymian integrable system
$\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},J,nD)\subset\mathrm{M}_{X,\mathrm{H}}(\mathrm{v}_{a},nC)$
constructed out of $\zeta_{X}^{-}$, into the Prymian integrable system
$\mathcal{P}_{C}(n,d,J_{C})\subset\mathcal{M}_{C}(n,d)$ given by the fixed
locus of the involution

(4.26)
$\begin{array}[]{cccc}\check{\lambda}_{J_{C},K}^{-}:&\mathcal{M}_{C}(n,d)&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathcal{M}_{C}(n,d)\\\
&(E,\varphi)&\longmapsto&(\zeta_{C}^{*}E^{*},-\zeta_{C}^{*}\varphi).\end{array}$

######  Remark 4.17.

For $n=1$, $\mathcal{P}_{C}(n,d,J_{C})$ is the cotangent bundle of a Prym
abelian variety associated to $\zeta_{C}$ (when $\zeta_{X}^{-}$ is ramified)
or a disjoint union of them (when $\zeta_{X}^{-}$ unramified).

####  4.4.2. The Arbarello–Saccà–Ferretti systems and Saccà Calabi-Yau’s

In this subsection, we consider a K3 surface endowed with an antisymplectic
involution $\zeta_{X}^{-}$ with empty fixed locus and whose quotient
$Y=X/\zeta_{X}$ is a smooth Enriques surface. The canonical divisor of an
Enriques surface $Y$ is not trivial but satisfies $2K_{Y}\sim 0$. Any smooth
curve $D\subset Y$ with positive self-intersection is big and nef, and so is
$D+K_{Y}$. Then, by Kodaira vanishing theorem,

$h^{1}(Y,D)=h^{2}(Y,D)=0,$

hence

$h^{0}(Y,D)=1+\frac{1}{2}D^{2}=g_{D},$

where $g_{D}$ is the genus of $D$. The curve $C:=D\times_{Y}X$ is equipped
with an unramified $2:1$ cover $\mathrm{q}_{C}\to D$ and an involution
$\zeta_{C}:C\to C$ without fixed points.

Observe that condition (2.21) is satisfied for every smooth curve $D\subset
Y$. Then, whenever (2.11) holds for a pair $\mathrm{J}$ and $\mathrm{v}_{a}$,
$\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},\mathrm{J},nD)$ has $2$ connected
components each of them an integrable system of dimension $2n^{2}(g_{D}-1)$
known as the Arbarello–Saccà–Ferretti integrable system, as described by these
authors in [ASF] when $n=1$. The Lagrangian subvariety
$\mathrm{N}_{X,\mathrm{H}}(\mathrm{v}_{a},nD)$ has dimension
$n^{2}(g_{C}-1)+1$. Furthermore, Saccà [Sa] studied the moduli space of pure
dimension $1$ sheaves over an Enriques surface
$\mathrm{M}_{Y,I}(\mathrm{w}_{b},nD)$ showing that it is Calabi–Yau. By
Proposition 2.11, $\mathrm{M}_{Y,I}(\mathrm{w}_{b},nD)/\mathbb{Z}_{2}$ is
birrational to $\mathrm{N}_{Y,\widetilde{I}}(\mathrm{v}_{2b},nD)$.

Observe that $L\cong K_{C}$ in this case, and $W\cong K_{D}L_{\gamma}$, where
$L_{\gamma}\in H^{1}(D,\mathbb{Z}_{2})$ is the $2$-torsion line bundle
obtained by restricting $K_{Y}$ to $D$. After Corollary 4.15
$\mathrm{M}_{Y,I}(\mathrm{w}_{b},nD)$ degenerates into
$\mathcal{M}^{L_{\gamma}K_{D}}_{D}(n,b-\delta/2)$ and
$\mathrm{N}_{X,\mathrm{H}}(\mathrm{v}_{a},nD)$ into
$\mathcal{N}_{C}(n,a-\delta)$, which is the fixed point locus of the
involution (4.25) constructed out of $\zeta_{C}$ unramified. It follows from
Corollary 4.16 that $\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},J,nD)$
degenerates into $\mathcal{P}_{C}(n,a-\delta,J_{C})$, which is the fixed point
locus of the involution (4.26), constructed out of $\zeta_{C}$ unramified.

####  4.4.3. Markusevich–Tickhomirov, Matteini and Sawon–Shen systems

Consider now an anti-symplectic involution $\zeta_{X}^{-}$ on the K3 surface
$X$ whose quotient $Y=X/\zeta_{X}^{-}$ is a del Pezzo surface of degree $d$.
In that case, by their defining property, the anti-canonical bundle $-K_{Y}$
is ample. In particular, $-(m+1)K_{Y}$ is big and nef for $m\geq 0$ and by
Kodaira vanishing theorem,

$h^{1}(Y,-mK_{Y})=h^{2}(Y,-mK_{Y})=0,$

so

$h^{0}(Y,-mK_{Y})=1+\frac{m^{2}+m}{2}\cdot d.$

Note that the genus of a smooth curve in the linear system $-mK_{Y}$ is
$1+(m^{2}-m)d/2$.

We recall that $\Delta$, the branching locus of $\mathrm{q}_{X}$, lies in
$|-2K_{Y}|$. Picking a smooth curve $D$ in the linear system $|-K_{Y}|$, hence
$g_{D}=1$, pulling-back to another smooth curve $C$ on $X$, we obtain a $2:1$
cover, $\mathrm{q}_{C}:C\to D$, ramified at the intersection with the
ramification locus of $\mathrm{q}_{X}$, that we denote by $R$. Observe that
the length of $R$ coincides with the intersection of $D$ and $\Delta$, so
$(-K_{Y})\cdot(-2K_{Y})=2d$, form where one can derive that $g_{C}=1+d$.

Markushevich and Tikhomirov [MT] gave the first construction of Prymian
intagrable system $\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},J,nC)$ starting
from a del Pezzo of degree $d=2$ and for $n=1$. In this case we have that
$\zeta_{C}$ sends a genus $3$ curve $C$, to $D$, an elliptic curve and
$\zeta_{C}$ is ramified at $4$ points and
$\mathrm{P}^{\mathrm{H}}_{X}(\mathrm{v}_{a},J,C)$ is a $4$-dimensional
symplectic $V$-manifold.

Matteini extended the construction of [MT], focusing on the cases of del Pezzo
surfaces of degree $d=1$ and $d=3$. In the case of $d=1$, $\zeta_{C}:C\to D$
is the cover of a genus $2$ curve over an elliptic curve, and Matteini found
that $\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},J,nC)$ is an elliptic K3
surface. In the $d=3$ case, $\zeta_{C}:C\to D$ is a $2:1$ cover of a genus $4$
curve onto an elliptic curve and
$\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},J,C)$ is a $6$-dimensional
symplectic $V$-manifold. Even in both cases only $n=1$ was considered by the
authors, their construction works for general $n$ as indicated by Matteini in
[Mn, Section 3.6].

Another $6$-dimensional Prym integrable system is considered by Sawon and Shen
in [SS2, Sh]. In this case $Y$ is a del Pezzo surface of degree $d=1$ and
$D\subset Y$ lies in $|-2K_{Y}|$, so $g_{D}=2$. The ramification divisor $R$
of the $2:1$ cover $\mathrm{q}_{C}:C\to D$ has length
$(-2K_{Y})\cdot(-K_{Y})=2$, hence $g_{C}=5$.

After Corollary 4.16 that $\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{a},J,nD)$
degenerates into $\mathcal{P}_{C}(n,a-\delta,J_{C})$, which is the fixed point
locus of the involution (4.26), constructed out of $\zeta_{C}$ ramified at
$\Delta_{C}=\Delta_{X}\cap X$.

One can also consider the natural Lagrangian subvarieties
$\mathrm{N}_{X,\mathrm{H}}(\mathrm{v}_{a},nD)$ associated to the mentioned
Prymian integrable systems. Note that in this case
$\mathrm{N}_{X,\widetilde{\mathrm{I}}}(\mathrm{v}_{2b},nD)$ are birrational to
$\mathrm{M}_{Y,\mathrm{I}}(\mathrm{v}_{b},nD)$. From Corolloary 4.15 we know
that $\mathrm{N}_{X,\mathrm{H}}(\mathrm{v}_{a},nD)$ degenerates into
$\mathcal{N}_{C}(n,a-\delta)$, which is the fixed point locus of the
involution (4.25), and $\mathrm{M}_{Y,\mathrm{I}}(\mathrm{v}_{b},nD)$
degenerates into $\mathcal{M}_{D}^{K_{D}Q}(n,b-(n^{2}-n)(g_{D}-1))$, where $Q$
is a line bundle on $D$ whose square is $\mathcal{O}_{D}(\Delta_{C})$.

####  4.4.4. Sawon–Shen degeneration into the
$\mathrm{Sp}(2m,\mathbb{C})$-Hitchin system

When $\zeta_{X}^{-}$ is an anti-symplectic involution on the K3 surface $X$
giving a del Pezzo surface $Y=X/\zeta_{X}^{-}$ of degree $d$, the ramification
divisor of $\mathrm{q}_{X}:X\to Y$ is a connected smooth projective curve
$\Delta$ of genus $g_{\Delta}=d+1$. Trivially, the restriction of $\zeta_{X}$
to $\Delta$ is the identity $\zeta_{\Delta}=\mathbf{1}_{\Delta}$. Setting
$\mathrm{v}_{\delta}=(0,2m\Delta,1-(4m^{2}-2m)(g_{\delta}-1))$,
$\mathrm{H}=\mathcal{O}_{X}(\ell\Delta)$ and $J=\mathcal{O}_{X}(-2m\Delta)$,
Sawon and Shen [SS1] have constructed a degeneration of the Prymian integrable
system $\mathrm{P}_{X,\mathrm{H}}(\mathrm{v}_{\delta},J,2m\Delta)$ into a
compactification of the $\mathrm{Sp}(2m,\mathbb{C})$-Hitchin system over
$\Delta$, given by the fixed locus of the involution

$\begin{array}[]{cccc}\check{\lambda}_{K^{-2m}_{\Delta},K}^{-}:&\mathcal{M}_{\Delta}(n,0)&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathcal{M}_{\Delta}(n,0)\\\
&(E,\varphi)&\longmapsto&(E^{*},-\varphi^{t}).\end{array}$

Note that Corollary 4.15 shows that the natural Lagrangian
$\mathrm{N}_{X,\mathrm{H}}(\mathrm{v}_{\delta})$ degenerates into the fixed
locus of the involution

$\begin{array}[]{cccc}\check{\zeta}_{K}^{-}:&\mathcal{M}_{\Delta}(n,0)&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathcal{M}_{\Delta}(n,0)\\\
&(E,\varphi)&\longmapsto&(E,-\varphi).\end{array}$

The morphism obtained by extension of structure group sends the moduli space
of $\mathrm{U}(m,m)$-Higgs bundles maps to the fixed locus of
$\check{\zeta}_{K}^{-}$. It is worth noticing that $\mathrm{U}(m,m)$ and
$\mathrm{Sp}(2m,\mathbb{C})$ are related under the Nadler-Langlads
correspondence [Na]. In [Hi3] Hitchin proposed that a $\mathrm{(BBB)}$-brane
constructed over the moduli space of $\mathrm{Sp}(2m,\mathbb{C})$-Higgs
bundles is Mirror dual to a $\mathrm{(BAA)}$-brane supported on the moduli
space of $\mathrm{U}(m,m)$-Higgs bundles (see Section 5). Evidence for this
duality was given by Hitchin himself in [Hi3] and by Hausel, Mellit and Pei in
[HMP].

##  5\. Branes and duality

Non-abelian Hogde theory [Co, Do, Hi2, Si1, Si2] implies that
$\mathcal{M}_{C}(n,d)$ is equipped with a hyperkähler structure
$(g,\Gamma_{1},\Gamma_{2},\Gamma_{3})$, where $\Gamma_{1}$ the complex
structure coming from the moduli space of Higgs bundles and $\Gamma_{2}$ from
the moduli of flat connections. The associated Kahler forms
$\omega_{i}(\,\cdot\,,\,\cdot\,):=g(\,\cdot\,,\Gamma_{i}(\,\cdot\,))$ combine
into a $\Gamma_{1}$-holomorphic symplectic form
$\omega_{2}+\mathrm{i}\omega_{3}$, which coincides with $\Omega_{0}$ up to
scaling. Similar constructions can be given for $i=2$ and $i=3$.

