# Generalized Hausdorff metric on $\displaystyle S_{b}$-metric space and some
fixed point results

Jayanta Sarkar  Department of Mathematical Sciences
Indian Institute of Technology (BHU)
Varanasi, India 221005<EMAIL_ADDRESS>,  Megha Pandey
Department of Mathematical Sciences
Indian Institute of Technology (BHU)
Varanasi, India 221005<EMAIL_ADDRESS>,  Tanmoy Som  Department
of Mathematical Sciences
Indian Institute of Technology (BHU)
Varanasi, India 221005<EMAIL_ADDRESS>and  B.S.Choudhury  Department
of Mathematics
Indian Institute of Engineering Science and Technology, Shibpur Howrah,711103,
India<EMAIL_ADDRESS>
###### Abstract.

In this paper, a metric on $\displaystyle S_{b}$-metric space analogous to the
Hausdorff metric has been introduced and some basic properties are obtained on
multi-valued $\displaystyle S_{b}$-metric space. Further, the fundamental
multi-valued contraction of Nadler(1962) has been extended to the
$\displaystyle S_{b}$-metric space setting, and two results have been
established. The entire study is supported by suitable examples.

###### Key words and phrases:

Complete metric space, Hausdorff metric space, Contraction, $\displaystyle
S_{b}$-metric space, Fixed point.

###### 2010 Mathematics Subject Classification:

$\displaystyle 54H10$, $\displaystyle 54H25$, $\displaystyle 47H10$.

##  1\. Introduction

The conceptualization of spaces or in particular generalization of metric
spaces and studying properties on them has consistently been an engrossing
area of maths. Moreover, studying fixed point theory on such spaces has always
been an interesting area of work for mathematicians due to its applications
not only in other areas of mathematics but also in a few other disciplines.
For example, the concept of $\displaystyle b$-metric space came into the
picture by Bakhtin [2], on which few researchers have developed some fixed
point results, however Czerwick [5] developed the extensions of Banach
contraction principle in a profuse manner by taking disparate contractive
conditions (follow [1],[4],[7],[9],[13],[19],[20],[24], and [21]).

Various mathematicians have explored the $\displaystyle S$-metric space and
manifested numerous consequences allied to the endurance of fixed
point,{[10],[11],[15],[16],[18], and [22]}.

Persuaded by the job of Bakhtin in [2], Souayah and Mlaiki [23] have
generalized the theory of $\displaystyle b$-metric space, presently known as
$\displaystyle S_{b}$-metric space and evinced certain fixed point results
under distinguished categories of contractions in a complete $\displaystyle
S_{b}$-metric space. Mlaiki [12] had further generalized the theory of
$\displaystyle S_{b}$-metric space to extended $\displaystyle S_{b}$-metric
space.

Our motive in this paper is to generalize Hausdorff metric for $\displaystyle
S_{b}$-metric space and to acquire fixed point of multi-valued operator on
$\displaystyle S_{b}$-metric space. Fixed point of multi-valued mapping on
metric spaces has been generalized in several ways. An initial credentials in
this direction is because of Nadler [14] where Banach contraction mapping
principle [3] is elongated to the domain of set-valued setting. In this paper,
we have generalized the Nadler [14] fixed point result to the occasion of
multi-valued mapping on this space.

All over of this paper, we designate by $\displaystyle\mathbb{N}$ the set of
natural numbers and by $\displaystyle\mathbb{R}$ the collection of real
numbers.

######  Definition 1.1.

_( $\displaystyle S_{b}$-metric space)[23]_. Let $\displaystyle\mathcal{H}$ be
a non-empty set and $\displaystyle s\geq 1$ be any real number. Define a map
$\displaystyle S_{b}:\mathcal{H}^{3}\rightarrow[0,\infty)$ such that for all
$\displaystyle u,w,v,t\in\mathcal{H},$ satisfies the properties

  1. i)

$\displaystyle S_{b}(u,w,v)=0$ if and only if $\displaystyle u=w=v$,

  2. ii)

$\displaystyle S_{b}(u,w,v)\leq s[S_{b}(u,u,t)+S_{b}(w,w,t)+S_{b}(v,v,t)].$

Then, the trio, $\displaystyle(\mathcal{H},S_{b},s)$, is known as
$\displaystyle S_{b}$-metric space.

Examples :

  1. (i)

Let $\displaystyle\mathcal{H}=\mathbb{R}$ and $\displaystyle
S_{b}(u,w,v)=s(\lvert u-v\rvert+\lvert w-v\rvert)$. Then,
$\displaystyle(\mathcal{H},S_{b},s)$ will be a $\displaystyle S_{b}$-metric
space for $\displaystyle s\in\mathbb{R}$ with $\displaystyle s\geq 1$.

  2. (ii)

Let $\displaystyle\mathcal{H}=\mathbb{R}^{n}$ and $\displaystyle\lVert.\rVert$
a norm on $\displaystyle\mathcal{H}$, then $\displaystyle
S_{b}(u,w,v)=s(\lVert w+v-2u\rVert+\lVert w-v\rVert)$ will be a $\displaystyle
S_{b}$-metric on $\displaystyle\mathcal{H}$ for $\displaystyle s\in\mathbb{R}$
with $\displaystyle s\geq 1.$

  3. (iii)

Let $\displaystyle\mathcal{H}$ be a non-empty set, $\displaystyle d$ be the
usual metric on $\displaystyle\mathcal{H}$, then $\displaystyle
S_{b}(u,w,v)=s(d(u,v)+d(w,v))$ is a $\displaystyle S_{b}$-metric space on
$\displaystyle\mathcal{H}$ for $\displaystyle s\in\mathbb{R}$ with
$\displaystyle s\geq 1.$

######  Remark 1.2.

If $\displaystyle S_{b}(u,u,w)=S_{b}(w,w,u)$ for all $\displaystyle
u,w\in\mathcal{H},$ then $\displaystyle(\mathcal{H},S_{b},s)$ is known as
symmetric $\displaystyle S_{b}$-metric space.

Everywhere in this paper, $\displaystyle\mathcal{H}$ will be a symmetric
$\displaystyle S_{b}$-metric space.

######  Definition 1.3.

[6][17] Assume $\displaystyle(\mathcal{H},S_{b},s)$ be a $\displaystyle
S_{b}$-metric space, then for $\displaystyle h\in\mathcal{H}$ and
$\displaystyle\bar{q}>0$ we defined open ball $\displaystyle
B_{S_{b}}(h,\bar{q})$ and closed ball $\displaystyle B_{S_{b}}[h,\bar{q}]$
with center $\displaystyle h$ and radius $\displaystyle\bar{q}$ as below :

$\displaystyle
B_{S_{b}}(h,\bar{q})=\left\\{u\in\mathcal{H}~{}:~{}S_{b}(h,h,u)<\bar{q}\right\\}~{}\text{and}~{}B_{S_{b}}[h,\bar{q}]=\left\\{t\in\mathcal{H}~{}:~{}S_{b}(h,h,u)\leq\bar{q}\right\\}~{}\text{respectively.}$

######  Definition 1.4.

[23][17] Let $\displaystyle(\mathcal{H},S_{b},s)$ is a $\displaystyle
S_{b}$-metric space and $\displaystyle\\{h_{n}\\}$ is a sequence in
$\displaystyle\mathcal{H}$. Then,

  1. i)

$\displaystyle\\{h_{n}\\}$ is called convergent iff there exists
$\displaystyle h\in\mathcal{H}$ such that $\displaystyle
S_{b}(h_{n},h_{n},h)\to 0$ when $\displaystyle n\to\infty$. We put down
$\displaystyle\lim\limits_{n\to\infty}h_{n}=h.$

  2. ii)

$\displaystyle\\{h_{n}\\}$ is called Cauchy sequence iff $\displaystyle
S_{b}(h_{n},h_{n},h_{m})\to 0$ when $\displaystyle n,m\to\infty.$

  3. iii)

$\displaystyle(\mathcal{H},S_{b})$ is said to be complete $\displaystyle
S_{b}$-metric space if every Cauchy sequence $\displaystyle\\{h_{n}\\}$
converges to a point $\displaystyle h\in\mathcal{H}$.

######  Definition 1.5.

_(Diameter of subset of a $\displaystyle S_{b}$-metric space)[23]_. Diameter
of a subset, $\displaystyle K$, of $\displaystyle S_{b}$-metric space,
$\displaystyle\mathcal{H}$, is define as follows

$\displaystyle D_{m}(K)=\sup\\{S_{b}(h,u,v)~{}:~{}h,u,v\in K\\}.$

We shall require the following concepts:
When $\displaystyle D_{m}(K)<\infty$ for any subset $\displaystyle K$ of a
$\displaystyle S_{b}$-metric space, $\displaystyle\mathcal{H}$, then
$\displaystyle K$ is pronounced to be a bounded subset of
$\displaystyle\mathcal{H}$.
A point $\displaystyle x\in K\subseteq\mathcal{H}$ is said to be limit point
of $\displaystyle K$ if there exists a sequence,
$\displaystyle\\{x_{n}\\}_{n\in\mathbb{N}}$, such that
$\displaystyle\lim\limits_{n\to\infty}x_{n}=x$.
A closed subset of a $\displaystyle S_{b}$-metric space is defined as a set
which contains all its limit points.

Our paper is organized as follows. The next section 2 is devoted to the
development of the Hausdorff metric like theory in the case of $\displaystyle
S_{b}$-metric space. We have concluded our paper with section 3, there we have
proved two fixed point results, first one generalizes Nadler’s fixed point
theorem [14] and another one is a conceptualization of Theorem 2 of Kikkawa
and Suzuki [8] followed by references.

##  2\. Multi-valued $\displaystyle S_{b}$-metric spaces

Let $\displaystyle(\mathcal{H},S_{b},s)$ be a $\displaystyle S_{b}$-metric
space for $\displaystyle s\geq 1$ and
$\displaystyle\mathcal{N}(X):=\\{P~{}:~{}P~{}\text{is a non-empty subset
of}~{}\mathcal{H}\\},\ \\\ \mathcal{CB}(\mathcal{H}):=\\{K~{}:~{}K~{}\text{is
a non-empty bounded and closed subset of}~{}\mathcal{H}\\}.\ \\\ $ For
$\displaystyle P,R,Q\in\mathcal{CB}(\mathcal{H})$ define

$\displaystyle\displaystyle D_{b}(h,R)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\inf\\{S_{b}(h,h,r):~{}r\in R\\},$ (2.1)
$\displaystyle\displaystyle N(\epsilon,P,R)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{h\in\mathcal{H}:~{}D_{b}(h,P)\leq\epsilon~{}\text{and}~{}D_{b}(h,R)\leq\epsilon\\},$
(2.2) $\displaystyle\displaystyle\mathcal{M}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{\epsilon>0:~{}P\subset
N(\epsilon,R,Q),~{}R\subset N(\epsilon,P,Q),~{}\text{and}~{}Q\subset
N(\epsilon,P,R)\\},$ (2.3) $\displaystyle\displaystyle
H_{b}(P,Q,R)=\max\Bigg{\\{}\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\},\max\left\\{\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},P),\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},Q)\right\\},\ $
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},P),\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},R)\right\\}\Bigg{\\}}.$ (2.4)

######  Lemma 2.1.

Assume $\displaystyle(\mathcal{H},S_{b},s)$ is a $\displaystyle S_{b}$-metric
space, $\displaystyle
H_{b}:(\mathcal{CB}(\mathcal{H}))^{3}\rightarrow[0,\infty)$ be a function
defined in (2.4), then for some $\displaystyle k\in\mathbb{R}$ with
$\displaystyle k\geq 1$, $\displaystyle H_{b}$ be a $\displaystyle
S_{b}$-metric on $\displaystyle\mathcal{CB}(\mathcal{H})$, i.e.,
$\displaystyle(\mathcal{CB}(\mathcal{H}),H_{b},k)$ is a $\displaystyle
S_{b}$-metric space.

Proof : To evince that $\displaystyle H_{b}$ is a $\displaystyle S_{b}$-metric
space, for every $\displaystyle P,R,Q,M\in\mathcal{CB}(\mathcal{H})$, it needs
to satisfy the following two conditions :

  1. (a)

$\displaystyle H_{b}(P,R,Q)=0~{}\text{iff}~{}P=R=Q$,

  2. (b)

$\displaystyle H_{b}(P,R,Q)\leq
k\\{H_{b}(P,P,M)+H_{b}(R,R,M)+H_{b}(Q,Q,M)\\}$.

Proof of (a). Let $\displaystyle P=R=Q$, then $\displaystyle H_{b}(P,R,Q)=0$
because every $\displaystyle\tilde{p}\in P$ satisfies $\displaystyle
D_{b}(\tilde{p},R)=0~{}\text{and}~{}D_{b}(\tilde{p},Q)=0$. Conversely, suppose
$\displaystyle H_{b}(P,R,Q)=0$, then from (2.4) we have

$\displaystyle\displaystyle D_{b}(\tilde{p},R)=0$ and
$\displaystyle\displaystyle D_{b}(\tilde{p},Q)=0~{}\text{for
all}~{}\tilde{p}\in P,$ $\displaystyle\displaystyle\implies P\subseteq R$ and
$\displaystyle\displaystyle P\subseteq Q.$ (2.5)

By similar argument, we obtain that

$\displaystyle\displaystyle R\subseteq P$ and $\displaystyle\displaystyle
R\subseteq Q$ (2.6) $\displaystyle\displaystyle Q\subseteq P$ and
$\displaystyle\displaystyle Q\subseteq R.$ (2.7)

Hence, from (2.5),(2.6), and (2.7) we conclude that $\displaystyle P=R=Q.$
Proof of (b). Since $\displaystyle D_{b}(h,R)=\underset{r\in
R}{\inf}\\{S_{b}(h,h,r)\\}$ by (2.1) and $\displaystyle(\mathcal{H},S_{b},s)$
is a $\displaystyle S_{b}$-metric space. Therefore, for every $\displaystyle
a\in M$, $\displaystyle\tilde{r}\in R$ and $\displaystyle\tilde{p}\in P$, we
have

$\displaystyle\displaystyle S_{b}(\tilde{p},\tilde{p},\tilde{r})\leq
s\\{S_{b}(\tilde{p},\tilde{p},a)+S_{b}(\tilde{p},\tilde{p},a)+S_{b}(\tilde{r},\tilde{r},a)\\}$
(2.8) $\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle\underset{\tilde{r}\in
R}{\inf}S_{b}(\tilde{p},\tilde{p},\tilde{r})\leq
s\\{2S_{b}(\tilde{p},\tilde{p},\tilde{r})+\underset{\tilde{r}\in
R}{\inf}S_{b}(\tilde{r},\tilde{r},a)\\}$ $\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle D_{b}(\tilde{p},R)\leq
s\\{2S_{b}(\tilde{p},\tilde{p},\tilde{r})+2\underset{\tilde{r}\in
R}{\inf}S_{b}(\tilde{r},\tilde{r},a)\\}$ $\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle D_{b}(\tilde{p},R)\leq
2s\\{S_{b}(\tilde{p},\tilde{p},a)+D_{b}(\tilde{r},A)\\}$
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq
2s\\{S_{b}(\tilde{p},\tilde{p},a)+H_{b}(R,R,M)\\}~{}\text{by \eqref{2.1}}$
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq 2s\\{\underset{a\in
A}{\inf}S_{b}(\tilde{p},\tilde{p},a)+H_{b}(R,R,M)\\}$
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq 2s\\{D_{b}(\tilde{p},A)+H_{b}(R,R,M)\\}$
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq 2s\\{H_{b}(P,P,M)+H_{b}(R,R,M)\\}~{}\text{by
\eqref{2.1}}$ $\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R)\leq 2s\\{H_{b}(P,P,M)+H_{b}(R,R,M)\\}~{}\text{since
$\displaystyle\tilde{p}$ is arbitrary.}$

By applying an analogous argument, we can prove that

$\underset{\tilde{p}\in P}{\sup}D_{b}(\tilde{p},Q)\leq
2s\\{H_{b}(P,P,M)+H_{b}(Q,Q,M)\\}.$ (2.9)

By using (2.8) and (2.9), we obtain

$\displaystyle\displaystyle\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\max\\{2s\\{H_{b}(P,P,M)+H_{b}(R,R,M)\\},$
$\displaystyle\displaystyle\hskip
28.45274pt2s\\{H_{b}(P,P,M)+H_{b}(Q,Q,M)\\}\\}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2s\\{H_{b}(P,P,M)+H_{b}(R,R,M)\\}$ $\displaystyle\displaystyle\hskip
28.45274pt+2s\\{H_{b}(P,P,M)+H_{b}(Q,Q,M)\\}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle 4s\\{H_{b}(P,P,M)+H_{b}(R,R,M)+H_{b}(Q,Q,M)\\},$
$\displaystyle\displaystyle i.e.,~{}\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle 4s\\{H_{b}(P,P,M)+H_{b}(R,R,M)$ (2.10)
$\displaystyle\displaystyle\hskip 113.81102pt+H_{b}(Q,Q,M)\\}.$

Similarly, we can show that

$\displaystyle\displaystyle\max\left\\{\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},P),\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},Q)\right\\}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle 4s\\{H_{b}(P,P,M)+H_{b}(R,R,M)$ (2.11)
$\displaystyle\displaystyle\hskip 113.81102pt+H_{b}(Q,Q,M)\\}~{}$
$\displaystyle\displaystyle\text{and},~{}\max\left\\{\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},P),\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},R)\right\\}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle 4s\\{H_{b}(P,P,M)+H_{b}(R,R,M)$ (2.12)
$\displaystyle\displaystyle\hskip 113.81102pt+H_{b}(Q,Q,M)\\}.~{}$

Thus, the inequalities (2.4), (2.10), (2.11), and (2.12) completes the proof
of (b).

######  Remark 2.2.

If $\displaystyle(\mathcal{H},S_{b},s)$ is a $\displaystyle S_{b}$-metric
space, then $\displaystyle(\mathcal{CB}(\mathcal{H}),H_{b},k)$ will be a
$\displaystyle S_{b}$-metric with $\displaystyle k=4s$.

######  Lemma 2.3.

Consider $\displaystyle(\mathcal{H},S_{b},s)$ be a $\displaystyle
S_{b}$-metric space, then for every $\displaystyle
P,R,Q\in\mathcal{CB}(\mathcal{H})$, $\displaystyle H_{b}(P,R,Q)$ defined by
(2.4), is equivalent to $\displaystyle
H_{b}(P,R,Q)=\underset{\epsilon>0}{\inf}\mathcal{M}$.

Proof : First assume that $\displaystyle H_{b}(P,R,Q)$ is defined by (2.4),
then for all $\displaystyle\tilde{p}\in P$, we have

$\displaystyle\displaystyle D_{b}(\tilde{p},R)\leq
H_{b}(P,R,Q)~{}\text{and}~{}D_{b}(\tilde{p},R)\leq H_{b}(P,R,Q)$ (2.13)
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle P\subset
N(H_{b}(P,R,Q),R,Q).$

Similarly, we have

$R\subset N(H_{b}(P,R,Q),P,Q)~{}\text{and}~{}Q\subset N(H_{b}(P,R,Q),P,R).$
(2.14)

Thus, from (2.13) and (2.14), we get $\displaystyle
H_{b}(P,R,Q)\in\mathcal{M}$.
Now for any $\displaystyle\epsilon\in\mathcal{M}$, we have

$\displaystyle\displaystyle A\subset N(\epsilon,R,Q)$ (2.15)
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq\epsilon~{}\text{and}~{}D_{b}(\tilde{p},Q)\leq\epsilon~{}\text{for
all}~{}\tilde{p}\in P$ $\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R)\leq\epsilon~{}~{}\text{and}~{}~{}\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\leq\epsilon$ $\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),~{}\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\}\leq\epsilon.$

Also by definition of $\displaystyle\mathcal{M}$ in (2.3), we have
$\displaystyle R\subset N(\epsilon,P,Q)$ and $\displaystyle Q\subset
N(\epsilon,P,R)$ this implies

$\displaystyle\displaystyle\max\left\\{\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},P),~{}\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},Q)\right\\}\leq\epsilon~{}\text{and}$ (2.16)
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},P),~{}\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},R)\right\\}\leq\epsilon.$ (2.17)

Thus, by using the (2.15), (2.16), (2.17) and $\displaystyle
H_{b}(P,R,Q)\in\mathcal{M}$, we get $\displaystyle
H_{b}(P,R,Q)=\inf\mathcal{M}.$
Conversely, assume $\displaystyle H_{b}(P,R,Q)=\inf\mathcal{M}$, then by
(2.3), we have

$\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq\epsilon~{}\text{and}~{}D_{b}(\tilde{p},Q)\leq\epsilon~{}\text{for
each}~{}\tilde{p}\in P~{}\text{and for all}~{}\epsilon\in\mathcal{M}$ (2.18)
$\displaystyle\displaystyle\implies$ $\displaystyle\displaystyle
D_{b}(\tilde{p},R)\leq\inf\mathcal{M}=H_{b}(P,R,Q)~{}\text{and}~{}D_{b}(p,Q)\leq\inf\mathcal{M}$
$\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R)\leq
H_{b}(P,R,Q)~{}~{}\text{and}~{}~{}\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\leq H_{b}(P,R,Q)$
$\displaystyle\displaystyle\implies$
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),~{}\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\}\leq H_{b}(P,R,Q).$

Similarly, we can show

$\displaystyle\displaystyle\max\left\\{\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},P),~{}\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},Q)\right\\}\leq H_{b}(P,R,Q)~{}\text{and}$ (2.19)
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},P),~{}\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},R)\right\\}\leq H_{b}(P,R,Q).$ (2.20)

Then, inequalities (2.18), (2.19) and (2.20) together, implies that

$\displaystyle\displaystyle\max\Bigg{\\{}\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\},\max\left\\{\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},P),\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},Q)\right\\},\ $
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},P),\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},R)\right\\}\Bigg{\\}}\leq H_{b}(P,R,Q).$ (2.21)

By (2.3), it follows that

$\displaystyle\displaystyle\max\Bigg{\\{}\max\left\\{\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},R),\underset{\tilde{p}\in
P}{\sup}D_{b}(\tilde{p},Q)\right\\},\max\left\\{\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},P),\underset{\tilde{r}\in
R}{\sup}D_{b}(\tilde{r},Q)\right\\},\ $
$\displaystyle\displaystyle\max\left\\{\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},P),\underset{\tilde{q}\in
Q}{\sup}D_{b}(\tilde{q},R)\right\\}\Bigg{\\}}\in\mathcal{M}.$ (2.22)

Hence, (2.21) and (2.22) together accomplishes the proof of Lemma 2.3.