Following [KW], a $\mathrm{(BAA)}$-brane in $\mathcal{M}_{C}(n,d)$ is a pair
$(N,\mathscr{G},\nabla_{\mathscr{G}})$ where $N$ is a subvariety which is
complex Lagrangian with respect to $\Omega_{0}$, and $(W,\nabla_{W})$ is a
flat bundle over $N$. A $\mathrm{(BBB)}$-brane in $\mathcal{M}_{C}(n,d)$ is
given by a pair $(P,\mathscr{F},\nabla_{\mathscr{F}})$, where $P$ is a
hyperholomorphic subvariety(i.e. holomorphic with respect to $\Gamma_{1}$,
$\Gamma_{2}$ and $\Gamma_{3}$) and $(\mathscr{F},\nabla_{\mathscr{F}})$ a
hyperholomorphic sheaf on $P$. This means that the connection
$\nabla_{\mathscr{F}}$ on the sheaf $\mathscr{F}$ is of type $(1,1)$ with
respect to all three Kähler structures $\Gamma_{1}$, $\Gamma_{2}$,
$\Gamma_{3}$. It is conjectured in [KW] that mirror symmetry interchanges
$\mathrm{(BBB)}$-branes with $\mathrm{(BAA)}$-branes within the Higgs moduli
space and in a certain limit, mirror symmetry is enhanced via a Fourier-Mukai
transform relative to the Hitchin fibration.

Note that the definition of $\mathrm{(BAA)}$ and $\mathrm{(BBB)}$-branes
extends naturally to any other hyperkähler varieties. The case of the moduli
space $\mathrm{M}_{S,\mathrm{H}}(\mathrm{v}_{a})$ of sheaves over a smooth
projective symplectic surface $S$ was considered in [FJM] by the author,
Jardim and Menet. It is also described in [FJM] the construction of the
$\mathrm{(BBB)}$-brane $\mathrm{N}_{S,\mathrm{H}}^{+}(\mathrm{v}_{a})$ arising
from the fixed locus of a natural involution associated to a sympletic
involution on the surface. Also, the authors studied the behaviour of these
natural branes under some correspondences. As a consequence of Propositions
4.5 and 4.9, we can describe these degenerations in the context of branes,
extending the description to $\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{a},J)$.

######  Corollary 5.1.

The $\mathrm{(BBB)}$-brane $\mathrm{N}_{S,\mathrm{H}}^{+}(\mathrm{v}_{a})$
(resp. the $\mathrm{(BAA)}$-brane
$\mathrm{N}_{S,\mathrm{H}}^{-}(\mathrm{v}_{a})$) arising from the fixed locus
of a natural involution associated to a sympletic (resp. anti-symplectic)
involution on a symplectic surface degenerate into the $\mathrm{(BBB)}$-brane
(resp. $\mathrm{(BAA)}$-brane) inside the Hitchin system
$\mathcal{N}_{C}^{+}(n,d)$ (resp. $\mathcal{N}_{C}^{-}(n,d)$), obtained as the
respective fixed locus of the involution $\check{\zeta}_{K}^{+}$ (resp.
$\check{\zeta}_{K}^{-}$), described in (3.15).

Similarly, the $\mathrm{(BBB)}$-brane obtained from
$\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{a},J)$ degenerates into the
$\mathrm{(BBB)}$-brane given by $\mathcal{P}_{C}(n,d,J_{C})$.

We now discuss the duality under Mirror symmetry of the branes associated to a
Prymian integrable sysem and the natural Lagrangian.

Consider the Mukai vector $\mathrm{v}_{0}$ on $S$, where we assume $a=0$, and
recall from Remark 2.9 that is is the pull-back of the Mukai vector
$\mathrm{w}_{0}$ on $T$. Following Remark 2.12, choose
$J_{0}=\mathcal{O}_{S}(-nC)$. In this section we show that certain open
subsets of the subvarieties $\mathrm{N}_{S,\mathrm{H}}(\mathrm{v}_{0},B)$ and
$\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{0},J_{0},B)$ are dual under
Fourier–Mukai transform.

Observe that the diagram,

$\textstyle{\mathrm{M}_{S,\mathrm{H}}(\mathrm{v}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{h}_{S}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{0},J_{0},B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\mathrm{N}_{S,\mathrm{H}}(\mathrm{v}_{0},B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\mathrm{M}_{S,\mathrm{H}}(\mathrm{v}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{h}_{S}}$$\textstyle{\mathrm{q}_{S}^{*}\\{B\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{A\\},}$

restricts, after Proposition 2.11 and Remark 2.16, to

$\textstyle{\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{h}_{S}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{Prym}\left(\mathrm{q}_{\mathcal{A}^{\mathrm{ssl}}}/{\\{B\\}^{\mathrm{ssl}}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\operatorname{Jac}_{\mathcal{B}^{\,\mathrm{ssl}}/\\{B\\}^{\mathrm{ssl}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\mathrm{q}}_{\mathcal{A}^{\mathrm{ssl}}}}$$\scriptstyle{\mu}$$\textstyle{\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{h}_{S}}$$\textstyle{\mathrm{q}_{S}^{*}\\{B\\}^{\mathrm{ssl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{A\\}^{\mathrm{sm}},}$

where we have dropped the systematic reference to degree $0$ in our notation
for Jacobians.

Let us recall that one can equip
$\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}}\times_{\\{A\\}^{\mathrm{sm}}}\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}}$
with a relative Poincaré bundle $\mathcal{U}_{\mathcal{A}^{\mathrm{sm}}}$.
Denoting by $\pi_{i}$ the projection from
$\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}}\times_{\\{A\\}^{\mathrm{sm}}}\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}}$
to the $i$-th factor, one can consider the relative Fourier–Mukai transform
[Mu1]

$\begin{array}[]{cccc}\Psi_{\mathcal{A}^{\mathrm{sm}}}:&\mathcal{D}^{b}(\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathcal{D}^{b}(\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}})\\\
&\mathcal{F}&\longmapsto&R\pi_{2,*}\left(\pi_{1}^{*}\mathcal{F}\otimes\mathcal{U}_{\mathcal{A}^{\mathrm{sm}}}\right),\end{array}$

and its inverse,

$\begin{array}[]{cccc}\Psi_{\mathcal{A}^{\mathrm{sm}}}^{-1}:&\mathcal{D}^{b}(\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}})&\stackrel{{\scriptstyle}}{{\longrightarrow}}&\mathcal{D}^{b}(\operatorname{Jac}_{\mathcal{A}^{\mathrm{sm}}/\\{A\\}^{\mathrm{sm}}})\\\
&\mathcal{G}&\longmapsto&R\pi_{1,*}\left(\pi_{2}^{*}\mathcal{G}\otimes\mathcal{U}^{-1}_{\mathcal{A}^{\mathrm{sm}}}\right).\end{array}$

After the choice of $J_{0}$ and (2.14) and the properties of Fourier–Mukai
transforms (see [Po] or instance), the composition is the auto-equivalence of
derived categories

$\Psi_{\mathcal{A}^{\mathrm{sm}}}\circ\Psi_{\mathcal{A}^{\mathrm{sm}}}^{-1}=\xi_{J_{0},S}[-g_{A}].$

One can provide a similar construction for $\mathcal{B}^{\mathrm{ssl}}$ in
$T$, giving rise to the Fourier–Mukai transform
$\Psi_{\mathcal{B}^{\mathrm{ssl}}}$. Recalling the morphism
$\widehat{\mathrm{q}}_{\mathcal{A}^{\mathrm{ssl}}}$ from (2.8), both
transforms satisfy (see [Po, (11.3.3)] for instance),

(5.1) $\Psi_{\mathcal{A}^{\mathrm{sm}}}\circ
R\widehat{\mathrm{q}}_{\mathcal{A}^{\mathrm{ssl}},*}\cong
L\mathrm{Nm}(\mathrm{q}_{\mathcal{A}^{\mathrm{ssl}}})^{*}\circ\Psi_{\mathcal{B}^{\mathrm{ssl}}},$

where the norm map $\mathrm{Nm}(\mathrm{q}_{\mathcal{A}^{\mathrm{ssl}}})$ is
the dual of $\widehat{\mathrm{q}}_{\mathcal{A}^{\mathrm{ssl}}}$.

######  Proposition 5.2.

Whenever $\mathrm{q}_{\mathcal{A}^{\mathrm{ssl}}}$ has ramification, the
structural sheaves of the subvarieties
$\left.\mathrm{N}_{S,\mathrm{H}}(\mathrm{v}_{0},B)\right|_{\\{B\\}^{\mathrm{ssl}}}$
and
$\left.\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{0},J_{0},B)\right|_{\\{B\\}^{\mathrm{ssl}}}$
are dual under a relative Fourier–Mukai transform.

If $\mathrm{q}_{\mathcal{A}^{\mathrm{ssl}}}$ is unramified, the structural
sheaf on a connected component of
$\left.\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{0},J_{0},B)\right|_{\\{B\\}^{\mathrm{ssl}}}$
is Fourier–Mukai dual to
$\left.\mathrm{N}_{S,\mathrm{H}}(\mathrm{v}_{0},B)\right|_{\\{B\\}^{\mathrm{ssl}}}$
equipped with the pair of line bundles obtained from pushing forward the
structural sheaf under $\widehat{\mathrm{q}}_{\mathcal{A}^{\mathrm{ssl}}.}$
direct sum of the trivial sheaf and the pull-back of a torsion $2$ line bundle
$\mathcal{L}\to\mathcal{B}^{\mathrm{ssl}}$.

###### Proof.

We start by the ramified case. Observe that
$R\widehat{\mathrm{q}}_{\mathcal{A}^{\mathrm{ssl}},*}\mathcal{O}_{\operatorname{Jac}_{\mathcal{B}}}$
coincides with the trivial sheaf over
$\mathrm{N}_{S,\mathrm{H}}(\mathrm{v}_{0},B)|_{\\{B\\}^{\mathrm{ssl}}}$ after
Proposition 2.11. Also,
$\Psi_{\mathcal{B}^{\mathrm{ssl}}}(\mathcal{O}_{\operatorname{Jac}_{\mathcal{B}}})$
is the relative sky-scraper sheaf at the identity, so its pull-back under
$\mathrm{Nm}(\mathrm{q}_{\mathcal{A}^{\mathrm{ssl}}})$ is precisely the
trivial sheaf over
$\mathrm{P}_{S,\mathrm{H}}(\mathrm{v}_{0},J_{0},B)|_{\\{B\\}^{\mathrm{ssl}}}$.
Then, the first statement follows easily from (5.1).

The second statement follows from the diagram (2.10) and the fact that the
preimage of the Norm map has two connected components. ∎

Recalling Proposition 5.2, one immediately deduces the following.

######  Corollary 5.3.

Whenever $\mathrm{q}_{C}$ has ramification, the structural sheaves of
$\mathcal{N}_{C}(n,d)|_{\\{B\\}^{\mathrm{ssl}}}$ and
$\mathcal{P}_{C}(n,d,J)|_{\\{B\\}^{\mathrm{ssl}}}$ are dual under a relative
Fourier–Mukai transform.

If $\mathrm{q}_{C}$ is unramified, the later is Fourier–Mukai dual to the
sheaf supported on $\mathcal{N}_{C}(n,d)|_{\\{B\\}^{\mathrm{ssl}}}$ given by
the direct sum of the trivial sheaf and the line bundle associated to the
unramified $2:1$ cover of the family of spectral curves.

In this context, Corollaries 4.15 and 4.16 can be seen, respectively, as a
degeneration of $\mathrm{(BAA)}$ and $\mathrm{(BBB)}$-branes, from the moduli
space of sheaves on symplectic surfaces into the Hitchin system.

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# Globally Convergent Visual-Feature Range Estimation with Biased Inertial
Measurements

Bowen Yi<EMAIL_ADDRESS>Chi Jin  Ian R. Manchester  Australian Centre
for Field Robotics, Sydney Institute for Robotics and Intelligent Systems, The
University of Sydney, NSW 2006, Australia  DJI Innovation Inc. Shenzhen
518057, China

###### Abstract

The design of a globally convergent position observer for feature points from
visual information is a challenging problem, especially for the case with only
inertial measurements and without assumptions of uniform observability, which
remained open for a long time. We give a solution to the problem in this paper
assuming that only the bearing of a feature point, and biased linear
acceleration and rotational velocity of a robot — all in the body-fixed frame
— are available. Further, in contrast to existing related results, we do not
need the value of the gravitational constant either. The proposed approach
builds upon the parameter estimation-based observer recently developed in
(Ortega et al., Syst. Control Lett., vol. 85, 2015) and its extension to
matrix Lie groups in our previous work. Conditions on the robot trajectory
under which the observer converges are given, and these are strictly weaker
than the standard persistency of excitation and uniform complete observability
conditions. Finally, as an illustration, we apply the proposed design to the
visual inertial navigation problem.