##  3\. Main Results

######  Theorem 3.1.

_(Generalization of Nadler’s fixed point theorem on $\displaystyle
S_{b}$-metric space)_.
Assume $\displaystyle(\mathcal{H},S_{b},s)$ be a complete metric space and
$\displaystyle\mathcal{CB}(\mathcal{H})$ be the collection of closed and
bounded subsets of $\displaystyle\mathcal{H}$ with metric $\displaystyle
H_{b}$ and let
$\displaystyle\mathfrak{T}:\mathcal{H}\rightarrow\mathcal{CB}(\mathcal{H})$ be
a continuous multi-valued map satisfying the condition,

$H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}g)\leq\alpha
S_{b}(h,h,g)~{}\text{for
all}~{}h,g\in\mathcal{H}~{}\text{and}~{}\alpha\in\left[0,\frac{1}{s}\right).$
(3.1)

Then, $\displaystyle\mathfrak{T}$ has a fixed point in
$\displaystyle\mathcal{H}.$

Proof: Consider $\displaystyle h_{0}\in\mathcal{H}$ and choose $\displaystyle
h_{1}\in\mathfrak{T}h_{0}$. Since
$\displaystyle\mathfrak{T}h_{0},~{}\mathfrak{T}h_{1}\in\mathcal{CB}(\mathcal{H})$,
therefore by Lemma 2.3 there exists $\displaystyle h_{2}\in\mathfrak{T}h_{1}$
such that

$\displaystyle\displaystyle S_{b}(h_{1},h_{1},h_{2})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
H_{b}(\mathfrak{T}h_{0},\mathfrak{T}h_{0},\mathfrak{T}h_{1})+\alpha,$
$\displaystyle\displaystyle\implies S_{b}(h_{1},h_{1},h_{2})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle\alpha
S_{b}(h_{0},h_{0},h_{1})+\alpha~{}\text{by \eqref{4.1}}.$ (3.2)
$\displaystyle\displaystyle\text{In the same way, there
exists}~{}h_{3}\in\mathfrak{T}h_{2},~{}\text{such that}$
$\displaystyle\displaystyle S_{b}(h_{2},h_{2},h_{3})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
H_{b}(\mathfrak{T}h_{1},\mathfrak{T}h_{1},\mathfrak{T}h_{2})+\alpha^{2}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle\alpha
S_{b}(h_{1},h_{1},h_{2})+\alpha^{2}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\alpha[\alpha
S_{b}(h_{0},h_{0},h_{1})+\alpha]+\alpha^{2}~{}\text{by \eqref{4.2}}$
$\displaystyle\displaystyle\implies S_{b}(h_{2},h_{2},h_{3})$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\alpha^{2}S_{b}(h_{0},h_{0},h_{1})+2\alpha^{2}.$

Ongoing in this fashion, we get a sequence
$\displaystyle\\{h_{i}\\}_{i\in\mathbb{N}}\subset\mathcal{H}$ such that
$\displaystyle h_{i+1}\in\mathfrak{T}h_{i}$ and
$\displaystyle S_{b}(h_{i},h_{i},h_{i+1})\leq
H_{b}(\mathfrak{T}h_{i-1},\mathfrak{T}h_{i-1},\mathfrak{T}h_{i})+\alpha^{i}~{}\text{for
all}~{}i\geq 1$.
Since

$\displaystyle\displaystyle S_{b}(h_{i},h_{i},h_{i+1})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
H_{b}(\mathfrak{T}h_{i-1},\mathfrak{T}h_{i-1},\mathfrak{T}h_{i})+\alpha^{i}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle\alpha
S_{b}(h_{i-1},h_{i-1},h_{i})+\alpha^{i}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\alpha[H_{b}(\mathfrak{T}h_{i-2},\mathfrak{T}h_{i-2},\mathfrak{T}h_{i-1})+\alpha^{i-1}]+\alpha^{i}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\alpha
H_{b}(\mathfrak{T}h_{i-2},\mathfrak{T}h_{i-2},\mathfrak{T}h_{i-1})+2\alpha^{i}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\alpha^{2}S_{b}(h_{i-2},h_{i-2},h_{i-1})+2\alpha^{i}$
$\displaystyle\displaystyle\vdots$ $\displaystyle\displaystyle\implies
S_{b}(h_{i},h_{i},h_{i+1})$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\alpha^{i}S_{b}(h_{0},h_{0},h_{1})+i\alpha^{i}~{}\text{for
all}~{}i\in\mathbb{N}.$ (3.3)

Therefore, for $\displaystyle m\geq n$, we deduce that

$\displaystyle\displaystyle S_{b}(h_{n},h_{n},h_{m})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
s\left[2S_{b}(h_{n},h_{n},h_{n+1})+S_{b}(h_{n+1},h_{n+1},h_{m})\right]~{}\text{
(by Remark \ref{r1})}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle
2sS_{b}(h_{n},h_{n},h_{n+1})+s\left[s\left[2S_{b}(h_{n+1},h_{n+1},h_{n+2})+S_{b}(h_{n+2}+h_{n+2}+h_{m})\right]\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2sS_{b}(h_{n},h_{n},h_{n+1})+2s^{2}S_{b}(h_{n+1},h_{n+1},h_{n+2})+s^{2}S_{b}(h_{n+2},h_{n+2},h_{m})$
$\displaystyle\displaystyle\vdots$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle
2sS_{b}(h_{n},h_{n},h_{n+1})+\cdots+2s^{m-n}S_{b}(h_{m-1},h_{m-1},h_{m})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2\Big{[}s\left\\{\alpha^{n}S_{b}(h_{0},h_{0},h_{1})+n\alpha^{n}\right\\}+s^{2}\\{\alpha^{n+1}S_{b}(h_{0},h_{0},h_{1})+(n+1)\alpha^{n+1}\\}$
$\displaystyle\displaystyle\hskip
85.35826pt+\cdots+s^{m-n}\left\\{\alpha^{m-1}S_{b}(h_{0},h_{0},h_{1})+(m-1)\alpha^{m-1}\right\\}\Big{]}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\Big{[}\\{s\alpha^{n}+\cdots+s^{m-n}\alpha^{m-1}\\}S_{b}(h_{0},h_{0},h_{1})+\\{sn\alpha^{n}+s^{2}(n+1)\alpha^{n+1}$
$\displaystyle\displaystyle\hskip
199.16928pt+\cdots+s^{m-n}(m-1)\alpha^{m-1}\\}\Big{]}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2(s\alpha^{n})(1-(s\alpha)^{m-n-1})}{1-s\alpha}S_{b}(h_{0},h_{0},h_{1})+s\alpha^{n}\\{n+(n+1)s\alpha$
$\displaystyle\displaystyle\hskip
156.49014pt+\cdots+(n+(m-n-1))(s\alpha)^{m-n-1}\\}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2s\alpha^{n}(1-(s\alpha)^{m-n-1})}{1-s\alpha}S_{b}(h_{0},h_{0},h_{1})+s\alpha^{n}\underset{i=1}{\sum^{m-n-1}}(n+i)(s\alpha)^{i}.$

This implies that

$S_{b}(h_{n},h_{n},h_{m})\leq\frac{2s\alpha^{n}(1-(s\alpha)^{m-n-1})}{1-s\alpha}S_{b}(h_{0},h_{0},h_{1})+s\alpha^{n}\underset{i=1}{\sum^{m-n-1}}(n+i)(s\alpha)^{i}.$
(3.4)

From the inequality (3.4), it follows that $\displaystyle\\{h_{i}\\}$ is a
Cauchy sequence. As $\displaystyle(\mathcal{H},S_{b},s)$ is complete,
$\displaystyle\\{h_{i}\\}$ converges to a point $\displaystyle
h\in\mathcal{H}$. Therefore, the sequence
$\displaystyle\\{\mathfrak{T}h_{i}\\}$ converges to
$\displaystyle\mathfrak{T}h$ and, since $\displaystyle
h_{i}\in\mathfrak{T}h_{i-1}$ for all $\displaystyle i$, it follows that
$\displaystyle h\in\mathfrak{T}h$. This establishes the theorem.

######  Example 3.2.

Let $\displaystyle\mathcal{H}=\mathbb{R}$ with $\displaystyle
S_{b}(h,u,w)=s\\{\lvert h-w\rvert+\lvert u-w\rvert\\}$ for some $\displaystyle
s\in\mathbb{R}$ with $\displaystyle s\geq 1$ and
$\displaystyle\mathfrak{T}:\mathcal{H}\rightarrow\mathcal{CB}(\mathcal{H})$
such that

$\displaystyle\mathfrak{T}h=\begin{cases}\left[\frac{h}{24s},~{}\frac{3h}{24s}+1\right],&h\geq
0\\\ \\{0\\},&h<0.\end{cases}$

Observe that

$\displaystyle H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}u)\leq\\\
\frac{1}{4s}S_{b}(h,h,u)~{}~{}\text{for every}~{}h,u\in\mathcal{H}$

and $\displaystyle\mathfrak{T}$ is continuous. Hence,
$\displaystyle\mathfrak{T}$ pleases all the constrains of Theorem 3.1,
therefore $\displaystyle\mathfrak{T}$ has a fixed point in
$\displaystyle\mathcal{H}$. Actually the set of all fixed points of
$\displaystyle\mathfrak{T}$ is
$\displaystyle\left[0,\frac{24s}{24s-3}\right].$

######  Theorem 3.3.

Consider $\displaystyle(\mathcal{H},S_{b},s)$ be a complete $\displaystyle
S_{b}$-metric space. Define a strictly decreasing function
$\displaystyle\eta:[0,\frac{1}{s})\rightarrow(\frac{1}{2s+1},\frac{1}{2s}]$

$\displaystyle\eta(r)=\frac{1}{s(2+r)},$

and let
$\displaystyle\mathfrak{T}:\mathcal{H}\rightarrow\mathcal{CB}(\mathcal{H})$ be
a mapping. Assume that there exists some $\displaystyle r\in[0,1)$ such that

$\eta(r)D_{b}(h,\mathfrak{T}h)\leq
S_{b}(h,h,t)~{}\text{implies}~{}H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}t)\leq
rS_{b}(h,h,t)~{}\text{for all}~{}h,t\in\mathcal{H}.$ (3.5)

Then, there exists $\displaystyle z\in\mathcal{H}$ such that $\displaystyle
z\in\mathfrak{T}z.$

Proof : Take a real number $\displaystyle r_{1}$ with $\displaystyle 0\leq
r_{1}<r<1$. Then, for each $\displaystyle u=u_{0}\in\mathcal{H}$ and
$\displaystyle u_{1}\in\mathfrak{T}u$, we have

$\displaystyle\displaystyle\eta(r)D_{b}(u,\mathfrak{T}u)$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\eta(r)S_{b}(u,u,u_{1})\leq S_{b}(u,u,u_{1})$
$\displaystyle\displaystyle\implies D_{b}(u_{1},\mathfrak{T}u_{1})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
H_{b}(\mathfrak{T}u,\mathfrak{T}u,\mathfrak{T}u_{1})\leq rS_{b}(u,u,u_{1})\leq
r_{1}S_{b}(u,u,u_{1})~{}\text{by \eqref{te4.2}}.$

Therefore, there exists $\displaystyle u_{2}\in\mathfrak{T}u_{1}$ such that
$\displaystyle S_{b}(u_{1},u_{1},u_{2})\leq r_{1}S_{b}(u,u,u_{1})$. Thus, we
get a sequence $\displaystyle\\{u_{n}\\}$ such that $\displaystyle
u_{n}\in\mathfrak{T}u_{n-1}$ and

$S_{b}(u_{n-1},u_{n-1},u_{n})\leq
r_{1}S_{b}(u_{n-2},u_{n-2},u_{n-1})\leq\cdots\leq
r_{1}^{n-1}S_{b}(u,u,u_{1}).$ (3.6)

Therefore, for $\displaystyle m\geq n$, we have

$\displaystyle\displaystyle S_{b}(u_{n},u_{n},u_{m})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
s[2S_{b}(u_{n},u_{n},u_{n+1})+S_{b}(u_{n+1},u_{n+1},u_{m})]~{}\text{( by
Remark \ref{r1})}$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle
2sS_{b}(u_{n},u_{n},u_{n+1})+s\left[s\left[2S_{b}(u_{n+1},u_{n+1},u_{n+2})+S_{b}(u_{n+2}+u_{n+2}+u_{m})\right]\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2sS_{b}(u_{n},u_{n},u_{n+1})+2s^{2}S_{b}(u_{n+1},u_{n+1},u_{n+2})+s^{2}S_{b}(u_{n+2},u_{n+2},u_{m})$
$\displaystyle\displaystyle\vdots$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle
2sS_{b}(u_{n},u_{n},u_{n+1})+\cdots+2s^{m-n}S_{b}(u_{m-1},u_{m-1},u_{m})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2sr_{1}^{n}S_{b}(u,u,u_{1})+2s^{2}r_{1}^{n+1}S_{b}(u,u,u_{1})+\cdots+2s^{m-n}r_{1}^{m-1}S_{b}(u,u,u_{1})~{}\text{by
\eqref{4.5}}$ $\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2sr_{1}^{n}[1+sr_{1}+(sr_{1})^{2}+\cdots+(sr_{1})^{m-n-1}]S_{b}(u,u,u_{1})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2sr_{1}^{n}(1-(sr_{1})^{m-n})}{1-sr_{1}}S_{b}(u,u,u_{1}).$

This implies

$S_{b}(u_{n},u_{n},u_{m})\leq\frac{2sr_{1}^{n}(1-(sr_{1})^{m-n})}{1-sr_{1}}S_{b}(u,u,u_{1}).$
(3.7)

From (3.7), it follows that $\displaystyle\\{u_{n}\\}$ is a Cauchy sequence.
Since $\displaystyle\mathcal{H}$ is a complete $\displaystyle S_{b}$-metric
space, $\displaystyle\\{u_{n}\\}$ converges to some point $\displaystyle
z\in\mathcal{H}$.

We next show that

$\displaystyle D_{b}(z,\mathfrak{T}h)\leq rS_{b}(z,z,h)~{}\text{for
each}~{}h\in\mathcal{H}/\\{z\\}.$

Since $\displaystyle u_{n}\to z$, there exists $\displaystyle k\in\mathbb{N}$
such that

$S_{b}(u_{n},u_{n},z)\leq\frac{1}{2s(1+4s)}S_{b}(h,h,z)~{}\text{for
all}~{}n\in\mathbb{N}~{}\text{and}~{}n\geq k.$ (3.8)

Then, we have

$\displaystyle\displaystyle\eta(r)D_{b}(u_{n},\mathfrak{T}u_{n})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
D_{b}(u_{n},\mathfrak{T}u_{n})\leq S_{b}(u_{n},u_{n},u_{n+1})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
s[2S_{b}(u_{n},u_{n},z)+S_{b}(u_{n+1},u_{n+1},z)]$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2s[S_{b}(u_{n},u_{n},z)+S_{b}(u_{n+1},u_{n+1},z)]$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\frac{4s}{2s(1+4s)}S_{b}(h,h,z)~{}\text{by
\eqref{4.7}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2s}S_{b}(h,h,z)-\frac{1}{2s(1+4s)}S_{b}(h,h,z)$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\frac{1}{2s}S_{b}(h,h,z)-S_{b}(u_{n},u_{n},z)~{}\text{by
\eqref{4.7}}$ $\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
S_{b}(u_{n},u_{n},h)$
$\displaystyle\displaystyle\implies\eta(r)D_{b}(u_{n},\mathfrak{T}u_{n})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
S_{b}(u_{n},u_{n},h).$

Therefore, by assumption it follows that $\displaystyle
H_{b}(\mathfrak{T}u_{n},\mathfrak{T}u_{n},\mathfrak{T}h)\leq
rS_{b}(u_{n},u_{n},h)$. This implies that $\displaystyle
D_{b}(u_{n+1},\mathfrak{T}h)\leq rS_{b}(u_{n},u_{n},h)$ for $\displaystyle
n\in\mathbb{N}$ with $\displaystyle n\geq k$. Letting $\displaystyle
n\to\infty$, we obtain

$D_{b}(z,\mathfrak{T}h)\leq rS_{b}(z,z,h)~{}\text{for
every}~{}h\in\mathcal{H}/\\{z\\}.$ (3.9)

We next prove that

$\displaystyle H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}z)\leq
rS_{b}(h,h,z)~{}\text{for every}~{}h\in\mathcal{H}.$

If $\displaystyle h=z$, then it holds obviously. Therefore, consider
$\displaystyle h\neq z$. Then, for every $\displaystyle n\in\mathbb{N}$, there
exists $\displaystyle y_{n}\in\mathfrak{T}h$ such that

$S_{b}(z,z,y_{n})\leq D_{b}(z,\mathfrak{T}h)+\frac{1}{ns}S_{b}(h,h,z).$ (3.10)

We have

$\displaystyle\displaystyle D_{b}(x,\mathfrak{T}h)$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle S_{b}(h,h,y_{n})$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
s[2S_{b}(h,h,z)+S_{b}(z,z,y_{n})]$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle
2sS_{b}(h,h,z)+sD_{b}(z,\mathfrak{T}h)+\frac{1}{n}S_{b}(h,h,z)~{}\text{by
\eqref{4.9}}$ $\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2sS_{b}(h,h,z)+srS_{b}(h,h,z)+\frac{1}{n}S_{b}(h,h,z)~{}\text{by \eqref{4.8}}$
$\displaystyle\displaystyle\implies D_{b}(h,\mathfrak{T}h)$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\left(2s+sr+\frac{1}{n}\right)S_{b}(h,h,z)~{}\text{for
each}~{}n\in\mathbb{N}$

and hence $\displaystyle\frac{1}{s(2+r)}D_{b}(h,\mathfrak{T}h)\leq
S_{b}(h,h,z)$. From (3.5), we have $\displaystyle
H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}h)\leq rS_{b}(h,h,z).$

Since

$\displaystyle
D_{b}(z,\mathfrak{T}z)=\lim\limits_{n\to\infty}D_{b}(u_{n+1},\mathfrak{T}z)\leq\lim\limits_{n\to\infty}H_{b}(\mathfrak{T}u_{n},\mathfrak{T}u_{n},\mathfrak{T}z)\leq\lim\limits_{n\to\infty}rS_{b}(u_{n},u_{n},z)=0$

and $\displaystyle\mathfrak{T}z\in\mathcal{CB}(\mathcal{H})$, we obtain
$\displaystyle z\in\mathfrak{T}z.$ This completes the proof.

######  Example 3.4.

Let $\displaystyle\mathcal{H}=\mathbb{R}$ such that $\displaystyle
S_{b}(h,u,w)=s\\{\lvert h-w\rvert+\lvert u-w\rvert\\}$, for $\displaystyle
s\in\mathbb{R}$ and $\displaystyle s\geq 1$, be a $\displaystyle S_{b}$-metric
on $\displaystyle\mathcal{H}$. Consider
$\displaystyle\mathfrak{T}:\mathcal{H}\rightarrow\mathcal{CB}(\mathcal{H})$ be
a set-valued map defined on $\displaystyle\mathcal{H}$ such that

$\displaystyle\mathfrak{T}h=\begin{cases}\left[\frac{h-1}{48s}-1,~{}\frac{3(h-1)}{48s}+2\right],&h\geq
0\\\ \\{0\\},&h<0.\end{cases}$

Then, observe that for all $\displaystyle h,u\in\mathcal{H}$, $\displaystyle
H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}u)\leq\frac{1}{8s}S_{b}(h,h,u)$.
Hence, it satisfies the condition of Theorem 3.3. Therefore, $\displaystyle T$
has a fixed point. Actually the set of all fixed points of $\displaystyle T$
is $\displaystyle\left[0,\frac{96s+3}{48s-3}\right]$.

Remark: Notice that using Theorem 3.3 there exists $\displaystyle
r\in\left[0,\frac{1}{s}\right)$ such that,

$\displaystyle H_{b}(\mathfrak{T}h,\mathfrak{T}h,\mathfrak{T}t)\leq
rS_{b}(h,h,t)~{}\text{for all}~{}h,t\in\mathcal{H}$

which implies that $\displaystyle\mathfrak{T}$ has a fixed point in
$\displaystyle\mathcal{H}$. This shows Theorem 3.1 is a generalization of
Theorem 3.3.

##  4\. Conclusion

We have generalized the Hausdorff metric for $\displaystyle S_{b}$ metric
space to obtain a fixed point of a multi-valued operator. In this article, we
have used the generalization of the Banach contraction mapping principle into
the domain of set-valued setting. In section 2.2, we defined the Hausdorff
metric in $\displaystyle S_{b}$ metric space and gave properties of it. We
have proved two fixed point results in section 2.3. The first one is Theorem
3.1, a generalization of Nadler’s fixed point theorem. The second one is
Theorem 3.3, which is the generalization of the theorem in article [8].

In the future, we may try to generalize the well-known fixed point results for
the case of $\displaystyle S_{b}$ metric space. Further, one can try to see
construct the set of Iterated function systems satisfying the condition given
in Theorem 3.1 and work on the fractal generated by these Iterated Function
Systems.

## Acknowledgements

The first author’s work is financially supported by the CSIR, India, with
grant no: 09/1217(13093)/2022-EMR-I.

The work is supported by MHRD Fellowship to the 2nd author as TA-ship at the
Indian Institute of Technology (BHU), Varanasi.

## Conflict of Interest

The authors have no conflict of interest in this paper.

## References

  * [1] H. Aydi, M. Bota, E. Karapınar, and S. Mitrović.  A fixed point theorem for set-valued quasi-contractions in b-metric spaces.  Fixed Point Theory and Applications, 2012(1):1–8, 2012.
  * [2] I. Bakhtin.  The contraction mapping in almost metric spaces, funct.  Funct. Ana. Gos. Ped. Inst. Unianowsk, 30:26–37, 1989.
  * [3] S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.  Fund. math, 3(1):133–181, 1922.
  * [4] M. Bota, A. Molnar, and C. Varga.  On Ekeland’s variational principle in $\displaystyle b$-metric spaces.  Fixed Point Theory, 12(2):21–28, 2011.
  * [5] S. Czerwik.  Contraction mappings in $\displaystyle b$-metric spaces.  Acta mathematica et informatica universitatis ostraviensis, 1(1):5–11, 1993.
  * [6] N. Dedovic, G. Kishore, D. Prasad, and J. Vujakovic.  Suzuki type fixed point results and applications in partially ordered sb-metric spaces.  Novi Sad J. Math, 49(2):123–138, 2019.
  * [7] A. Dubey, R. Shukla, and R. Dubey.  Some fixed point results in $\displaystyle b$-metric spaces.  Asian journal of mathematics and applications, 2014, 2014.
  * [8] M. Kikkawa and T. Suzuki.  Three fixed point theorems for generalized contractions with constants in complete metric spaces.  Nonlinear Analysis: Theory, Methods & Applications, 69(9):2942–2949, 2008.
  * [9] M. Kir and H. Kiziltunc.  On some well known fixed point theorems in b-metric spaces.  Turkish journal of analysis and number theory, 1(1):13–16, 2013\.
  * [10] N. Mlaiki.  Common fixed points in complex S-metric space.  Adv. Fixed Point Theory, 4(4):509–524, 2014.
  * [11] N. Mlaiki.  $\displaystyle\psi$-contractive mapping on S-metric space.  Mathematical Sciences Letters, 4(1):9, 2015.
  * [12] N. Mlaiki.  Extended $\displaystyle{S}_{b}$-metric spaces.  J. Math. Anal, 9:124–135, 2018.
  * [13] A. Mukheimer.  $\displaystyle\alpha$-$\displaystyle\psi$-$\displaystyle\phi$-contractive mappings in ordered partial b-metric spaces.  J. Nonlinear Sci. Appl, 7:168–179, 2014.
  * [14] S. Nadler et al.  Multi-valued contraction mappings.  Pacific Journal of Mathematics, 30(2):475–488, 1969.
  * [15] K. Prudhvi.  Fixed point results in S-metric spaces, univer.  J. Comput. Math, 3:19–21, 2015.
  * [16] S. Sedghi and N. Shobe.  A common unique random fixed point theorems in S-metric spaces.  Journal of prime research in Mathematics, 7:25–34, 2011.
  * [17] S. Sedghi, N. Shobkolaei, M. Shahraki, and T. Došenović.  Common fixed point of four maps in s-metric spaces.  Mathematical Sciences, 12(2):137–143, 2018.
  * [18] S. Sedghi and N. Van Dung.  Fixed point theorems on S-metric spaces.  Matematički Vesnik, (255):113–124, 2014.
  * [19] W. Shatanawi.  Fixed point theory for contractive mappings satisfying-maps in-metric spaces.  Fixed point theory and Applications, 2010:1–9, 2010.
  * [20] W. Shatanawi and A. Pitea.  Some coupled fixed point theorems in quasi-partial metric spaces.  Fixed point theory and applications, 2013(1):1–15, 2013.
  * [21] S. Shukla.  Partial $\displaystyle b$-metric spaces and fixed point theorems.  Mediterranean Journal of Mathematics, 11(2):703–711, 2014.
  * [22] A. Singh and N. Hooda.  Coupled fixed point theorems in S-metric spaces.  International Journal of Mathematics and Statistics Invention, 2(4):33–39, 2014.
  * [23] N. Souayah and N. Mlaiki.  A fixed point theorem in $\displaystyle{S}_{b}$-metric spaces.  J. Math. Computer Sci, 16:131–139, 2016.
  * [24] C. Vetro, S. Chauhan, E. Karapınar, and W. Shatanawi.  Fixed points of weakly compatible mappings satisfying generalized $\displaystyle\varphi$-weak contractions.  Bulletin of the Malaysian Mathematical Sciences Society, 38(3):1085–1105, 2015.