###### keywords:

Observers Design; Nonlinear Systems; Range Estimation; Robotics

††thanks: This paper was not presented at any IFAC meeting. This paper is
supported by the Australian Research Council.

, ,

##  1 Introduction

Determination of the position of feature points from visual measurements is a
classical problem in several fields, including machine vision, robotics and
control [2, 4, 7, 8, 17, 32]. However, its mathematical model does not fall
into the canonical forms that have been comprehensively studied in the
nonlinear observer community [5], thus being of theoretical interest. With a
single monocular camera, estimating the position of a feature point is
equivalent to estimating its range or depth. There are two general classes of
methodologies: batch methods [11] and observer design [8, 12, 14]. In this
paper, we focus on the latter for its simpler on-line computations.

In [13], the authors adopt a change of coordinate to the inverse of depth,
which has since become a popular formulation, and propose an identifier based
observer. Its main drawback is high-gain injection thus yielding sensitivity
to measurement noise. Thereafter, many nonlinear observers were proposed using
different constructive tools. A less complicated observer design based on
Lyapunov analysis is proposed in [9], which guarantees global asymptotic
stability (GAS) under an instantaneous observability assumption on the robot
trajectory. An extension to paracatadioptric camera can be found in [12],
where a sliding-mode observer is proposed for both affine and non-affine
systems. In [14], a reduced-order observer is proposed based on the widely
used immersion and invariance (I&I) technique [3], achieving semi-global
convergence under the same condition in [9]. To address visual servoing, a
simple feature depth observer is introduced in [8], which builds upon a well-
known lemma from adaptive control. This design requires a persistency of
excitation (PE) condition — weaker than the condition in [9, 14] — however,
with only local convergence guaranteed. More recently it is shown that by
selecting alternative outputs we may obtain a linear time-varying (LTV) model
[10, 17], for which Kalman-Bucy or Riccati observer is applicable to achieve
global stability under some PE conditions. In the last decade, practical
considerations related to this problem have also received significant
attention [23, 26, 27].

In all the above works, it is assumed a measurement of linear velocity
available in observer design. However, a practically important scenario is
that the robot is equipped with inertial measurement unit (IMU), which
provides measurements of linear acceleration rather than velocity. In spite of
intensive research efforts, we are unaware of any observer for the extension
to this case. The main challenge relies on that the unknown attitude — living
in the special orthogonal group — appears in the dynamics of the body-fixed
velocity.

In this paper, we give the first constructive solution to the problem,
presenting a novel globally exponentially convergent position observer for a
single feature point. The constructive tool we adopt is the recently-
introduced parameter estimation-based observer (PEBO) in [21] and its
extension to matrix Lie groups in our previous work [31] on the simultaneous
localization and mapping (SLAM) problem. The central idea in PEBO is to
translate the estimation of a (time-varying) system state into one of
estimating constant parameters. Another merit is that, unlike all existing
solutions, the proposed observer neither requires instantaneous observability
[14], uniform complete observability [17], nor PE conditions [8], and its
convergence is achieved under a much weaker interval excitation (IE)
condition.

Our second contribution is applying the proposed scheme to visual inertial
navigation, arising for in Global Positioning System (GPS)-denied environment,
which is concerned with fusing the information from IMUs and cameras, using
the inertial coordinates of some feature points in a known map. Many types of
Kalman filters have became the _de facto_ standard algorithms in industrial
applications [19], which, however, are intrinsically local. In order to
enlarge domain of attraction, many researchers from the nonlinear control
community have made important contributions in recent years [4, 30]. In this
paper, we provide a simple, almost globally asymptotically convergent solution
only requiring the trajectory being IE. The relaxed excitation condition is
especially important for the case with unknown gravitational constant and
sensor bias, for which it is quite strict to impose the uniformity of
excitation with respect to time.

Notation. We use generally $\epsilon_{t}$ to represent exponentially decaying
terms with proper dimensions. $0_{n}\in\mathbb{R}^{n}$ and $0_{n\times
m}\in\mathbb{R}^{n\times m}$ denote the zero column vector of dimension $n$
and the zero matrix of dimension $n\times m$, respectively. We use $p$ to
represent the differential operator $p:={d\over dt}[\cdot]$, and $|\cdot|$ as
the Euclidean norm of a vector. Given a square matrix $A\in\mathbb{R}^{n\times
n}$, the Frobenius norm is defined as $\|A\|=\sqrt{\mbox{tr}(A^{\top}A)}$, and
$\mbox{adj}(A)$ and $\det(A)$ denote its adjugate matrix and determinant,
respectively. We use $SO(3)=\\{R\in\mathbb{R}^{3\times
3}|R^{\top}R=I_{3},~{}\det(R)=1\\}$ to represent the special orthogonal group,
and ${\mathfrak{so}}(3)$ is its Lie algebra. Given a variable $R\in SO(3)$, we
use $|R|_{I}$ to represent the normalized distance on $SO(3)$ with
$|R|_{I}^{2}:={1\over 4}\mbox{tr}(I_{3}-R)$. For any
$x\in\mathbb{R}^{3}/\\{0\\}$, its projector is defined as
$\Pi_{x}:=I_{3}-{1\over|x|^{2}}xx^{\top}.$ Given $a\in\mathbb{R}^{3}$, we
define the operator $(\cdot)_{\times}$ as
$a_{\times}:=\begin{bmatrix}0&-a_{3}&a_{2}\\\ a_{3}&0&-a_{1}\\\
-a_{2}&a_{1}&0\end{bmatrix}\in{\mathfrak{so}}(3).$

##  2 Model and Problem Formulation

Figure 1: Coordinate systems of a robot observing fixed points

A single monocular camera provides 2D images of its environment, and we focus
the 3D position estimation of _point_ features extracted from images in the
view of a camera on the mobile robot, which is equipped with IMUs. To be
precise, in the body-fixed frame $\\{\mathcal{B}\\}$ the observed feature
point is denoted by $z\in\mathbb{R}^{3}$, satisfying $z=R^{\top}({}^{I}z-x)$
with unknown position $x\in\mathbb{R}^{3}$ of the robot in the inertial frame
$\\{\mathcal{I}\\}$, and $R\in SO(3)$ is the attitude of $\\{\mathcal{B}\\}$
with respect to $\\{\mathcal{I}\\}$. Here, the constant vector
${}^{I}z\in\mathbb{R}^{3}$ represents the feature position in
$\\{\mathcal{I}\\}$. The dynamics of $z$ is given by

$\dot{z}=-\Omega_{\times}z-v,$ (1)

in which $\Omega\in\mathbb{R}^{3}$ is the rotational velocity, and
$v\in\mathbb{R}^{3}$ is the linear velocity, both in the body-fixed frame
$\\{\mathcal{B}\\}$; see Fig. 1. The kinematics of the robot is given by

$\displaystyle\dot{x}={}^{I}v,\quad\dot{R}=R\Omega_{\times}$ (2)

in which ${}^{I}v$ is the velocity in $\\{\mathcal{I}\\}$, i.e. ${}^{I}v:=Rv.$
We have

$\dot{v}=-\Omega_{\times}v+a+b_{a}+R^{\top}g,$ (3)

in which $a\in\mathbb{R}^{3}$ is the “apparent acceleration” representing all
non-gravitational forces in $\\{\mathcal{B}\\}$ measured by IMUs,
$g:=[0,0,\mathsf{g}]^{\top}$ is the gravity vector in $\\{\mathcal{I}\\}$, and
$b_{a}\in\mathbb{R}^{3}$ is the constant sensor bias. In this paper, we
consider the _spherical projection_ model of the camera. In this case, the
output is the bearing of the feature point in the body-fixed frame
$\\{\mathcal{B}\\}$, i.e.

$y={z\over|z|}\in\mathbb{S}^{2},$ (4)

for non-zero $z$. We make the following assumption.

######  Assumption 1.

The origin of the robot never coincides with the feature point, i.e.
$|z(t)|\neq 0$ for all $t\geq 0$. Besides, $\Omega$ and $a$ guarantee the
systems state bounded over time.

The proposed approach is applicable to other image-based estimation problems
with different projection models. Now let us recall some definitions and a
technical lemma as follows. Note that the IE condition is called “exciting
over a finite time interval” in [28, pp. 108]. The swapping lemma for stable
filters is useful for dealing with unavailable “dirty derivatives” to generate
linear regressors.

######  Definition 1.

Given a bounded signal $\phi:\mathbb{R}_{+}\to\mathbb{R}^{n}$, it is 1) PE if
$\int_{t}^{t+T}\phi(s)\phi^{\top}(s)ds\succeq\delta I_{n},~{}\forall t\geq 0$
for some $T>0,\delta>0$; 2) IE if there exist $t_{0},t_{c}\geq 0$ such that
$\int_{t_{0}}^{t_{0}+t_{c}}\phi(s)\phi^{\top}(s)ds\succeq\delta I_{n}$ for
some $\delta>0$.

######  Lemma 1.

(Swapping lemma [24]) For $C^{1}$-differential signals $x,y:\mathbb{R}_{\geq
0}\to\mathbb{R}$, the following holds

${\alpha\over p+\alpha}[xy]=y{\alpha\over p+\alpha}[x]-{1\over
p+\alpha}\left[\dot{y}{\alpha\over p+\alpha}[x]\right]$ (5)

for any $\alpha>0$ with $p:={d/dt}$.

We are interested in the position estimation problem below.

######  Problem 1.

Consider the system (1)-(3) with $\Omega$ and $y$ available.

  * P1

If the velocity $v$ is measurable, design an observer
$\dot{\zeta}=N(\zeta,\Omega,v,y)$, $\hat{z}=H(\zeta,\Omega,v,y)$; or

  * P2

If only $a$ is measurable with $b_{a}$ and $g$ unknown, design an observer
$\dot{\zeta}=N(\zeta,\Omega,a,y),~{}(\hat{z},\hat{v})=H(\zeta,\Omega,a,y)$

with the observer state $\zeta\in\mathbb{R}^{n_{\zeta}}$ and the mappings $N$
and $H$ of proper dimensions, such that $\lim_{t\to\infty}|\hat{z}(t)-z(t)|=0$
for a class of robot trajectories.

##  3 Position Observer with Velocity Information

To start with, in this section we give a solution to P1 as a motivating
design, and in the next section, we will focus on P2 without the information
of $v$.

###  3.1 Generation of Linear Regression Models

For P1 we show that position estimation is equivalent to identification of a
constant scalar parameter. In the following we present a new parameterization
to the range, and show how to generate a linear regression equation (LRE).

######  Proposition 3.1.

Consider the dynamics (1) and the dynamic extension

$\dot{\xi}=-y^{\top}v$ (6)

with arbitrary $\xi(0)\in\mathbb{R}$. Then, the position $z$ satisfies

$z(t)=[\xi(t)+\theta]y,\quad\forall t\geq 0,$ (7)

in which the unknown constant $\theta\in\mathbb{R}$ verifies the LRE

$y_{\tt R}=\phi\theta+\epsilon_{t}$ (8)

with an exponentially decaying term $\epsilon_{t}$, measurable signals

$\displaystyle\phi$ $\displaystyle~{}:=~{}G_{1}[y]+\alpha
G_{2}[\Omega_{\times}y]$ (9) $\displaystyle y_{\tt R}$
$\displaystyle~{}:=~{}-G_{2}[\alpha\Pi_{y}v+y^{\top}v\phi]-\phi\xi,$

the filters $G_{1}[\cdot]:={\alpha p\over p+\alpha}[\cdot]$ ,
$G_{2}[\cdot]:={1\over p+\alpha}[\cdot],$ and $\alpha>0$.

######  Proof 3.2.

From $|z|^{2}=z^{\top}z$ and (1), we have
$|z|\dot{{\aoverbrace[L1R]{|z|}}}=z^{\top}(-\Omega_{\times}z-v)=z^{\top}v,$
and thus the bearing of $z$ satisfies ${d\over dt}|z|=-y^{\top}v.$ Note that
the time derivative of state in the above coordinate is measurable. Thus, the
dynamic extension (6) guarantees $|z(t)|-\xi(t)=\theta,~{}\forall t\geq 0,$
with $\xi$ an available signal, in which the constant scalar parameter
$\theta$ is defined as $\theta:=|z(0)|-\xi(0)$. Hence, we have verified (7).