# A Spatiotemporal Hand-Eye Calibration for Trajectory Alignment in
Visual(-Inertial) Odometry Evaluation

Zichao Shu1, Lijun Li1, Rui Wang2 and Zetao Chen1 Manuscript received:
December 12, 2023; Revised: March 7, 2024; Accepted: March 30, 2024. This
paper was recommended for publication by Editor Sven Behnke upon evaluation of
the Associate Editor and Reviewers’ comments. 1Zichao Shu, Lijun Li and Zetao
Chen are with the Advanced Display and Sensing Research Center, Yongjiang
Laboratory, China. {zichao-shu; lijun-li<EMAIL_ADDRESS>Wang is
with the Mixed Reality & AI Lab-Zürich, Microsoft, Switzerland.
<EMAIL_ADDRESS>Object Identifier (DOI): see top of this page.

###### Abstract

A common prerequisite for evaluating a visual(-inertial) odometry (VO/VIO)
algorithm is to align the timestamps and the reference frame of its estimated
trajectory with a reference ground-truth derived from a system of superior
precision, such as a motion capture system. The trajectory-based alignment,
typically modeled as a classic hand-eye calibration, significantly influences
the accuracy of evaluation metrics. However, traditional calibration methods
are susceptible to the quality of the input poses. Few studies have taken this
into account when evaluating VO/VIO trajectories that usually suffer from
noise and drift. To fill this gap, we propose a novel spatiotemporal hand-eye
calibration algorithm that fully leverages multiple constraints from screw
theory for enhanced accuracy and robustness. Experimental results show that
our algorithm has better performance and is less noise-prone than state-of-
the-art methods.

###### Index Terms:

Calibration and Identification, Performance Evaluation and Benchmarking,
Visual-Inertial SLAM.

##  I INTRODUCTION

Visual(-inertial) odometry (VO/VIO) is known to provide state estimation of
motion devices and has a wide range of application domains, such as robotics,
extended reality, and autonomous driving. The performance evaluation is a
fundamental task in VO/VIO research and application, where metrics are
typically quantified by evaluating the estimated trajectory from VO/VIO with
respect to the ground-truth. Commonly, the ground-truth trajectory can be
obtained by tracking the motion device simultaneously with a system of
superior precision, e.g., using a motion capture (MoCap) system, laser
tracker, etc [1, 2, 3]. There are two main problems when comparing the
estimated trajectory against the ground-truth: the trajectory pair is usually
on different clock domains (thus with non-corresponding timestamps) and
expressed in different global and local reference frames. While well-known
methods such as the Umeyama algorithm [4] can align the global frames, the
spatiotemporal alignment, which calculates the offsets of timestamps and local
frames of the trajectory pair, still needs meticulous handling.

The spatiotemporal alignment problem above can be modeled as a classic hand-
eye calibration problem: given the local frames of the ground-truth and
estimated trajectory as the hand and the eye respectively, calculate the
timestamp offset and estimate the homogeneous transformation between them.
While the essence of hand-eye calibration problem has been well addressed in
numerous studies [5, 6, 7, 8, 9, 10], the accuracy and robustness may still be
compromised in practical applications. The error introduced in this step will
affect the transformation of the ground-truth trajectory, thereby exerting a
substantial influence on the subsequent evaluation metrics.

###  I-A Motivation

In this work, we consider the scenario in which only the trajectory
information is available. This is common among commercial consumer devices,
such as extended reality headsets or home robots, where the original raw
sensor data used to derive the device trajectories are not accessible by
users. Existing hand-eye calibration algorithms can be categorized into two
distinct approaches: tightly-coupled and loosely-coupled. The former typically
joints raw data from the sensors such as images with information of the
calibration boards [6, 7, 8] or IMU measurements [1, 2], and optimizes the
result in a maximum likelihood estimation (MLE) framework. The latter, on the
other hand, directly calculates the offset between the hand and the eye based
on their independently estimated poses [9, 10]. While tightly-coupled
approaches can theoretically achieve higher accuracy and are used in well-
known benchmarks such as EuRoC [1] and TUM-VI [2], they are not applicable in
cases where only the trajectory information is available. Loosely-coupled
approaches can perform calibration in the pose-only condition, but due to the
ubiquitous noise and accumulated error in the VO/VIO estimation, the accuracy
and robustness of existing methods are generally insufficient.

###  I-B Contribution

In this letter, we propose a novel loosely-coupled spatiotemporal hand-eye
calibration method tailored for VO/VIO evaluation. This method demonstrates
robustness against noise and accumulated error in the input trajectories. For
time alignment, we improve the correlation analysis of the screw invariant and
obtain synchronized trajectories. For spatial calibration, we construct linear
equations using local relative poses based on rotational constraint to fully
utilize the motion information, rather than the naive global or inter-frame
strategies [9, 8, 10]. Additionally, we introduce a well-designed robust
kernel based on the screw theory to stabilize the linear solution. These
operations are iteratively completed within the random sample consensus
(RANSAC) framework to recover inlier data. Finally, we design a nonlinear
optimization tool to jointly refine the time offset and the linear extrinsic
solution. To validate the effectiveness of our algorithm, we conduct
experiments on public and simulated datasets, as well as our own datasets
collected by a virtual reality (VR) headset with VIO capability and a MoCap
system (see in Fig. 1).

Figure 1: Our spatiotemporal hand-eye calibration platform and the convention
of the reference frames. The global frame for MoCap trajectory is denoted as
G, and the local frame is referenced to a specific tracking marker indicated
as H. The global frame for VIO trajectory is denoted as W, and the local frame
E coincides with the IMU body frame. During this process, dashed lines
represent transformations that change over time, while solid lines indicate
static offsets.

The rest of this letter is organized as follows. Section II reviews related
work on spatiotemporal hand-eye calibration. The new method is described in
Section III and its performance is evaluated in Section IV. Section V
concludes the letter.

##  II RELATED WORK

Spatiotemporal hand-eye calibration based on different strategies is a widely
studied area. In our review, we briefly discuss the related work that shares
the same strategy as the proposed method, i.e., the loosely-coupled methods,
and motivate the design adapted in our work.

###  II-A Spatial Hand-Eye Calibration

In our application scenario, the loosely-coupled hand-eye calibration can be
formulated as $\mathbf{AX}$ = $\mathbf{XB}$ [11], where $\mathbf{A}$ and
$\mathbf{B}$ are the hand and the eye poses between two frames respectively,
and $\mathbf{X}$ is the unknown homogeneous transformation between the hand
and the eye. The solution for $\mathbf{X}$ can be categorized into two
approaches: either separately or simultaneously solving the rotation and
translation parts for the transformation.

One of the earliest separated approaches was presented by Shiu and Ahmad [11].
They represented rotation using angle-axis and proposed a closed-form solution
for the $\mathit{\mathbf{AX}}$ = $\mathit{\mathbf{XB}}$ formulation. Tsai and
Lenz [12] proposed a similar but simplified method to improve computational
efficiency, which has been widely used to this day (e.g. in OpenCV). Later
methods focused on utilizing various parameterizations of the rotation, such
as angle-axis [13], quaternion [14], Lie algebra [15], and Kronecker product
[16], to achieve more efficient solution. Although separated methods are
computationally efficient, they are error-prone due to the independence
assumption between the rotation and translation, which are actually
nonlinearly coupled [17].

In contrast, the simultaneous methods calibrate the rotation and translation
parts jointly. Representative methods include screw motion [18], Kronecker
product [19], dual quaternion [20] and dual tensor [21], which use alternative
analytical parameterizations to express the complete homogeneous
transformation. Additionally, there are also algorithms based on numerical
optimization [22, 23, 24]. While the computational cost may increase, they are
generally more accurate than the separated methods. In addition to improving
accuracy, recent researchers have also focused on enhancing the robustness of
algorithms across various applications [25, 10, 9, 26]. However, these efforts
were mostly focused on scenarios characterized by comparatively high-precision
pose data. This differs from our use case where the trajectories are
substantially affected by noise or accumulated error. In this paper, we
utilize a dual quaternion scheme similar to [20, 10] and construct a robust
linear solving system using multiple constraints from screw theory to address
the challenges from the VO/VIO trajectory.

###  II-B Temporal Alignment

Given that the hand and the eye sensors usually operate on different clocks,
the temporal correspondence between $\mathbf{A}$ and $\mathbf{B}$ is usually
unknown. The time alignment prior to spatial calibration is thus necessary.
Kelly and Sukhatme [27] considered the problem to be a registration task and
solved it by utilizing the iterative closest point algorithm. Based on the
discrete Fourier transform (DFT) theory, the time alignment of trajectories
can be converted to the correlation analysis between two invariant signals
derived from screw motion. This simple and effective method has been widely
used in data synchronization for hand-eye calibration [10, 28, 29, 30].
However, the precision of this method is limited by the temporal resolution of
the correlation function, with low-frequency data resulting in reduced time
alignment accuracy. Alternatively, the TUM-VI benchmark [2] achieves time
alignment by utilizing information from an error function, which is calculated
through a grid search between the motion invariants. Additionally, a parabolic
fitting is applied to the error function to enhance the precision of the time
alignment. Unlike the grid search method in the time domain employed in TUM-
VI, our approach builds upon the more commonly used DFT-based approach. To
overcome the limitation posed by data frequency, we adopt a similar technique
inspired by TUM-VI.

In some studies of continuous-time state estimation, by parameterizing the
state variables as continuous-time functions, it is possible to achieve
simultaneous spatiotemporal multi-sensor calibration within a MLE framework
[31, 32, 33]. These high-precision methods can be easily extended to the pose-
only scenario, but they are sensitive to the initial condition. In our work,
we use the result of our linear calibration as the initial guess and perform a
further refinement.

##  III METHODOLOGY

To better illustrate our algorithm, we take our hand-eye calibration platform
as an example (see in Fig. 1). The unit dual quaternions
$\hat{\mathbf{q}}\in\mathbb{DQ}$ is used to represent the homogeneous
transformation. A dual quaternion $\hat{\mathbf{q}}$ has the form
$\hat{\mathbf{q}}=\mathbf{q}+\epsilon\mathbf{q}^{\prime}=\left(\mathbf{q},\mathbf{q}^{\prime}\right)$,
where Hamiltonian quaternions $\mathbf{q}$ and $\mathbf{q}^{\prime}$ are the
standard part and the dual part of $\hat{\mathbf{q}}$ respectively, and
$\epsilon$ is the infinitesimal unit satisfying $\epsilon^{2}=0$. Our goal is
to estimate the time offset $\Delta t_{\scriptscriptstyle\mathrm{HE}}$ and the
extrinsic $\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}$ between the MoCap
(hand) trajectory $\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}}$ and the
estimated VIO (eye) trajectory
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}}$. Fig. 2 provides an
overview of the proposed algorithm, which comprises three modules. The main
steps of the algorithm will be described in the remaining of this section.

Figure 2: Flowchart of the proposed spatiotemporal hand-eye calibration, where
the green and blue parallelograms represent the inputs and outputs
respectively, and the orange rectangles represent the critical processing
steps.

###  III-A Time Alignment

Figure 3: Illustration of time alignment. The time offset can be determined by
performing a quadratic polynomial curve fitting around the maximum
(highlighted in gray) of the correlation function and obtaining the index of
the maximum. This method enables synchronization of angular velocity at a
finer granularity.

In order to process poses from sensors with different clocks, the first
crucial step is to align the two sets of timestamps. To achieve this, we can
leverage the constraint from screw motion, i.e., based on the equality of the
angular velocities $\omega_{\scriptscriptstyle\mathrm{GH}}$ and
$\omega_{\scriptscriptstyle\mathrm{WE}}$ of trajectories
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}}$ and
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}}$, which is independent of
calibration parameters. This simplifies the time alignment to the
synchronization of angular velocity signals. Based on theory of DFT, the
correlation between two time domain signals is greater when they exhibit
higher similarity. Therefore, the synchronization involves finding the time
shift, $\tau_{\mathrm{shift}}$, where the correlation function reaches its
maximum:

$\tau_{\mathrm{shift}}=\underset{\mathrm{index}}{\operatorname{argmax}}\left(\operatorname{Corr}\left(\omega_{\scriptscriptstyle\mathrm{GH}},\omega_{\scriptscriptstyle\mathrm{WE}}\right)\right),$
(1)

where $\operatorname{Corr}\left(\cdot\right)$ is the correlation function.

Given the discrete character of the angular velocity signals, the correlation
function also appears discrete in the time domain, with the precision of the
obtained time shift depending on the temporal resolution of the function.
Assuming the data around the maximum follows a quadratic polynomial
distribution, we perform curve fitting to refine our time shift similar to
[2]. As illustrated in Fig. 3, this approach allows us to accurately determine
the index of maximum correlation and trace it back to the corresponding time
offset, $\Delta t_{\scriptscriptstyle\mathrm{HE}}$.

###  III-B Linear Calibration

Given the time aligned trajectories
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}}$ and
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}}$, we can perform spatial
hand-eye calibration. As shown in Fig. 1, for any hand-eye motion from $i$ to
$j$ in the trajectories, we have:

$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}_{i}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}_{i}}^{-1}=\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}_{j}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}_{j}}^{-1}.$
(2)

Using the relative transformation between two poses, i.e.,
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}=\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}_{i}}^{-1}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}_{j}}$
and
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}=\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}_{i}}^{-1}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}_{j}}$,
(2) can be rewritten in the form of $\mathbf{AX}$ = $\mathbf{XB}$ as
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}=\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}$.
This fundamental equation can be divided into the standard and dual parts,
yielding:

$\displaystyle{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}-{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}$
$\displaystyle=0,$ (3)
$\displaystyle{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{\prime}{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}-{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}^{\prime}+{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\prime}-{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\prime}{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}$
$\displaystyle=0.$

Due to the redundancy of the scalar part of the dual quaternion, we set
$\mathbf{r}=\bigl{(}{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\bigr{)}_{v}$,
$\mathbf{r}^{\prime}=\bigl{(}{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{\prime}\bigr{)}_{v}$,
$\mathbf{s}=\bigl{(}{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\bigr{)}_{v}$
and
$\mathbf{s}^{\prime}=\bigl{(}{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}^{\prime}\bigr{)}_{v}$,
where $\left(\cdot\right)_{v}$ denotes the vector part of the quaternion. The
linear equation for extrinsic calibration derived from a single motion can be
written as:

$\begin{bmatrix}\mathbf{r}-\mathbf{s}&\left(\mathbf{r}+\mathbf{s}\right)^{\wedge}&\mathbf{0}_{3\times
1}&\mathbf{0}_{3\times 3}\\\
\mathbf{r}^{\prime}-\mathbf{s}^{\prime}&\left(\mathbf{r}^{\prime}+\mathbf{s}^{\prime}\right)^{\wedge}&\mathbf{r}-\mathbf{s}&\left(\mathbf{r}+\mathbf{s}\right)^{\wedge}\end{bmatrix}\begin{bmatrix}{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\\\
{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\prime}\end{bmatrix}=0,$ (4)

where $\left(\cdot\right)^{\wedge}$ denotes the antisymmetric matrix of a
vector, and the coefficient matrix with dimensions of $6\times 8$ will be
denoted as $\mathbf{S}$. With $n\geq 2$ motions, we can stack $\mathbf{S}$ to
obtain a $6n\times 8$ matrix with rank 6 in the noise-free case as:

$\mathbf{M}=\left[\mathbf{S}_{1}^{\mathrm{T}},\mathbf{S}_{2}^{\mathrm{T}},\ldots,\mathbf{S}_{n}^{\mathrm{T}}\right]^{\mathrm{T}}.$
(5)

The singular value decomposition (SVD) algorithm is then used to find the
linear least squares solution of the extrinsic with the constraint of the unit
dual quaternion. For more detailed information about the fundamental
principles of dual quaternion-based hand-eye calibration, please refer to
[20].

SVD is known to be sensitive to noise and outliers. In the following, we will
propose three strategies to enhance the accuracy and robustness of the
algorithm.

####  III-B1 Relative poses construction

As shown in (4), the equation entirely relies on the relative poses of the
hand-eye motion, therefore, the relative poses construction method will
significantly affect the quality of the solution. Conventional methods either
use a global or an inter-frame strategy. The former fixes a certain frame and
calculates the relative poses of the remaining frames with respect to it.
However, this method is prone to coupling the trajectory drift in the VO/VIO
scenario. The latter calculates the relative poses between two successive
frames, but may suffer from the noise caused by insufficient motion. Moreover,
for a relative pose with pure translation, the matrix in (4) will degenerate
and cannot constrain the dual part of the extrinsic.

Figure 4: Illustration of relative poses construction, different methods are
represented by three different lines, and ours shown as the solid green line.

We use a rotationally constrained approach to select the hand-eye keyframes to
construct the relative poses, aiming to mitigate the error coupling and
solution degeneracy. Specifically, one can keep searching forward from a frame
$\hat{\mathbf{q}}_{i}$ in the hand or eye trajectory, and build the relative
pose when a frame $\hat{\mathbf{q}}_{j}$ satisfies the constraint:

$2\arccos\left(\left(\hat{\mathbf{q}}_{i}^{-1}\hat{\mathbf{q}}_{j}\right)_{w}\right)\geq\eta,$
(6)

where $\left(\cdot\right)_{w}$ denotes the scalar part of the standard
component of the dual quaternion, and $\eta$ is an adjustable threshold which
we set to 5 degrees. Fig. 4 illustrates the construction of relative poses
using both conventional methods and our proposed method for comparison.

####  III-B2 Robust kernel

Despite the effort to construct high-quality relative poses, they still
contain varying degrees of noise. To quantify the weights of different
relative poses, we propose a robust kernel in the linear system construction.

For a unit dual quaternion $\hat{\mathbf{q}}$, its scalar part is defined as:

$\operatorname{Scalar}\left(\hat{\mathbf{q}}\right)=\frac{\left(\hat{\mathbf{q}}+\hat{\mathbf{q}}^{-1}\right)}{2},$
(7)

which can be expressed in the form of a vector, written as:

$\displaystyle\operatorname{Scalar}\left(\hat{\mathbf{q}}\right)$
$\displaystyle=\left[\omega,\mathbf{0}_{1\times
3},\omega^{\prime},\mathbf{0}_{1\times 3}\right]^{\mathrm{T}}$ (8)
$\displaystyle=\Bigl{[}\cos\frac{\theta}{2},\mathbf{0}_{1\times
3},-\frac{d}{2}\sin\frac{\theta}{2},\mathbf{0}_{1\times
3}\Bigr{]}^{\mathrm{T}},$

where $\omega$ and $\omega^{\prime}$ are the scalar parts of the standard part
and the dual part of the dual quaternion respectively, $\theta$ denotes the
rotation angle, and $d$ represents the translation norm of the screw motion.

The screw concatenation between the hand and eye can be further transformed
from the form of $\mathbf{AX}$ = $\mathbf{XB}$ to
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}=\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}$.
Based on the definition of the scalar part in (7), we have:

$\displaystyle\operatorname{Scalar}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\right)$
$\displaystyle=\frac{1}{2}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}+\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{-1}\right)$
(9)
$\displaystyle=\frac{1}{2}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}+\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}^{-1}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}\right)$
$\displaystyle=\frac{1}{2}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}+\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}^{-1}\right)\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}$
$\displaystyle=\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\operatorname{Scalar}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right)\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}$
$\displaystyle=\operatorname{Scalar}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right)\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}$
$\displaystyle=\operatorname{Scalar}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right),$

which demonstrates that the scalar parts of the hand-eye relative pose pair
are completely equal in the absence of noise. According to (8), this
constraint can be transformed into the equality of rotation angles in local
frames and translation norms along the principal axes of rotation, known as
the screw congruence theorem.

We design the function as (10) to evaluate the quality of the hand-eye
relative poses based on the scalar part of the dual quaternion in (8). The
function yields a result of 1 when the screw motion constraint in (9) is
strictly satisfied, while deviating from 1 as the noise increases.

$\displaystyle
E_{i}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}},\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right)$
(10) $\displaystyle=$ $\displaystyle
E_{i}^{\prime}\left(\operatorname{Scalar}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\right),\operatorname{Scalar}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\max\left(\left|\omega_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\right|,\left|\omega_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right|\right)}{\min\left(\left|\omega_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\right|,\left|\omega_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right|\right)}+\frac{\max\bigl{(}\bigl{|}\omega_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{\prime}\bigr{|},\bigl{|}\omega_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}^{\prime}\bigr{|}\bigr{)}}{\min\bigl{(}\bigl{|}\omega_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{\prime}\bigr{|},\bigl{|}\omega_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}^{\prime}\bigr{|}\bigr{)}}\right).$

Based on (10), we can define the robust kernel $W$ as:

$W_{i}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}},\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right)=\exp\left(\mu\left(1-E_{i}\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}},\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\right)^{2}\right)\right),$
(11)

where the parameter $\mu$ is an adjustable magnification factor, which we set
to 5. Hence, the linear calibration matrix in (5) can be robustified for
better numerical stability:

$\mathbf{M}_{\mathrm{r}}=\left[W_{1}\mathbf{S}_{1}^{\mathrm{T}},W_{2}\mathbf{S}_{2}^{\mathrm{T}},\ldots,W_{n}\mathbf{S}_{n}^{\mathrm{T}}\right]^{\mathrm{T}}.$
(12)

####  III-B3 Outlier elimination

During the calibration process, it is also important to identify and discard
outliers that exhibit significant error. We incorporate the RANSAC algorithm
which utilizes an iterative sampling strategy, and has the ability to recover
the inliers from noisy data.

Note that at least two pairs of relative poses are required to construct the
linear calibration system. For each iteration, we randomly sample a pose pair
subset consisting of two members and determine inliers by using the
quantitative criterion as:

$\displaystyle e_{1}$
$\displaystyle=\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{-1}\right)_{\operatorname{rot}}<\varphi,$
(13) $\displaystyle e_{2}$
$\displaystyle=\left(\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{-1}\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}^{-1}\right)_{\operatorname{trans}}<\psi,$

where $\left(\cdot\right)_{\operatorname{rot}}$ and
$\left(\cdot\right)_{\operatorname{trans}}$ are the rotation angle and
translation norm of the dual quaternion, respectively. Inliers are required to
have both indicators in (13) less than the thresholds $\varphi$ and $\psi$,
which we set to 0.5 degrees and 0.02 m experimentally.

Input: Time aligned hand-eye trajectories
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{GH}}$,
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{WE}}$.

Output: Hand-eye extrinsic
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\ast}$.