In the remainder, we show how to derive the LRE (8). By calculating the time
derivatives of (4), we obtain

$\displaystyle\dot{y}$
$\displaystyle={1\over|z|^{2}}\Big{[}(-\Omega_{\times}z-v)|z|+zy^{\top}v\Big{]}$
$\displaystyle=-\Omega_{\times}y-{1\over|z|}v+{1\over|z|}yy^{\top}v=-\Omega_{\times}y-{1\over|z|}\Pi_{y}v,$

equivalently,

$r(\dot{y}+\Omega_{\times}y)=-\Pi_{y}v,$ (10)

where we have defined the variable $r=|z|$ for convenience. Noting the
unavailability of the derivative $\dot{y}$, we thus apply the filter
${\alpha\over p+\alpha}$ to both sides of (10) with $\alpha>0$. For the first
term, we utilize the swapping lemma for filters — see Lemma 1 — and then
obtain

$\displaystyle{\alpha\over p+\alpha}[r\dot{y}]$ $\displaystyle~{}=~{}r{\alpha
p\over p+\alpha}[y]-{1\over p+\alpha}\left[\dot{r}{\alpha\over
p+\alpha}[\dot{y}]\right]+\epsilon_{t}$ $\displaystyle~{}=~{}r{\alpha p\over
p+\alpha}[y]+{1\over p+\alpha}\left[y^{\top}v{\alpha\over
p+\alpha}[\dot{y}]\right]+\epsilon_{t},$

in which the exponentially decaying term $\epsilon_{t}$ arises from the
initial conditions of filters. To be precise, the state-space model of the
filter ${\alpha\over\alpha+p}[\cdot]$ is given by $\dot{\bf y}=-\alpha{\bf
y}+\alpha u$ with input $u$ and output ${\bf y}$. For non-zero ${\bf y}(0)$,
it yields an exponentially decaying term $e^{-\alpha t}{\bf y}(0)$. Therefore,
we have

$\displaystyle r{\alpha p\over p+\alpha}[y]+{1\over
p+\alpha}\left[y^{\top}v{\alpha\over p+\alpha}[\dot{y}]\right]+{\alpha\over
p+\alpha}[r\Omega_{\times}y]$ $\displaystyle=-{\alpha\over
p+\alpha}[\Pi_{y}v]+\epsilon_{t},$

then

$\displaystyle r\phi+G_{2}[y^{\top}v\phi]=-\alpha
G_{2}[\Pi_{y}v]+\epsilon_{t}.$ (11)

Substituting $r=\xi+\theta$ into the above equation, we have the LRE (8).
$\blacksquare$

Thanks to the algebraic relation (7), we have translated, via designing the
dynamic extension (6), the estimation of the position $z$ into the on-line
consistent identification of $\theta$ from the LRE (8). Note that $\theta$ is
a scalar constant, and $\phi$ is a column vector, and thus the least squares
problem is solvable if $\phi(t_{\star})$ is non-zero for some $t_{\star}>0$.
We make the following assumption.

######  Assumption 2.

There exists $t_{\star}>0$ such that $|\phi(t_{\star})|\neq 0.$

###  3.2 Position Observer Design

Based on the LRE (8), we design in this section two globally exponentially
convergent position observers to address P1, i.e., a gradient observer and a
PEBO. In the former a PE condition is imposed to guarantee convergence; in
contrast for the latter, we relax the requirement significantly.

######  Proposition 3.3.

(Gradient position observer) Consider (1) with $v$ and $y$ available. If
$\phi^{\top}$ in (9) is PE, then the observer

$\displaystyle\dot{\hat{r}}$
$\displaystyle=-y^{\top}v-\gamma\phi^{\top}\Big{(}\phi\hat{r}+G_{2}[y^{\top}v\phi]+\alpha
G_{2}[\Pi_{y}v]\Big{)}$ (12) $\displaystyle\hat{z}$ $\displaystyle=\hat{r}y$

with $\gamma>0$, provides a globally exponentially convergent estimate to the
position $z$, i.e.,

$\lim_{t\to\infty}|\hat{z}(t)-z(t)|=0\quad\mbox{(exp.)}.$ (13)

######  Proof 3.4.

See Appendix. $\blacksquare$

The methodology of gradient observers may date back to [25] for general
nonlinear models, which has been extended to several applications, e.g.,
electric motors [22] and pose estimation [16]. Its key step is to obtain a
regression model of unknown states, and PE conditions are imposed to achieve
uniform convergence. In many cases, it may not be guaranteed for given robot
trajectories. To address this, we design a PEBO under a weaker excitation
requirement.

######  Proposition 3.5.

(Position PEBO) Considering the dynamics (1) with $v$ and $y$ available, the
observer consisting of (6) and

$\displaystyle\dot{\zeta}$ $\displaystyle~{}=~{}\phi^{\top}y_{\tt
R}-\phi^{\top}\phi\zeta$ (14) $\displaystyle\dot{\omega}$
$\displaystyle~{}=~{}-\phi^{\top}\phi\omega,\quad\omega(0)=1$
$\displaystyle\dot{\hat{\theta}}$
$\displaystyle~{}=~{}\gamma[(\zeta-\omega\zeta_{0})-(1-\omega)\hat{\theta}]$
$\displaystyle\hat{z}$ $\displaystyle~{}=~{}(\xi+\hat{\theta})y$

with $\zeta(0)=\zeta_{0}$, $y_{\tt R},\phi$ defined in (9), and the filters
$G_{1}[\cdot]$ and $G_{2}[\cdot]$ starting from zero initial conditions,
guarantees the global exponential convergence (13) if Assumption 2 holds.

######  Proof 3.6.

From Proposition 3.1, we have $r\equiv\xi+\theta.$ We need to verify
$\hat{\theta}\to\theta$ as $t\to\infty$. The following analysis is motivated
by the proof in our previous work [31, Proposition 1], where an LTV filter is
designed to generate PE regressors from the ones only satisfying IE; see [29,
6] for extensions. It yields

$\displaystyle{d\over dt}(\zeta-\theta)=~{}\phi^{\top}y_{\tt
R}-\phi^{\top}\phi\zeta=-\phi^{\top}\phi(\zeta-\theta),$

in which we have used (8), and the fact that $\theta$ is a constant. It is
underlined that, by setting zero initial conditions of the filters
$G_{1}[\cdot]$ and $G_{2}[\cdot]$, the decaying term $\epsilon_{t}$ disappears
in (8); see Remark 3.8 on how to implement it. It is easy to get
$\zeta-\theta=\omega(\zeta_{0}-\theta)$ with
$\omega(t)=\exp(-\int_{0}^{t}\phi^{\top}(s)\phi(s)ds)$, then yielding a new
linear regression model

$\zeta-\omega\zeta_{0}=(1-\omega)\theta.$ (15)

Now, let us show that the new regression $(1-\omega)$ is PE if Assumption 2
holds. From the continuity of the signal $\phi$, Assumption 2 implies the
existence of a sufficiently small parameter $\epsilon>0$ such that
$\int_{t_{\star}}^{t_{\star}+\epsilon}\phi^{\top}(s)\phi(s)ds>0.$ Hence, we
conclude that $\phi^{\top}$ is IE. The solution of $\omega$ satisfies for
$t\in[0,t_{\star}+\epsilon]$, $1-\omega(t)\geq 0$; and for
$t>t_{\star}+\epsilon$

$\displaystyle 1-\omega(t)$
$\displaystyle=1-\exp\left(-\int_{0}^{t}\phi^{\top}(s)\phi(s)ds\right)$
$\displaystyle\geq
1-\exp\left(-\int_{t_{\star}}^{t_{\star}+\epsilon}\phi^{\top}(s)\phi(s)ds\right)>\delta_{0},$

for some constant $\delta_{0}>0$. Hence, the regression $(1-\omega)$ in (15)
is non-negative and PE. We define the parameter estimation error
$\tilde{\theta}:=\hat{\theta}-\theta$, the dynamics of which is

$\dot{\tilde{\theta}}=-\gamma(1-\omega)\tilde{\theta},\quad\gamma>1.$ (16)

Using the Cauchy–Schwarz inequality for integrals, we conclude that
$(1-\omega)^{1\over 2}$ is PE. The global exponential stability of the LTV
dynamics (16) is established [24, Thm 2.5.1]. Then, we have
$\lim_{t\to\infty}|\xi(t)+\hat{\theta}(t)-r(t)|=0$ exponentially. Invoking
state boundedness, we complete the proof. $\blacksquare$

######  Remark 3.7.

In the new regressor (15), the signal $(1-\omega)\in[0,1)$ — intuitively, this
value characterizes how “strong” the excited signal is — as a result limiting
the convergence speed of the observer. A possible way to overcome this issue
is to mix (15) with the original regressor (8), and then obtaining a new LRE
${y}_{\tt R}^{\prime}={\phi}^{\prime}\theta,$ with ${y}_{\tt
R}^{\prime}:=\zeta-\omega\zeta_{0}+k_{p}\phi^{\top}y_{\tt R}$ and the new
regression $\phi^{\prime}:=(1-\omega)+k_{p}|\phi|^{2}$, which is not
necessarily less than one. Here, the gain $k_{p}>0$ plays the role to make a
tradeoff on the trust of historical and current information. The last equation
in the observer (14) may be modified accordingly to accelerate convergence
speed.

######  Remark 3.8.

We underline that the third equation in the observer (14), in which we have
used both the PE condition of $(1-\omega)$ as well as its positiveness after
$t_{\star}+\epsilon$, is not a gradient flow. The gradient flow is given by
${d\over
dt}{\hat{\theta}}=\gamma(1-\omega)[(\zeta-\omega\zeta_{0})-(1-\omega)\hat{\theta}],$
which also provides a globally exponentially convergent estimate to $\theta$
under the IE assumption. We refer the reader to [24] for on-line gradient
descent algorithms. Besides, in Proposition 3.5 it is necessary to carefully
select the initial conditions of filters $G_{1}[\cdot]$ and $G_{2}[\cdot]$ to
deal with the term $\epsilon_{t}$ for the IE case. It can be implemented as
the state space models $\dot{\bf y}=-\alpha{\bf y}+{\bf u}$ with ${\bf
y}(0)=0$ for $G_{2}[\cdot]$, and $\dot{\bf x}=-\alpha{\bf x}+\alpha^{2}{\bf
u},~{}{\bf y}=-{\bf x}+\alpha{\bf u}$ with ${\bf x}(0)=\alpha{\bf u}(0)$ for
$G_{1}[\cdot]$, in which ${\bf x},~{}{\bf u}$ and ${\bf y}$ denote the
internal state, input and output of the filters, respectively.

##  4 Position-Velocity Observer with Biased Acceleration Measurement

In this section, we present the main result of the paper, i.e., a solution to
P2, for which we design a position-velocity observer with availability of only
the acceleration.