1
$\left\\{\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{E}_{i}\scriptscriptstyle\mathrm{E}_{j}},\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{H}_{i}\scriptscriptstyle\mathrm{H}_{j}}\right\\}\in\mathcal{C}\leftarrow$
Construct the relative poses based on the rotational constraint in (6);

2 while  _not reached the iteration limit_ do

3 Subset $\mathcal{D}\leftarrow$ Randomly sampling from $\mathcal{C}$;

4 Construct $\mathbf{M}_{12\times 8}$ based on $\mathcal{D}$ using (5);

5 $\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\mathrm{init}}\leftarrow$
Apply SVD to M;

6 foreach  _pose pair $\in$ $\mathcal{C}$_ do

7 $\mathcal{G}\leftarrow$ Select inliers using (13) with
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\mathrm{init}}$;

8

9 end foreach

10 Construct $\mathbf{M}_{2n\times 8}^{\prime}$ with robust kernel based on
$\mathcal{G}$ using (12);

11
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\mathrm{refine}}\leftarrow$
Apply SVD to $\mathbf{M}^{\prime}$;

12
$\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\ast}\leftarrow\hat{\mathbf{q}}_{\scriptscriptstyle\mathrm{HE}}^{\mathrm{refine}}$
with the smallest $\frac{\sigma_{7}}{\sigma_{6}}$ value;

13 end while

Algorithm 1 Robust linear hand-eye calibration

Additionally, we need to assess the quality of the solutions during the
iteration. Given that the matrix $\mathbf{M}_{\mathrm{r}}$ in (12) possesses a
two-dimensional null space, the singular values, $\sigma_{7}$ and
$\sigma_{8}$, which are the last two in the descending diagonal matrix
$\Sigma$ from the SVD result $\mathbf{M}_{\mathrm{r}}=\mathbf{U\Sigma
V}^{\mathrm{T}}$, are expected to be zero in the absence of noise. However,
when noise is present, these two singular values, corresponding to the noise,
disproportionately increase compared to the remaining singular values. The
ratio of $\sigma_{7}$ to the third-to-last singular value, $\sigma_{6}$, can
serve as an indicator of this disproportionality, and a lower ratio suggests a
reduced impact of noise. Consequently, we can establish a direct metric for
quantifying the quality of the solutions, rather than relying on traditional
measures such as the number of inliers or the root mean square error (RMSE) of
the solutions with respect to inliers. The specific implementation of the
linear spatial hand-eye calibration within the RANSAC framework is detailed in
Algorithm 1.

###  III-C Batch Estimation

Despite the fact that our linear calibration method achieves good accuracy and
robustness in the experiments of Section IV, it can be further improved by
introducing the correlation between the temporal and spatial calibration
parameters. In this section we follow the continuous-time batch estimation
methods proposed in [32, 33] and provide an estimator within the rigorous
theoretical framework of MLE, to jointly optimize the time offset and the
spatial transformation. Since the estimator is hard to converge globally, we
used the results derived in Section III-A and III-B as the initial guesses.

Specifically, the original hand trajectory is parameterized using a B-spline
functions $\mathbf{T}_{\mathrm{GH}}\left(t\right)$, with the translation part
represented by a B-spline in three-dimensional vector space, and the rotation
part parameterized by a B-spline on $SO\left(3\right)$. For the B-spline
$\mathbf{q}\left(t\right)$ on $SO\left(3\right)$ with order $\xi$, knots
$\left\\{t_{i}|i\in\left\\{1,2,3,\dots,N\right\\}\right\\}$, and satisfying
$N\geq 2\xi$, the function can be defined in each subinterval
$\left\\{t\in\left[t_{i},t_{i+1}\right)|\xi\leq i\leq N-\xi\right\\}$ as:

$\mathbf{q}\left(t\right)=\mathbf{q}_{l\left(i\right)}\prod_{j=1}^{\xi-1}\mathbf{EXP}\biggl{(}\Bigl{(}\sum_{k=\eta}^{i}f_{k}\Bigr{)}\mathbf{LOG}\left(\mathbf{q}_{\eta-1}^{-1}\mathbf{q}_{\eta}\right)\biggl{)},$
(14)

where $l\left(i\right)=i-\xi+1$ and $\eta=l\left(i\right)+j$, $f$ and
$\mathbf{q}$ represent the B-spline basis functions and the control vertices
in unit quaternion form, respectively. $\mathbf{EXP}$ denotes the mapping from
the Lie algebra to the Lie group, while $\mathbf{LOG}$ represents the inverse
process.

The parameters determined by our estimator include
$\mathbf{T}_{\scriptscriptstyle\mathrm{GH}}\left(t\right)$, homogeneous
transformation $\mathbf{T}_{\scriptscriptstyle\mathrm{HE}}$, and time offset
$\Delta t_{\scriptscriptstyle\mathrm{HE}}$. With the observations of hand-eye
trajectories, we minimize the negative log-likelihood function as:

$\displaystyle g$
$\displaystyle=\sum_{h=1}^{H-1}\rho\left(\left\|d\left(\mathbf{T}_{\scriptscriptstyle\mathrm{GH}}\left(t_{h}\right),\mathbf{T}_{\scriptscriptstyle\mathrm{GH}}\left(t_{h+1}\right),\mathbf{T}_{\scriptscriptstyle\mathrm{GH}_{h}},\mathbf{T}_{\scriptscriptstyle\mathrm{GH}_{h+1}}\right)\right\|_{\Sigma_{\scriptscriptstyle\mathrm{H}}}^{2}\right)$
(15)
$\displaystyle+\sum_{e=1}^{E-1}\rho\left(\left\|d\left(\mathbf{T}_{\scriptscriptstyle\mathrm{WE}}\left(t_{e}\right),\mathbf{T}_{\scriptscriptstyle\mathrm{WE}}\left(t_{e+1}\right),\mathbf{T}_{\scriptscriptstyle\mathrm{WE}_{e}},\mathbf{T}_{\scriptscriptstyle\mathrm{WE}_{e+1}}\right)\right\|_{\Sigma_{\scriptscriptstyle\mathrm{E}}}^{2}\right),$

where $H$ and $E$ denote the number of pose observations of the hand and the
eye respectively, and $\rho\left(\cdot\right)$ is the Huber loss function. Due
to the rigid connection between the hand and the eye, we have
$\mathbf{T}_{\scriptscriptstyle\mathrm{WE}}\left(t\right)=\mathbf{T}_{\scriptscriptstyle\mathrm{WG}}\mathbf{T}_{\scriptscriptstyle\mathrm{GH}}\left(t-\Delta
t_{\scriptscriptstyle\mathrm{HE}}\right)\mathbf{T}_{\scriptscriptstyle\mathrm{HE}}$.
Meanwhile, we define the residual function in a relative form to eliminate the
influence of the unknown $\mathbf{T}_{\scriptscriptstyle\mathrm{WG}}$ as:

$d\left(\mathbf{T}_{i},\mathbf{T}_{i+1},\mathbf{T}_{i}^{\prime},\mathbf{T}_{i+1}^{\prime}\right)=\mathbf{LOG}\Bigl{(}\mathbf{T}_{i}^{-1}\mathbf{T}_{i+1}\left(\mathbf{T}_{i}^{\prime-1}\mathbf{T}_{i+1}^{\prime}\right)^{-1}\Bigr{)}.$
(16)

For the above cost function, the Levenberg–Marquardt algorithm is used to
obtain the refined calibration result.

##  IV EXPERIMENTAL RESULTS

###  IV-A Ablation Studies

To validate the effectiveness of the key improvement strategies proposed in
our methodology, we conduct comparative experiments using the simulated
datasets. Specifically, to obtain the required data, we model a real motion
trajectory using the B-spline. Subsequently, we extract two trajectories from
the model at 100Hz and 20Hz to simulate MoCap and VO/VIO trajectories for
hand-eye calibration. Throughout the process, we introduce frame-wise
cumulative error into each eye pose to simulate the noise and drift typically
present in the VO/VIO trajectory. The standard deviations for translation and
rotation errors are divided into ten levels, ranging from 0 to 5 mm and 0 to
0.2 degrees, respectively.

We first test the time alignment algorithm proposed in Section III-A to verify
the enhancement provided by the quadratic polynomial curve fitting in the
correlation analysis compared to the baseline approach used in [10]. In this
experiment, we introduce random delays into the eye trajectories derived from
the B-spline to simulate the time offsets. To facilitate the correlation
function computation, standardizing the frequencies of the trajectory pair is
necessary. Specifically, we adjust the frequencies to match those of hand and
eye trajectories, which are respectively 100Hz and 20Hz. As shown in Fig. 5,
correlation analysis-based time alignment methods exhibit minimal sensitivity
to trajectory noise. However, limited by the temporal resolution of the
correlation function, time alignment error of the baseline method is
essentially determined by the trajectory frequency. Acquiring the high-
frequency trajectory is challenging and requires significant computational
effort. By implementing the curve fitting strategy, we can determine the
maximum index of the correlation function with greater accuracy, leading to
higher precision in time alignment, even at low frequency.

Figure 5: Performance comparison of different time alignment methods under
various noise levels. We report the mean and standard deviation of the time
alignment error.  Figure 6: Performance comparison of different strategies
used in linear calibration under various noise levels. We calculate the
translation error and the rotation error separately, and report the mean and
standard deviation.

Additionally, we evaluate the effectiveness of the strategies for spatial
linear calibration proposed in Section III-B. The three relative poses
construction methods, namely global, inter-frame, and rotationally
constrained, are comparatively validated, and different thresholds in (6) are
explored within the proposed rotationally constrained approach. Meanwhile, we
test the impact of the robust kernel (RB). In this experiment, the calibration
error is used as a quantitative metric for assessment. Fig. 6 indicates that
the calibration errors for all scenarios trend to increase as noise levels
rise. Nonetheless, our method, which constructs rotationally constrained
relative poses yields smaller error due to its ability to derive the linear
equation with higher signal-to-noise ratio. At the same time, the rotational
constraint threshold also affects the accuracy of the solution, with 5 degrees
proving to be optimal in our tests. Furthermore, our robust kernel strategy
demonstrates superior robustness in the presence of increasing noise, as
indicated by the relatively small rise in calibration error and standard
deviation.

###  IV-B Overall Evaluation

We evaluate the performance of our hand-eye calibration algorithm using real-
world datasets, including public VIO datasets, and datasets collected by our
system (see in Fig. 1). For comparison, we also evaluate two other state-of-
the-art (SOTA) linear hand-eye calibration algorithms based on dual quaternion
and random sampling, as implemented in [10], namely RANSAC scalar-based inlier
check (RS) and RANSAC classic (RC). We use the well-known VIO algorithm
OpenVINS [34] to estimate the trajectory of the eye, while the trajectory of
the hand is derived from the ground-truth system.

Figure 7: Performance comparison of different linear calibration methods on
public datasets. We select two sequences in each scenario and count the
distribution of the translation error and the rotation error separately.

We first test on the TUM-VI room scenario [2] as well as machine hall (MH) and
Vicon room (V) scenarios provided by EuRoC benchmark [1]. All three scenarios
provide complete raw data of ground-truth trajectories, and achieve high-
precision hand-eye calibration through the tightly-coupled method. We align
the estimated VIO trajectories with the raw ground-truth trajectories using
different hand-eye calibration algorithms. For evaluation metrics, we compare
with the calibration results provided by the public datasets, and calculate
the translation error and the rotation error. As shown in Fig. 7, we first
evaluate our linear calibration (LC). The comparison results demonstrate that
our algorithm achieves the highest accuracy and repeatability in all
sequences. In challenging sequences such as MH_04 and V2_03, where the VIO
algorithm struggles to deliver high-quality trajectories, traditional
calibration algorithms are prone to fail, but our algorithm handles these
situations effectively.

TABLE I: Detailed Performance of the Proposed Methods and the Trajectory Metrics on Public Datasets Sequences | Error of our LC | Error of our BE | Original metrics | Our metrics
---|---|---|---|---
Time (ms) | Trans (m) | Rot (deg) | Time (ms) | Trans (m) | Rot (deg) | APE (m) | ARE (deg) | APE (m) | ARE (deg)
Room_1 | 1.393 | 0.003 | 0.284 | 0.198 | 0.004 | 0.106 | 0.059 | 1.593 | 0.059 | 1.593
Room_6 | 1.266 | 0.024 | 0.675 | 0.296 | 0.016 | 0.199 | 0.085 | 1.686 | 0.085 | 1.683
MH_01 | 3.430 | 0.032 | 0.596 | 0.883 | 0.018 | 0.641 | 0.156 | 1.884 | 0.157 | 1.683
MH_04 | 5.404 | 0.031 | 0.927 | 3.691 | 0.018 | 0.749 | 0.161 | 0.950 | 0.157 | 0.561
V1_01 | 3.718 | 0.053 | 4.444 | 1.982 | 0.049 | 4.479 | 0.061 | 5.520 | 0.047 | 1.064
V2_03 | 3.092 | 0.015 | 0.280 | 2.135 | 0.007 | 0.118 | 0.095 | 1.157 | 0.095 | 1.139

(a)

(b)

Figure 8: AR visualization of the EuRoC V1_01 sequence, featuring a virtual
box rendered in the scene. This visualization utilizes (a) transformed ground-
truth poses provided by the EuRoC benchmark and (b) transformed ground-truth
poses based on the raw pre-calibrated trajectory and our calibration result.
The red dashed circle highlights the difference in consistency between the
virtual and real elements when using different poses.