###  4.1 Generation of Linear Regression Models

We are interested in the estimation of position $z$, linear velocity $v$ and
the bias $b_{a}$, which are collected in the vector

$X:=\mbox{col}(z,v,b_{a})\in\mathbb{R}^{9}.$

######  Proposition 4.9.

Consider the dynamics (1)-(3), and design the following dynamic extension

$\displaystyle\dot{Q}$ $\displaystyle=Q\Omega_{\times}$ (17)
$\displaystyle\dot{\xi}$
$\displaystyle=A(y,\Omega,Q)\xi+B(a),\quad\xi(0)=0_{10}$
$\displaystyle\dot{\Psi}$
$\displaystyle=A(y,\Omega,Q)\Psi,\quad\Psi(0)=I_{10}$

with $Q(0)\in SO(3)$, and

$A:=\begin{bmatrix}0&-y^{\top}&0_{3}^{\top}&0_{3}^{\top}\\\
0_{3}&-\Omega_{\times}&I_{3}&Q^{\top}\\\ 0_{6}&\ldots&\ldots&0_{6\times
3}\end{bmatrix}\quad B:=\begin{bmatrix}0\\\ a\\\ 0_{6}\end{bmatrix}.$ (18)

Then, the state $X$ satisfies the algebraic equation

$X=T(y)[\xi+\Psi\theta]$ (19)

with $T(y):=\mbox{diag}\\{y,[I_{6},0_{6\times 3}]\\},$ in which the unknown
constant vector $\theta$ admits the LRE $y_{\tt N}=\psi^{\top}\theta,$ with

$\displaystyle y_{\tt N}$ $\displaystyle~{}:=~{}-\phi T_{1}\xi-
G_{2}\big{[}(\phi y^{\top}+\alpha\Pi_{y})T_{2}\xi\big{]}$ (20)
$\displaystyle\psi$ $\displaystyle~{}:=~{}\left(\phi
T_{1}\Psi\right)^{\top}+G_{2}\big{[}(\phi
y^{\top}+\alpha\Pi_{y})T_{2}\Psi\big{]}^{\top},$

$\phi$ defined in (9), the filters $G_{1}[\cdot],~{}G_{2}[\cdot]$ starting
from zero, the parameter $\alpha>0$, and the matrices
$T_{1}:=\begin{bmatrix}1&~{}0_{1\times
9}\end{bmatrix},~{}T_{2}:=\begin{bmatrix}0_{3\times 1}~{}&I_{3}~{}&0_{3\times
6}\end{bmatrix}.$

######  Proof 4.10.

From the analysis in Section 3, the full dynamics is given by (2)-(3) and
$\dot{r}=-y^{\top}v$. Define the error between $Q$ and $R$ in $SO(3)$ as
$E(R,Q):=RQ^{\top},$ and we have

$\displaystyle\dot{{\aoverbrace[L1R]{E(R,Q)}}}$
$\displaystyle:=\dot{R}Q^{\top}-RQ^{-1}\dot{Q}Q^{-1}$
$\displaystyle=R\Omega_{\times}Q^{\top}-RQ^{\top}Q\Omega_{\times}Q^{\top}=0.$

Hence, there exists a constant matrix $Q_{c}\in SO(3)$ satisfying

$R(t)=Q_{c}Q(t),~{}\forall t\geq 0,$

with $Q_{c}:=R(0)Q(0)^{\top}$. Then, we have the parameterization to the last
term in the dynamics of $v$ as

$R(t)^{\top}g=Q(t)^{\top}g_{c},~{}\forall t\geq 0$

with a new _constant unknown_ vector $g_{c}\in\mathbb{R}^{3}$ defined as
$g_{c}:=Q_{c}^{\top}g$.

Now considering that $b_{a}$ is a constant bias, and defining the extended
state $\chi:=\mbox{col}(r,v,b_{a},g_{c})\in\mathbb{R}^{10},$ the plant
dynamics may be compactly rewritten as

$\dot{\chi}=A(y,\Omega,Q)\chi+B(a).$ (21)

Comparing the dynamics (21) and (17) and following the idea of generalized
(G)PEBO [20], it yields
$\frac{d}{dt}(\chi-\xi)=A(y,\Omega,Q)\big{[}\chi-\xi\big{]}.$ The matrix
$\Psi(t)\Psi(s)^{\top}$ can be viewed as the state transition matrix for this
LTV system from $s$ to $t$. Hence, this yields

$\displaystyle\chi-\xi=\Psi[\chi_{0}-\xi(0)]\quad\implies$
$\displaystyle\quad\chi=\xi+\Psi\chi_{0}$ (22)
$\displaystyle\overset{\theta:=\chi_{0}}{\implies}$
$\displaystyle\quad\chi=\xi+\Psi\theta,$

where in the second equation we have used $\xi(0)=0$. From the proof of
Proposition 3.1, we have

$r\phi+G_{2}[(\phi y^{\top}+\alpha\Pi_{y})v]=0$ (23)

with $\phi$ defined in (9), for the case with zero filtering initial
conditions. From the definitions of $\chi$ and $X$ we have $X=T(y)\chi,$ thus
verifying the algebraic equation (19). Substituting the last equation of (22)
into (23), we then have

$\displaystyle\quad\phi T_{1}\chi+G_{2}[(\phi
y^{\top}+\alpha\Pi_{y})T_{2}\chi]=0$ $\displaystyle\implies$
$\displaystyle\quad\phi T_{1}(\xi+\Psi\theta)+G_{2}[(\phi
y^{\top}+\alpha\Pi_{y})T_{2}(\xi+\Psi\theta)]=0$ $\displaystyle\implies$
$\displaystyle\quad y_{\tt N}=\psi^{\top}\theta$

with the signals $y_{\tt N}$ and $\psi$ given in (20). $\blacksquare$

The above regression model is motivated by an extension of PEBO from Euclidean
space to matrix Lie groups in our previous work [31]. By gathering the unknown
but constant matrices $Q_{c}\in SO(3)$ and $g\in\mathbb{R}^{3}$ in a new
vector $g_{c}$, it provides an effective way to deal with the scenario with
unknown attitude and gravitational constant, e.g., in aerospace applications.

###  4.2 Position-Velocity Observer Design

In the above subsection we present a novel linear regression model with
respect to the constant vector $\theta$, the estimation of which is sufficient
to reconstruct the state $X$. Now, the remaining task is to estimate $\theta$
stemmed from Proposition 4.9. In the following we provide a globally
convergent position-velocity observer with biased inertial measurements for
the robot trajectory _not_ satisfying the PE condition.

######  Proposition 4.11.

Consider the dynamics (1)-(3) under Assumption 1, and the filtered signal

$\hat{\theta}={\mathcal{H}}[(y_{\tt N},\psi)]$ (24)

with $y_{\tt N},\psi$ defined in Proposition 4.9 and filter ${\mathcal{H}}$
given by

$\left\\{\begin{aligned} \dot{\Phi}&=-\rho\Phi+\psi\psi^{\top}\\\
\dot{Y}&=-\rho Y+\psi y_{\tt N}\\\
\dot{\zeta}&=\Delta\mathbb{Y}-\Delta^{2}\zeta\\\
\dot{\omega}&=-\Delta^{2}\omega,\quad\omega(0)=1\\\
\dot{\hat{\theta}}&=\gamma\big{[}(\zeta+k_{p}\Delta\mathbb{Y})-(1-\omega+k_{p}\Delta^{2})\hat{\theta}\big{]}\end{aligned}\right.$
(25)

with the gains $\rho>0$, $\gamma>0$ and $k_{p}>0$, $\Delta:=\det\\{\Phi\\}$,
$\mathbb{Y}:=\mbox{adj}\\{\Phi\\}Y$, $n=\dim\\{\theta\\}$, and the initial
conditions $\zeta(0)=0_{n},Y(0)=0_{n}$ and $\Phi(0)=0_{n\times n}$. Then, the
observer consisting of (17), (25) and the estimate
$\hat{X}=T(y)(\xi+\Psi\hat{\theta})$ with $T(y)$ defined in Proposition 4.9,
guarantees global exponential convergence
$\lim_{t\to\infty}|\hat{X}(t)-X(t)|=0$ from any initial guess
$\hat{\theta}(0)\in\mathbb{R}^{n}$, assuming that $\psi$ is IE.

######  Proof 4.12.

According to Proposition 4.9, the system dynamics (1) and (3), together with
(17) admits the LRE $y_{\tt N}=\psi^{\top}\theta$. Following the dynamic
regressor extension and mixing (DREM) technique [1], after going through an
LTV filter — the first two equations in (25) — we obtain ${d\over
dt}(Y-\Phi\theta)=-\rho(Y-\Phi\theta)$ with $\rho>0$ from
$Y(0)-\Phi(0)\theta=0$, thus obtaining the Kreisselmeier’s extended LRE [15]

$Y=\Phi\theta$ (26)

with $Y\in\mathbb{R}^{n}$ and $\Phi\in\mathbb{R}^{n\times n}$. After pre-
multiplying the adjugate matrix $\mbox{adj}\\{\Phi\\}$ to the both sides of
(26), we get the decoupled regressors $\mathbb{Y}=\Delta\theta$ with a
_scalar_ regression $\Delta$, which is now in the same form as in Proposition
3.5. Without loss of generality, we assume $\psi$ is IE in the interval
$[0,t_{c}]$ with $t_{c}>0$, i.e.,
$\int_{0}^{t_{c}}\psi(s)\psi(s)^{\top}ds\succeq\delta I$ for some $\delta>0$.
From $\dot{\Phi}=-\rho\Phi+\psi\psi^{\top}$ with $\Phi(0)=0_{n\times n}$, we
have

$\displaystyle\Phi(t_{c})~{}={}$
$\displaystyle\int_{0}^{t_{c}}e^{-\rho(t_{c}-s)}\psi(s)\psi(s)^{\top}ds$
$\displaystyle~{}\succeq{}$ $\displaystyle e^{-\rho
t_{c}}\int_{0}^{t_{c}}\psi(s)\psi(s)^{\top}ds$ $\displaystyle~{}\succeq{}$
$\displaystyle\delta e^{-\rho t_{c}}I.$

From the definition $\Delta$ being the determinant of $\Phi$, it yields

$\displaystyle\psi\in\mbox{ IE}$
$\displaystyle\implies\quad\Delta\in\mbox{IE}$
$\displaystyle\implies\quad(1-\omega+k_{p}\Delta^{2})\in\mbox{PE},$

where the second implication can be proved using the same arguments as
Proposition 3.5 thus omitted. On the other hand, we can verify that the vector
$\theta$ satisfies the LRE

$\zeta+k_{p}\Delta\mathbb{Y}=(1-\omega+k_{p}\Delta^{2})\theta.$ (27)

Now, we obtain a new LRE (27), satisfying the PE condition, from the IE
regressor (19). By defining the estimation error
$\tilde{\theta}=\hat{\theta}-\theta$, we have

$\dot{\tilde{\theta}}=-\gamma(1-\omega+k_{p}\Delta^{2})\tilde{\theta}.$ (28)

Since $(1-\omega+k_{p}\Delta^{2})$ is PE and non-negative for $k_{p}>0$, the
LTV system (28) is globally exponentially stable at the origin. By invoking
the identity (19) and state boundedness from Assumption 1, it completes the
proof. $\blacksquare$

##  5 Application to Visual-Inertial Navigation

We now apply the proposed observer to the vision-aided inertial navigation
systems as an illustration. In this problem, there are several feature points
in the camera view field, whose inertial positions ${}^{I}z_{i}$ are known in
advance. We assume that some data association algorithm has been used to
determine if these observations correspond to landmarks in the given map. Note
that a known map makes it significantly simpler than the SLAM problem, which
is of more practical interests; see for example [17]. The camera may provide
the bearing measurement of features in the body-fixed frame

$y_{i}={z_{i}\over|z_{i}|}=R^{\top}{{}^{I}z_{i}-x\over|{}^{I}z_{i}-x|},\quad
i\in\mathcal{N}:=\\{1,\ldots,n\\}.$ (29)

We adopt the same notations in Section 2, except the subindex for feature
points to distinguish them.

######  Problem 2.

(Navigation observer) Consider (2)-(3) with measurements $(a,\Omega)$ and
$y_{i}$. Both the bias $b_{a}\in\mathbb{R}^{3}$ and the gravity vector
$g\in\mathbb{R}^{3}$ are unknown. Assume the positions ${}^{I}z_{i}$ in
$\\{\mathcal{I}\\}$ are constant and known. Design an observer to
asymptotically estimate the pose $(R,x)$.

A simple solution is to use the proposed range observer to reconstruct the
position of feature points in $\\{\mathcal{B}\\}$, and then solve the
localization problem with “full position measurement”. We need the following.

######  Assumption 3.

There exist $i,j\in\mathcal{N}\backslash\\{n\\}$ such that
${}^{I}\eta_{i}\times{}^{I}\eta_{j}\neq 0,~{}i\neq j$ with the vectors
${}^{I}\eta_{i}:={}^{I}z_{i+1}-{}^{I}z_{i}$.

######  Proposition 5.13.

Consider the navigation observer consisting of the ranges observer

$\displaystyle\dot{Q}$ $\displaystyle~{}=~{}Q\Omega_{\times}$
$\displaystyle\dot{\xi}$
$\displaystyle~{}=~{}A_{e}(\Omega,Q,\bar{y})\xi+B_{e}(a),\quad\xi(0)=0$ (30)
$\displaystyle\dot{\Psi}$
$\displaystyle~{}=~{}A_{e}(\Omega,Q,\bar{y})\Psi,\quad\Psi(0)=I$ (31)
$\displaystyle\hat{\theta}$ $\displaystyle~{}=~{}{\mathcal{H}}(y_{\tt
N},\psi),~{}\hat{v}=T_{v}(\xi+\Psi\hat{\theta}),$ $\displaystyle\hat{z}_{i}$
$\displaystyle~{}=~{}T_{z,i}(\xi+\Psi\hat{\theta})y_{i}$ (32)

with $\bar{y}:=\mbox{col}(y_{1}^{\top},\ldots,y_{n}^{\top})$ and

$\displaystyle y_{\tt N}$ $\displaystyle~{}:=~{}-\Lambda_{\phi}T_{r}\xi-
G_{2}[(\Lambda_{\phi}\bar{y}+\alpha\mathbf{\Pi})T_{v}\xi]$ (33)
$\displaystyle\psi$
$\displaystyle~{}:=~{}(\Lambda_{\phi}T_{r}\Psi)^{\top}+G_{2}[(\Lambda_{\phi}\bar{y}+\alpha\mathbf{\Pi})T_{v}\Psi]^{\top},$

the filter ${\mathcal{H}}$ defined in (24)-(25),
$\mathbf{\Pi}:=\mbox{col}(\Pi_{y_{1}},\ldots,\Pi_{y_{n}})$, the matrices
$T_{r}:=[0_{n\times 9}~{}I_{n}]$, $T_{v}:=[I_{3}\quad 0_{3\times(6+n)}]$,
$T_{z,i}:=[0_{1\times 9}\quad\mathsf{e}_{i}^{\top}]$, $\mathsf{e}_{i}$ being
the $i$-th canonical basis in $\mathbb{R}^{n}$,
$\Lambda_{\phi}:=\mbox{diag}\\{\phi_{1},\ldots,\phi_{n}\\}$, $\phi_{i}$ in (9)
for the $i$-th feature point, and

$A_{e}:=\begin{bmatrix}-\Omega_{\times}&I_{3}&Q^{\top}&0_{3\times n}\\\
\mbox{---}&&0_{6\times(n+9)}&\mbox{---}\\\ -y_{1}^{\top}&0&&\\\
\vdots&&\ddots&\\\
-y_{n}^{\top}&&&0\end{bmatrix},~{}B_{e}:=\begin{bmatrix}a\\\ 0\\\ \vdots\\\
0\end{bmatrix},$ (34)

cascading to the (full-position) localization observer

$\displaystyle\dot{\hat{Q}}_{c}$
$\displaystyle~{}=~{}-w_{\times}\hat{Q}_{c},\quad\hat{R}=\hat{Q}_{c}Q,$
$\displaystyle w$ $\displaystyle~{}=~{}{1\over
2}{\textstyle\sum_{i\in\mathcal{N}\backslash\\{n\\}}}k_{i}{}^{I}\eta_{i}\times(\hat{Q}_{c}Q\hat{\eta}_{i})$
(35) $\displaystyle\dot{\hat{x}}$
$\displaystyle~{}=~{}\hat{Q}_{c}Q\hat{v}+{\textstyle\sum_{i\in\mathcal{N}}\sigma_{i}}({}^{I}z_{i}-\hat{x}-\hat{Q}_{c}Q\hat{z}_{i})$

with $\hat{\eta}_{i}=\hat{z}_{i+1}-\hat{z}_{i}$ and the gains
$k_{i},\sigma_{i},\alpha>0$. If $\psi$ is IE and Assumption 3 holds, then the
convergence $\lim_{t\to\infty}[\|\hat{R}(t)-R(t)\|+|\hat{x}(t)-x(t)|]=0$ holds
almost globally.

######  Proof 5.14.

It is given in Appendix. $\blacksquare$

The proposed navigation observer is implemented in a _modular_ manner, i.e.,
it constitutes of a ranges observer (31) and a full-position localization
observer (35). To be precise, we first use (31) to reconstruct the full
position “measurement”, which is then utilized in the localization observer
(35). We figure out that the “integrated” ranges observer (31) can be replaced
by several “individual” observers in Propositions 4.9-4.11 for each feature
point.

##  6 Simulation Results

(a) Range estimate $\hat{r}$ (IE)

(b) Position estimate error $|\hat{z}-z|$ (PE)

(c) Position estimate error $|\hat{z}-z|$ (IE)

Figure 2: Comparison among the PEBO, the I&I observer in [3], the feature
depth observer (FDO) in [8], the LTV Kalman filter in [17] and the Riccati
observer in [10] with linear velocity information (with noise)

(a) Range estimate $\hat{r}$

(b) Velocity estimation error

(c) Bias estimation error

Figure 3: Performance of the proposed position-velocity observer with biased
acceleration measurement (IE, with noise)

(a) Position estimation error

(b) Attitude estimation error

(c) Velocity estimation error

Figure 4: Performance of the proposed navigation observer with biased
acceleration measurement (IE, noise-free)

In this section, we present simulation results to illustrate the properties of
the proposed observers. Full details and additional figures can be found in
Appendix. First, consider the case of measurable velocity in Section 3 with
two trajectories, one being PE and the other being IE but not PE. In Fig. 2,
we observe that the estimate from the PEBO with measurement noise ultimately
converges to a small neighbourhood of its true value, showing robustness _vis-
à-vis_ noise. We also simulated the I&I observer [14], the feature depth
observer [8], the LTV Kalman filter [17] and the Riccati observer [10] all of
which require some PE conditions. Since the observers in [8, 14] are concerned
with the depth estimation, we draw them together for comparison using the IE
trajectory in Fig. 2(a). These observers have different state spaces, so we
chose the initial range estimates as close as possible to make a fair
comparison. Note that both the I&I and the feature depth observers have large
ultimate errors since their excitation conditions are not satisfied. In Figs.
2(b)-2(c), we compare to the position observers in [17] and [10] using both
the PE and IE trajectories. As expected, these three observers admit similar
accuracy for PE trajectories; however, for IE trajectories, the proposed PEBO
is superior to the others.

Now let us consider position and velocity estimation in Section 4, testing
with an IE but not PE trajectory and noisy measurements. The initial
conditions for the observer are set as $\hat{\theta}(0)=[0_{1\times 9},10]$
with a small sensor bias. Fig. 3 shows convergence of all estimation errors to
a neighborhood of zero.

Lastly, we evaluate the performance of the navigation observer in Proposition
5.13 with three known landmarks in $\\{\mathcal{I}\\}$. The same robot
trajectory and conditions were adopted. The behaviour is shown in Fig. 4 in
the absence of measurement noise. The pose (including the position and the
attitude) estimate converge its true value after a short transient stage,
showing high estimation performance.

##  7 Concluding Remarks

We have proposed a nonlinear observer for the problem of range estimation of a
visual feature, using only its bearing and linear acceleration measurements,
by means of the recently developed PEBO methodology. The design is applicable
to the case with unknown sensor bias and gravitational constant, and can
achieve global exponential convergence under a weak IE condition on the robot
trajectory. The trade off is that the design requires a more complicated
observer structure than other methods. To illustrate its utility, we apply the
observer to the problem of visual inertial navigation, providing a novel
almost globally convergent solution.

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## Appendix

##  Appendix A Proof of Proposition 3.3

Proof. Define the estimation error of range as $\tilde{r}:=\hat{r}-r$ with
$r=|z|$, the dynamics of which is given by

$\dot{\tilde{r}}=-\gamma\phi^{\top}\left(\phi\hat{r}+G_{2}[y^{\top}v\phi]+\alpha
G_{2}[\Pi_{y}v]\right).$

Invoking (11), we have

$\dot{\tilde{r}}=-\gamma\phi^{\top}\phi\tilde{r}+\epsilon_{t},$ (36)

in which the exponentially decaying term $\epsilon_{t}$ is caused by initial
conditions of the stable filters $G_{1}[\cdot]$ and $G_{2}[\cdot]$.

Now we use the standard perturbation analysis to show that $\tilde{r}\to 0$ as
$t\to\infty$. Consider an auxiliary (nominal) system

$\dot{\tilde{r}}=-\gamma\phi^{\top}\phi\tilde{r},$ (37)

for which the necessary and sufficient condition of the global exponential
stability (GES) of the origin is that $\phi^{\top}$ is PE [24, Thm 2.5.1]. For
the LTV system (37), invoking the converse Lyapunov theorem [Kha02, Thm 4.14],
there exist a smooth function $V(\tilde{r},t)$ and constants
$c_{i}\in\mathbb{R}_{>0}$ ($i=1,\ldots,4$) satisfying

$\displaystyle c_{1}|\tilde{r}|^{2}\leq V(\tilde{r},t)$ $\displaystyle\leq
c_{2}|\tilde{r}|^{2}$ (38) $\displaystyle{\partial V\over\partial t}+{\partial
V\over\partial\tilde{r}}(-\gamma\phi^{\top}\phi\tilde{r})$ $\displaystyle\leq-
c_{3}|\tilde{r}|^{2}$ $\displaystyle\left|{\partial
V\over\partial\tilde{r}}(\tilde{r},t)\right|$ $\displaystyle\leq
c_{4}|\tilde{r}|.$

Consider the Lyapunov-like function

$W(\tilde{r},t)=V(\tilde{r},t)+V_{\varepsilon}(t)$

with

$V_{\varepsilon}:=\int_{t}^{+\infty}{c_{4}^{2}\over
2c_{3}}|\epsilon_{t}(s)|^{2}ds,$

which is well-defined since $\epsilon_{t}$ is exponentially decaying — thus
square integrable. Calculating the time derivative of $W(\tilde{r},t)$ along
the trajectory of the _perturbed_ system (36), we have

$\displaystyle\dot{W}$ $\displaystyle={\partial V\over\partial t}+{\partial
V\over\partial\tilde{r}}(-\gamma\phi^{\top}\phi\tilde{r}+\epsilon_{t})-{c_{4}^{2}\over
2c_{3}}|\epsilon_{t}(t)|^{2}$ $\displaystyle\leq-
c_{3}|\tilde{r}|^{2}+\left|{\partial
V\over\partial\tilde{r}}\right||\epsilon_{t}|-{c_{4}^{2}\over
2c_{3}}|\epsilon_{t}|^{2}$ $\displaystyle\leq-{c_{3}\over 2}|\tilde{r}|^{2}$
$\displaystyle\leq-{c_{3}\over 2c_{2}}(W-V_{\varepsilon}(t)),$

where in the third inequality we have used $|{\partial
V\over\partial\tilde{r}}||\epsilon_{t}|\leq{c_{3}\over
2}|\tilde{r}|^{2}+{c_{4}^{2}\over 2c_{3}}|\epsilon_{t}|^{2}$, as well as the
last equality in (38). Using the comparison lemma to the inequality
$\dot{W}\leq-{c_{3}\over 2c_{2}}(W-V_{\varepsilon}(t))$ and the fact
$V_{\varepsilon}\to 0$ exponentially due to its definition, we have both
$W(\tilde{r},t)$ and $\tilde{r}$ converging to zero exponentially fast as
$t\to\infty$. Invoking the algebraic relation $z=ry$, as well as the
boundedness assumption of $y$, it completes the proof. $\blacksquare$

##  Appendix B Proof of Proposition 5.13

Proof. In the problem set of navigation, the full system dynamics is given by

$\displaystyle\dot{R}$ $\displaystyle=R\Omega_{\times}$ (39)
$\displaystyle\dot{v}$ $\displaystyle=-\Omega_{\times}v+a+b_{a}+R^{\top}g$
$\displaystyle\dot{r}_{1}$ $\displaystyle=-y_{1}^{\top}v$
$\displaystyle~{}\vdots$ $\displaystyle\dot{r}_{n}$
$\displaystyle=-y_{n}^{\top}v$

With the dynamic extension $\dot{Q}=Q\Omega_{\times}$, we define the constant
vector

$g_{c}:=Q(0)R(0)^{\top}g,$

and the full-state variable

$\chi:=\begin{bmatrix}v^{\top}&b_{a}^{\top}&g_{c}^{\top}&r_{1}&\ldots&r_{n}\end{bmatrix}^{\top}$

with $r_{i}:=|z_{i}|$ ($i\in\mathcal{N}$). Then, we have

$\dot{\chi}=A_{e}(\Omega,Q,\bar{y})\chi+B_{e}(a)$

with the matrices $A_{e}$ and $B_{e}$ defined in (34). It yields

$\dot{{\aoverbrace[L1R]{\chi-\xi}}}~{}=~{}A(\bar{y},\Omega,Q)[\chi-\xi],$

thus $\chi=\xi+\Psi\theta$ with $\theta:=\chi_{0}$.

From (23), we have

$\Lambda_{\phi}T_{r}(\xi+\Psi\theta)+G_{2}\left[(\Lambda_{\phi}\bar{y}+\alpha\mathbf{\Pi})T_{v}(\xi+\Psi\theta)\right]=0$

and

$\bar{y}=\begin{bmatrix}y_{1}^{\top}\\\ \vdots\\\
y_{n}^{\top}\end{bmatrix},\quad\mathbf{\Pi}=\begin{bmatrix}\Pi_{y_{1}}\\\
\vdots\\\ \Pi_{y_{n}}\end{bmatrix}$

in which we have defined $\phi_{i}$ as (9) for the $i$-th feature point with a
slight abuse of notation using subscripts. Then, we obtain the linear
regressor $y_{\tt N}=\psi^{\top}\theta$ with (33). Following Proposition 4.11,
if $\psi$ is IE, then $\hat{\theta}={\mathcal{H}}[y_{\tt N},\psi]$ provides a
globally exponentially convergent estimate to $\theta$. Invoking state
boundedness from Assumption 1, it yields for all $i\in\mathcal{N}$

$\lim_{t\to\infty}\left|\begin{bmatrix}\hat{v}(t)-v(t)\\\
\hat{z}_{i}(t)-z_{i}(t)\end{bmatrix}\right|=0\quad\mbox{(exp.)}.$ (40)

The second step is to show that $\hat{Q}_{c}\in SO(3)$ is an almost global
asymptotic estimate to $Q_{c}$, which is similar to attitude estimation in
[18] but with a constant matrix $Q_{c}$ rather than a time-varying one. Define
its estimation error as

$\tilde{Q}_{c}:=Q_{c}\hat{Q}_{c}^{\top},$

and its time derivative is given by111The operator $\mbox{skew}(\cdot)$ is
defined as $\mbox{skew}(A):=\frac{1}{2}(A-A^{\top})$ for a square matrix $A$.

$\displaystyle\dot{\tilde{Q}}_{c}$
$\displaystyle=\tilde{Q}_{c}\sum_{i=1}^{n-1}{k_{i}\over
2}[(\hat{Q}_{c}Q\hat{\eta}_{i})({}^{I}\eta_{i})^{\top}-{}^{I}\eta_{i}(\hat{Q}_{c}Q\hat{\eta}_{i})^{\top}]$
(41)
$\displaystyle=\tilde{Q}_{c}\sum_{i=1}^{n-1}k_{i}\mbox{skew}[\tilde{Q}_{c}^{\top}{}^{I}\hat{\eta}_{i}({}^{I}\eta_{i})^{\top}]$
$\displaystyle=\tilde{Q}_{c}\mbox{skew}[\tilde{Q}_{c}^{\top}M]+\mathcal{E}(t)$

where in the second equation we have used the fact $[c\times
d]_{\times}=dc^{\top}-cd^{\top}$ for any $c,d\in\mathbb{R}^{n}$, and define
the constant matrix
$M:=\sum_{i=1}^{n-1}k_{i}{}^{I}\eta_{i}({}^{I}\eta_{i})^{\top}$, the vectors
${}^{I}\hat{\eta}_{i}=R\hat{\eta}_{i}$, and

$\mathcal{E}:=\tilde{Q}_{c}\sum_{i=1}^{n-1}k_{i}\mbox{skew}[\tilde{Q}_{c}^{\top}R\tilde{\eta}_{i}({}^{I}\eta_{i})^{\top}]$

with exponentially decaying terms $\tilde{\eta}_{i}:=\hat{\eta}_{i}-\eta_{i}$,
which satisfy $\eta_{i}\to 0$ as $t\to 0$ exponentially fast invoking (40).
Since both $\tilde{Q}_{c}$ and $R$ living in a compact space $SO(3)$ and
${}^{I}\eta_{i}$ being constant, there exist two constants $a_{0},a_{1}>0$
such that

$|\mbox{tr}(\mathcal{E}^{\top}(t)M)|\leq a_{0}e^{-a_{1}t}=:\varepsilon(t).$
(42)

We consider the time-varying Lyapunov function

$V(\tilde{Q}_{c},t)=\sum_{i=1}^{n-1}k_{i}|{}^{I}\eta_{i}|^{2}-\mbox{tr}(\tilde{Q}_{c}^{\top}M)+\int_{t}^{\infty}|\varepsilon(s)|ds$

which is a non-negative function and well defined, since $\varepsilon$ is
absolutely integrable, i.e.

$\int_{0}^{\infty}|\varepsilon(s)|ds<+\infty.$

Now let us show non-negativeness of $V(\tilde{Q}_{c},t)$. Since the last term
$\int_{t}^{\infty}|\varepsilon(s)|ds\geq 0$, we focus on the first two terms.
According to the definition of matrix $M$, invoking that ${}^{I}\eta_{i}$ are
constant vectors, we have

$\displaystyle\sum_{i=1}^{n-1}k_{i}|{}^{I}\eta_{i}|^{2}$
$\displaystyle-\mbox{tr}(\tilde{Q}_{c}^{\top}M)$
$\displaystyle=\sum_{i=1}^{n-1}k_{i}|{}^{I}\eta_{i}|^{2}\left[1-\mbox{tr}(\tilde{Q}_{c}^{\top}{{}^{I}\eta_{i}\over|{}^{I}\eta_{i}|}({{}^{I}\eta_{i}\over|{}^{I}\eta_{i}|})^{\top})\right].$

Noting that $\tilde{Q}_{c}\in SO(3)$ and
${{}^{I}\eta_{i}\over|{}^{I}\eta_{i}|}$ is a unit vector, we have
$\mbox{tr}(\tilde{Q}_{c}^{\top}{{}^{I}\eta_{i}\over|{}^{I}\eta_{i}|}({{}^{I}\eta_{i}\over|{}^{I}\eta_{i}|})^{\top})\leq
1$. As a result, we verify that $\tilde{V}(\tilde{Q}_{c},t)\geq 0$ .

The time derivative of $V(\tilde{Q}_{c},t)$ is given by

$\displaystyle\dot{V}$
$\displaystyle=-\mbox{tr}\left(\mbox{skew}(\tilde{Q}_{c}^{\top}M)^{\top}\tilde{Q}_{c}^{\top}M+\mathcal{E}^{\top}M\right)-|\varepsilon(t)|$
$\displaystyle=-\big{\|}\mbox{skew}\big{(}\tilde{Q}_{c}^{\top}M\big{)}\big{\|}^{2}-\mbox{tr}\big{(}\mathcal{E}^{\top}M\big{)}-|\varepsilon(t)|$
$\displaystyle\leq-\big{\|}\mbox{skew}\big{(}\tilde{Q}_{c}^{\top}M\big{)}\big{\|}^{2},$

in which we have used (42) in the last inequality. Then, it yields

$\int_{0}^{\infty}\big{\|}\mbox{skew}\big{(}\tilde{Q}_{c}^{\top}(s)M\big{)}\big{\|}^{2}ds<+\infty.$

Invoking the boundedness of the time derivative of
$\mbox{skew}\big{(}\tilde{Q}_{c}^{\top}M)$ and using Babalat’s lemma, we
conclude that all the trajectories converge to the invariant set

$\Omega_{e}:=\\{\tilde{Q}_{c}\in
SO(3)~{}|~{}\mbox{skew}\big{(}\tilde{Q}_{c}^{\top}M\big{)}=0\\}.$

Following the similar procedure of the proof in [18, Theorem 5.1], we can show
that the set $\Omega_{e}$ has a locally exponentially stable equilibrium
$\tilde{Q}_{c}=I_{3}$, and other three isolated unstable equilibria
$\mathbb{Q}_{i}$ ($i=1,2,3$) on $SO(3)$, and there are no poles on the
imaginary axis. By using the non-autonomous version of Hartman-Grobman theorem
[AW00], the observation error dynamics (41) is topologically equivalent to an
LTV dynamics in a small neighbourhood of these three unstable equilibria. As a
result, only some very specific trajectories, from a zero Lebesgue measure set
${\mathcal{M}}_{\epsilon}$, ultimately converge to the unstable equilibria
$\mathbb{Q}_{i}$ ($i=1,2,3$). Combining the local exponential stability of the
equilibrium $I_{3}$, thus it yields the almost global asymptotic stability of
the error dynamics (41) at $\tilde{Q}_{c}=I_{3}$. It is equivalent to show
that

$\forall\tilde{Q}_{c}(0)\in
SO(3)\backslash{\mathcal{M}}_{\epsilon},\quad\lim_{t\to+\infty}\|\hat{Q}_{c}(t)-Q_{c}\|=0,$
(43)

and $\hat{Q}_{c}(t)\in SO(3),\;\forall t\geq 0$. Namely, $\hat{R}$ provides an
asymptotically convergent estimate to $R$ as well.

The last part of the proof is to show the convergence of position estimate
$\hat{x}$. Now define its estimation error as $\tilde{x}:=\hat{x}-x,$ and it
yields

$\displaystyle\dot{\tilde{x}}$
$\displaystyle=\hat{R}\hat{v}-Rv+\sum_{i=1}^{n}\sigma_{i}({}^{I}z_{i}-\hat{x}-\hat{R}\hat{z}_{i})$
$\displaystyle=\sum_{i=1}^{n}\sigma_{i}({}^{I}z_{i}-\hat{x}-Rz_{i})+\epsilon_{t}$
$\displaystyle=\sum_{i=1}^{n}\sigma_{i}({}^{I}z_{i}-\hat{x}-RR^{\top}({}^{I}z_{i}-x))+\epsilon_{t},$

thus

$\dot{\tilde{x}}=-\sigma_{\Sigma}\cdot\tilde{x}+\epsilon_{t}$

with $\sigma_{\Sigma}:=\sum_{i=1}^{n}\sigma_{i}>0$ and an exponentially
decaying term $\epsilon_{t}$, where in the second equation we have used the
convergence (40) and (43), as well as the compactness of $SO(3)$ and the
boundedness of $v$. The dynamics of $\tilde{x}$ is a linear time-invariant
(LTI), stable system perturbed by $\epsilon_{t}$, thus also being globally
exponentially stable. It completes the proof. $\blacksquare$

##  Appendix C Simulation details

In this section, we provide additional details to the simulation results in
Section 6.

Position estimation with velocity measurement. We consider two trajectories:
the first trajectory being PE, i.e.,

$x_{1}=\begin{bmatrix}\cos({t\over 2})\\\ {1\over 4}\sin(t)\\\ -{\sqrt{3}\over
4}\sin(t)\end{bmatrix},\quad\Omega_{1}=\begin{bmatrix}\sin(0.1+\pi)\\\
0.5\sin(2t)\\\ 0.1\sin(0.3t+{\pi\over 3})\end{bmatrix}$

from the initial attitude $R(0)=I_{3}$; and the second being IE but not PE
with the same scenario as the first one but decaying after $4$s, i.e.,

$\begin{bmatrix}{}^{I}v_{2}\\\ \Omega_{2}\end{bmatrix}=\left\\{\begin{aligned}
&\begin{bmatrix}\dot{x}_{1}(t)\\\ \Omega_{1}(t)\end{bmatrix},&t\in[0,4]\\\
&\begin{bmatrix}e^{-5(t-4)}\dot{x}_{1}(t)\\\
e^{-5(t-4)}\Omega_{1}(t)\end{bmatrix},&t\geq 4.\end{aligned}\right.$

The parameters and initial conditions in the observers are selected as

$\alpha=1,~{}\gamma=50,~{}\hat{r}(0)=0,~{}\xi(0)=0,~{}\hat{\theta}(0)=0.$

For this case, We give some simulation results complementary to Fig. 2. In
Fig. 5 we present the estimation performance of the proposed gradient observer
(12) and the PEBO (14) under two robot trajectories. For the first trajectory,
both achieve satisfactory performance; but for the IE trajectory there is a
significant ultimate estimation error from the gradient observer. In the
feature depth observer [8], we set $k_{i}=1$ ($i=1,\ldots,3$) and
$\hat{x}(0)=[0.2~{}0.2~{}0.5]^{\top}$. In the I&I observer [14] we select the
initial guess as $0.5$ and the gain $\lambda=0.2$. Even though states in these
three observers live in different spaces, we tried to make the initial range
estimates as close as possible to make a fair comparison. In Figs. 2(b)-2(c),
we also compare to the LTV Kalman filter [17] and the Riccati observer [10].
The initial conditions of position estimates are both set as
$\hat{z}(0)=0_{3}$. See also the last figure in Fig. 6.

(a) Range estimate $\hat{r}$ (PE)

(b) Position estimate $\hat{z}$ (PE)

(c) Range estimate $\hat{r}$ (IE)

(d) Position estimate $\hat{z}$ (IE)

Figure 5: Performance of the proposed position observers under PE and IE
trajectories, i.e., the gradient observer (12) and the PEBO (14) for the case
with unbiased velocity information and noisy measurement

Position estimation with biased acceleration measurement. In the second group
of simulations, we consider position and velocity estimation with bearing and
biased linear acceleration measurable. The robot trajectory we use is given by
for $t\in[0,20]$s

${}^{I}a=\begin{bmatrix}-0.5\cos(0.5t)\\\ -0.5\sin(t)\\\ {\sqrt{3}\over
4}\sin(t)\end{bmatrix},\quad\Omega=\begin{bmatrix}0.2\sin(0.1t+\pi)\\\
0.1\sin(0.2t)\\\ 0.1\sin(0.3t+{\pi\over 3})\end{bmatrix}$

and $t\geq 20$s

${}^{I}a=0_{3},\quad\Omega=0_{3}.$

Such a trajectory only guarantees the regression being IE rather than PE. The
parameters and initial conditions for the observer in Section 4 are set as

$\alpha=2,~{}\gamma=100,~{}\rho=0.4,~{}k_{p}=500,~{}\hat{\theta}(0)=[0,\ldots,0,10]^{\top}.$

We consider a small sensor bias $b_{a}=[0.09,0.10,0.11]^{\top}$ and the
feature point is located at ${}^{I}z=[-2,1,3]^{\top}$. We show in the second
sub-figure of Fig. 6 the position $z$ and its estimate $\hat{z}$ as complement
to Fig. 3.

Vision-aided inertial navigation. Finally, we evaluate the performance of the
visual inertial navigation observer in Proposition 5.13. Three known landmarks
in the inertial frame are located at $[-2,1,3]^{\top},[-2,2,1]^{\top}$ and
$[1,1,1]^{\top}$. The same robot trajectory and parameters as in the last
subsection were adopted, except $\alpha=1$ and $k_{p}=10^{3}$. The behaviour
is shown in Fig. 4 and the last sub-figure of Fig. 6 in the absence of
measurement noise. Note that, for visualization, in Fig. 6 we draw in the
inertial frame $\\{\mathcal{I}\\}$ the attitude and its estimate by means of
the unit vector along $x$-axis of the body-fixed frame $\\{\mathcal{B}\\}$.

(a) Position estimate $\hat{z}$ from PEBO, I&I observer and feature depth
observer (complementary to Fig. 2(a))

(b) Position estimate $\hat{z}$ (complementary to Fig. 3(a))

(c) Attitude estimate $\hat{R}$ (complementary to Fig. 4(b))

Figure 6: Some complementary figures: Estimated trajectories and their true
values

## References

  * [Kha02] H. K. Khalil. Nonlinear Systems, Third edition, 2002.
  * [AW00] B. Aulbach and T. Wanner. The Hartman-Grobman theorem for carathéeodory-type differential equations in Banach spaces. Nonlinear Anal. Theory Methods Appl., 40(1-8):91–104, 2000.


3cm3cm3cm3cm

# A Finiteness Theorem in Galois Cohomology

Dylon Chow

###### Abstract.

We prove the finiteness of the kernel of the localization map in the Galois
cohomology of a connected reductive group over a global field.

##  1\. Introduction

The purpose of this note is to prove:

######  Theorem 1.

(Borel-Serre, Borel-Prasad, Nisnevich). Let $G$ be a connected reductive group
defined over a global field $F$. Let $V$ denote the set of places of $F$. The
fibers of the canonical map

$\lambda\colon H^{1}(F,G)\to\prod_{v\in V}H^{1}(F_{v},G)$

are finite.

In more geometric language, this says that given a principal homogeneous space
$X$ over $F$ for $G$, the principal homogeneous spaces over $F$ for $G$ which
are isomorphic to $X$ over $F_{v}$ for all $v$ form a finite number of
isomorphism classes over $F$. In the case when $F$ is a number field, this is
[4, Theorem 7.1]. By a twisting argument, it suffices to prove that the kernel
of $\lambda$ is finite. A proof that the kernel of $\lambda$ is finite is
given in [1, Theorem 6.8]. As Platonov and Rapinchuk point out (see p.316 of
[11]), there is a fairly standard proof in the case when $G$ is a torus, but
the proof of Borel and Serre in the general case is very different and appeals
to the reduction theory of adele groups.

On p.176 of [3], Borel and Prasad gave a proof in the function field case for
any connected reductive group $G$. They first prove the theorem for semisimple
groups in [2, Theorem B.1]. For arbitrary reductive groups, they use the
finiteness of the class number of a semisimple group [2, Proposition 3.9], the
proof of which uses lower bounds on covolumes of arithmetic subgroups.

In this note, we give a totally different proof of the theorem. We take as
given the case when $G$ is a torus. We prove the theorem in two steps. The
first step is the case when the derived group of $G$ is simply connected. Then
we use a $z$-extension to prove the theorem in general.

## Acknowledgements

We would like to thank Mikhail Borovoi, Gopal Prasad and Ramin Takloo-Bighash
for helpful comments.

##  2\. Proof of the Theorem

First we record the statement of Theorem 1 for tori:

######  Proposition 2.

Theorem 1 holds when $G$ is a torus.

###### Proof.

This follows from the global Nakayama-Tate duality theorem. See [11], Theorem
6.3 and the discussion preceding it. This reference only states Nakayama-Tate
duality for number fields, but the same proof works for function fields. See
[13] for a proof of Nakayama-Tate duality. ∎

In the statement below, a local field is a finite extension of either
${\mathbb{R}},{\mathbb{Q}}_{p}$, or $\mathbb{F}_{p}((t))$ for a prime $p$.

######  Proposition 3.

  1. (1)

If $H$ is a connected reductive group over a local field $k$, then
$H^{1}(k,H)$ is finite.

  2. (2)

If $H$ is a semisimple simply connected group over a nonarchimedean local
field $k$, then $H^{1}(k,H)=1$.

  3. (3)

Let $H$ be a semisimple, simply connected algebraic group over a global field
$k$. The canonical map

$H^{1}(k,H)\to\prod_{v}H^{1}(k_{v},H)$

is injective.

###### Proof.

  1. (1)

In characteristic $0$, this is proven in [4]. For a textbook reference, see
[11, Theorem 6.14]. In characteristic $p>0$, see [12, III.4.3, Remark 2].

  2. (2)

This is due to Bruhat-Tits. See [5, Thm. in section 4.7].

  3. (3)

In the number field case, this was proved in [7] except for the case when $G$
has a factor of type $E_{8}$, which was proved in [6]. The function field case
is proved in [8] \- in fact, in this case, $H^{1}(k,H)=0$.

∎

Now we proceed to the proof of Theorem 1. The first step is to handle the case
when the derived group of $G$ is simply connected.

######  Proposition 4.

If the derived group of $G$ is simply connected, then Theorem 1 holds.

###### Proof.

We consider the short exact sequence

$1\to G_{\text{der}}\to G\to D\to 1,$

where $D$ denotes the quotient of $G$ by its derived group $G_{\text{der}}$.
This gives us a commutative diagram with exact rows

${H^{1}(F,G_{\text{der}})}$${H^{1}(F,G)}$${H^{1}(F,D)}$${\prod_{v}H^{1}(F_{v},G_{\text{der}})}$${\prod_{v}H^{1}(F_{v},G)}$${\prod_{v}H^{1}(F_{v},D).}$$\scriptstyle{\rho}$$\scriptstyle{\alpha}$$\scriptstyle{\sigma}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\sigma^{\prime}}$

We want to show that $\text{ker}(\beta)$ is finite. We have

$\text{ker}(\beta)\subset\sigma^{-1}(\text{ker}(\gamma)),$

and by Proposition 2, $\text{ker}(\gamma)$ is finite. Proposition 3 tells us
that $H^{1}(F_{v},G_{\text{der}})$ is trivial for all finite places $v$, that
$H^{1}(F_{v},G_{\text{der}})$ is finite for all infinite places $v$, and that
$\alpha$ is injective. Together, these statements imply that
$H^{1}(F,G_{\text{der}})$ is finite. Thus $\text{ker}(\sigma)=\text{im}(\rho)$
is also finite. Now we use a standard twisting argument (see section I.5.4 of
[12]). Let $\zeta\in\text{ker}(\gamma)$, and choose a 1-cocycle $z$ that
represents a class in $\sigma^{-1}(\zeta)$. By twisting the sequence above by
$z$, we obtain a short exact sequence

$1\to(G_{z})_{\text{der}}\to G_{z}\to D\to 1.$

This induces a map $H^{1}(F,G_{z})\to H^{1}(F,D)$ whose kernel is
$\sigma^{-1}(\zeta)$ (after identifying $H^{1}(F,G)$ with $H^{1}(F,G_{z})$).
Therefore, for every element $\zeta\in\text{ker}(\gamma)$,
$\sigma^{-1}(\zeta)$ is finite. and so $\text{ker}(\beta)$ is finite.

∎

We have proved Theorem 1 when $G_{\text{der}}$ is simply connected. Now we use
$z$-extensions to prove it in general. We choose a finite Galois extension
$E/F$ splitting $G$ and an extension

$1\to Z\to G^{\prime}\to G\to 1$

such that $Z$ is an induced torus (i.e., it is obtained by Weil restriction of
scalars from a split $E$-torus) and the derived group of $G^{\prime}$ is
simply connected. Such an extension exists - see, for instance, the discussion
in the proof of Lemma A.1 of [9]. This gives a commutative diagram with exact
rows

${0}$${H^{1}(F,G^{\prime})}$${H^{1}(F,G)}$${H^{2}(F,Z)}$${0}$${\prod_{v}H^{1}(F_{v},G^{\prime})}$${\prod_{v}H^{1}(F_{v},G)}$${\prod_{v}H^{2}(F_{v},Z).}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\beta}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\gamma_{3}}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$

The $0$ on the leftmost term in the top row comes from the fact that the
$F$-torus $Z$ is induced. Since
$Z\times_{\text{Spec}(F)}\text{Spec}(F^{\prime})$ is also induced over
$F^{\prime}$ for all fields $F^{\prime}\supset F$, the leftmost term in the
bottom row is also $0$. We will use the following lemma:

######  Lemma 5.

The homomorphism $H^{2}(F,Z)\to\prod_{v}H^{2}(F_{v},Z)$ is injective.

###### Proof.

The torus $Z$ is the Weil restriction of some split torus over $E$, say
$Z=R_{E/F}(S)$ where $S\cong\mathbb{G}_{m}^{r}$. By Shapiro’s lemma we have

$H^{2}(F,Z)\cong H^{2}(E,(\mathbb{G}_{m,E})^{r}).$

Let $v$ be a place of $F$. Then we have (see [10], p.58)

$\displaystyle(R_{E/F}(S))_{F_{v}}$ $\displaystyle\cong
R_{E\otimes_{F}F_{v}}(S_{E\otimes_{F}F_{v}})$
$\displaystyle\cong\left(\prod_{w\lvert
v}R_{E_{w}/F_{v}}(\mathbb{G}_{m})\right)^{r}.$

Therefore,

$\displaystyle H^{2}(F_{v},Z)$ $\displaystyle\cong\prod_{w\lvert
v}H^{2}(F_{v},R_{E_{w}/F_{v}}(\mathbb{G}_{m})^{r})$
$\displaystyle=\prod_{w\lvert v}H^{2}(E_{w},\mathbb{G}_{m})^{r}.$

Thus the map $H^{2}(F,Z)\to\prod_{v}H^{2}(F_{v},Z)$ becomes

$(\text{Br}(E))^{r}\to(\prod_{w\in V_{E}}\text{Br}(E_{w}))^{r},$

which is injective by global class field theory. ∎

Now we return to the proof of the theorem. We want to show that
$\text{ker}(\gamma_{2})$ is finite. Suppose that $x\in\text{ker}(\gamma_{2})$.
Then $\gamma_{3}(\beta(x))=0$. By Lemma 5, $\beta(x)=0$. Thus
$x\in\text{im}(\alpha)$, say $\alpha(w)=x$. But then
$\alpha^{\prime}(\gamma_{1}(w))=0$, and so $\gamma_{1}(w)=0$. Therefore,
$\text{ker}(\gamma_{2})\subset\alpha(\text{ker}(\gamma_{1}))$. Since
$\text{ker}(\gamma_{1})$ is finite by Proposition 4, $\text{ker}(\gamma_{2})$
is also finite. This completes the proof of Theorem 1.

## References

  * [1] A. Borel.  Some finiteness properties of adele groups over number fields.  Publications Mathématiques de l’IHÉS, 16:5–30, 1963.
  * [2] A. Borel and G. Prasad.  Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups.  Publications Mathématiques de l’IHÉS, 69:119–171, 1989.
  * [3] A. Borel and G. Prasad.  Addendum : Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups.  Publications Mathématiques de l’IHÉS, 71:173–177, 1990.
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