Table I details the performance of our algorithm in each sequence, including
the time alignment errors and the results of our batch estimation (BE).
Additionally, to analyze the impact of hand-eye calibration on VIO trajectory
evaluation, we use our calibration results to transform the raw ground-truth
trajectories and calculate the absolute positional error (APE) and absolute
rotational error (ARE) [3], which are important metrics in VIO evaluation. As
a comparison, we also calculate the original metrics using the transformed
ground-truth provided by the public datasets. The obtained results demonstrate
that our LC effectively accomplishes spatiotemporal hand-eye calibration, and
our BE can further optimizes the result to achieve higher accuracy. In most of
the sequences, our calibration algorithm provides accurate transformations for
ground-truths with minimal impact on evaluation metrics. However, it is
interesting to note that in some sequences, particularly V1_01, our algorithm
exhibits some errors. This may stem from the inherent inaccuracies in the
calibration results of the benchmarks, corroborated by a similar conclusion in
[34]. To provide a more intuitive illustration, we render a virtual box on the
images of the V1_01 sequence using two different transformed ground-truth
trajectories. The first is provided by the EuRoC benchmark and the second is
obtained based on our hand-eye calibration result. Inaccurate hand-eye
calibration can lead to misalignment between the virtual and the real elements
in this augmented reality (AR) application. As shown in the comparison results
in Fig. 8, the virtual box is more consistent with the real world in our AR
result, i.e., implying higher calibration accuracy.

Figure 9: Experimental setup for evaluating the hand-eye calibration using a relative translation method. The VR headset remains stationary, and the tracking markers can be translated along the red arrow.  TABLE II: Performance Comparison of Different Calibration Methods on Self-Collected Datasets. We Count the Translation Error at Different Relative Translation Distances Sequences |  | Error of
---
RS (m)
| Error of
---
RC (m)
| Error of
---
our_LC (m)
| Error of
---
our_BE (m)
0.1m_easy | 0.035 | 0.028 | 0.007 | 0.003
0.2m_easy | 0.026 | 0.022 | 0.005 | 0.003
0.3m_easy | 0.049 | 0.034 | 0.007 | 0.002
0.1m_difficult | 0.041 | 0.045 | 0.007 | 0.004
0.2m_difficult | 0.056 | 0.038 | 0.006 | 0.004
0.3m_difficult | 0.062 | 0.047 | 0.010 | 0.004

In real-world system, evaluating hand-eye calibration is inherently difficult,
as the ground-truth of offset is not available. To address this, we employ a
relative approach using self-collected data. The experimental setup, as
illustrated in Fig. 9, involved mounting a VR headset and tracking markers on
the same object for motion. After obtaining a set of hand-eye trajectories, we
translate the tracking markers in a specified direction to gather additional
calibration data. We compute the norm of the relative translation using the
extrinsics derived from the calibrations performed before and after the
translation. Calibration error is then determined by comparing this norm
against the high-precision, directly measured result.

Table II presents the errors obtained from different hand-eye calibration
methods when the tracking markers are translated by 0.1 m, 0.2 m, and 0.3 m,
respectively. We categorize the scenarios into easy (texture-rich and slow-
moving) and difficult (texture-less and fast-moving) in the context of VIO, to
test the robustness of the calibration algorithms. The experimental results
show that our algorithm performs best in all sequences. Additionally, both our
LC and BE achieve millimeter-level calibration accuracy and are less affected
by trajectory error.

##  V CONCLUSIONS

In this letter, we propose an improved spatiotemporal hand-eye calibration
algorithm for trajectory alignment in VO/VIO evaluation. Aiming to optimize
for VO/VIO scenarios, we have designed multiple strategies based on screw
theory to enhance both the accuracy and robustness of our proposed algorithm.
The validation experiments demonstrate that our algorithm outperforms SOTA
methods, exhibiting superior accuracy while effectively mitigating the
influence of noise. Our method is well poised to be applied in the evaluation
of modern VO/VIO algorithms. Nevertheless, our algorithm is less effective
over an extended period. This limitation arises partly because our calibration
algorithm processes the entire trajectory, leading to inefficiency with long
sequences. Furthermore, our algorithm presupposes a constant time offset, an
assumption unsuitable over a long time. In future work, we will focus on
spatiotemporal hand-eye calibration tailored for long-duration trajectories.
The aim is to develop an algorithm that can process extensive data efficiently
and address the problem of time offset drift over an extended period.

## References

  * [1] M. Burri, J. Nikolic, P. Gohl, T. Schneider, J. Rehder, S. Omari, M. W. Achtelik, and R. Siegwart, “The $\mathrm{E}$u$\mathrm{R}$o$\mathrm{C}$ micro aerial vehicle datasets,” _The International Journal of Robotics Research_ , vol. 35, no. 10, pp. 1157–1163, 2016.
  * [2] D. Schubert, T. Goll, N. Demmel, V. Usenko, J. Stückler, and D. Cremers, “The $\mathrm{TUM\ VI}$ benchmark for evaluating visual-inertial odometry,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2018, pp. 1680–1687.
  * [3] J. Li, B. Yang, D. Chen, N. Wang, G. Zhang, and H. Bao, “Survey and evaluation of monocular visual-inertial $\mathrm{SLAM}$ algorithms for augmented reality,” _Virtual Reality & Intelligent Hardware_, vol. 1, no. 4, pp. 386–410, 2019.
  * [4] S. Umeyama, “Least-squares estimation of transformation parameters between two point patterns,” _IEEE Transactions on Pattern Analysis & Machine Intelligence_, vol. 13, no. 4, pp. 376–380, 1991.
  * [5] J. Jiang, X. Luo, Q. Luo, L. Qiao, and M. Li, “An overview of hand-eye calibration,” _The International Journal of Advanced Manufacturing Technology_ , vol. 119, no. 1-2, pp. 77–97, 2022.
  * [6] E. Pedrosa, M. Oliveira, N. Lau, and V. Santos, “A general approach to hand-eye calibration through the optimization of atomic transformations,” _IEEE Transactions on Robotics_ , vol. 37, no. 5, pp. 1619–1633, 2021.
  * [7] K. Koide and E. Menegatti, “General hand-eye calibration based on reprojection error minimization,” _IEEE Robotics and Automation Letters_ , vol. 4, no. 2, pp. 1021–1028, 2019.
  * [8] I. Ali, O. Suominen, A. Gotchev, and E. R. Morales, “Methods for simultaneous robot-world-hand-eye calibration: A comparative study,” _Sensors_ , vol. 19, no. 12, p. 2837, 2019.
  * [9] S. Sarabandi, J. M. Porta, and F. Thomas, “Hand-eye calibration made easy through a closed-form two-stage method,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 3679–3686, 2022.
  * [10] F. Furrer, M. Fehr, T. Novkovic, H. Sommer, I. Gilitschenski, and R. Siegwart, “Evaluation of combined time-offset estimation and hand-eye calibration on robotic datasets,” in _Field and Service Robotics: Results of the 11th International Conference_. Springer, 2018, pp. 145–159.
  * [11] Y. C. Shiu and S. Ahmad, “Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form $\mathrm{AX}$ = $\mathrm{XB}$,” _IEEE Transactions on Robotics and Automation_ , vol. 5, no. 1, pp. 16–29, 1989.
  * [12] R. Y. Tsai and R. K. Lenz, “Real time versatile robotics hand/eye calibration using 3$\mathrm{D}$ machine vision,” in _1988 IEEE International Conference on Robotics and Automation_. IEEE, 1988, pp. 554–561.
  * [13] C. C. Wang, “Extrinsic calibration of a vision sensor mounted on a robot,” _IEEE Transactions on Robotics and Automation_ , vol. 8, no. 2, pp. 161–175, 1992.
  * [14] J. C. Chou and M. Kamel, “Finding the position and orientation of a sensor on a robot manipulator using quaternions,” _The International Journal of Robotics Research_ , vol. 10, no. 3, pp. 240–254, 1991.
  * [15] F. C. Park and B. J. Martin, “Robot sensor calibration: Solving $\mathrm{AX}$ = $\mathrm{XB}$ on the $\mathrm{E}$uclidean group,” _IEEE Transactions on Robotics and Automation_ , vol. 10, no. 5, pp. 717–721, 1994.
  * [16] R. Liang and J. Mao, “Hand-eye calibration with a new linear decomposition algorithm,” _Journal of Zhejiang University-SCIENCE A_ , vol. 9, no. 10, pp. 1363–1368, 2008.
  * [17] H. Nguyen and Q. C. Pham, “On the covariance of $\mathrm{X}$ in $\mathrm{AX}$ = $\mathrm{XB}$,” _IEEE Transactions on Robotics_ , vol. 34, no. 6, pp. 1651–1658, 2018.
  * [18] H. H. Chen, “A screw motion approach to uniqueness analysis of head-eye geometry,” in _1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition_. IEEE Computer Society, 1991, pp. 145–146.
  * [19] N. Andreff, R. Horaud, and B. Espiau, “On-line hand-eye calibration,” in _Second International Conference on 3-D Digital Imaging and Modeling_. IEEE, 1999, pp. 430–436.
  * [20] K. Daniilidis, “Hand-eye calibration using dual quaternions,” _The International Journal of Robotics Research_ , vol. 18, no. 3, pp. 286–298, 1999.
  * [21] D. Condurache and A. Burlacu, “Orthogonal dual tensor method for solving the $\mathrm{AX}$ = $\mathrm{XB}$ sensor calibration problem,” _Mechanism and Machine Theory_ , vol. 104, pp. 382–404, 2016.
  * [22] K. H. Strobl and G. Hirzinger, “Optimal hand-eye calibration,” in _2006 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2006, pp. 4647–4653.
  * [23] J. Heller, D. Henrion, and T. Pajdla, “Hand-eye and robot-world calibration by global polynomial optimization,” in _2014 IEEE International Conference on Robotics and Automation_. IEEE, 2014, pp. 3157–3164.
  * [24] Z. Zhao, “Simultaneous robot-world and hand-eye calibration by the alternative linear programming,” _Pattern Recognition Letters_ , vol. 127, pp. 174–180, 2019.
  * [25] J. Schmidt, F. Vogt, and H. Niemann, “Robust hand-eye calibration of an endoscopic surgery robot using dual quaternions,” in _Pattern Recognition: 25th DAGM Symposium_. Springer, 2003, pp. 548–556.
  * [26] J. Wu, Y. Sun, M. Wang, and M. Liu, “Hand-eye calibration: 4-$\mathrm{D}$ procrustes analysis approach,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 69, no. 6, pp. 2966–2981, 2019.
  * [27] J. Kelly and G. S. Sukhatme, “A general framework for temporal calibration of multiple proprioceptive and exteroceptive sensors,” in _Experimental Robotics: The 12th International Symposium on Experimental Robotics_. Springer, 2014, pp. 195–209.
  * [28] M. K. Ackerman, A. Cheng, B. Shiffman, E. Boctor, and G. Chirikjian, “Sensor calibration with unknown correspondence: Solving $\mathrm{AX}$ = $\mathrm{XB}$ using $\mathrm{E}$uclidean-group invariants,” in _2013 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2013, pp. 1308–1313.
  * [29] H. Li, Q. Ma, T. Wang, and G. S. Chirikjian, “Simultaneous hand-eye and robot-world calibration by solving the $\mathrm{AX}$ = $\mathrm{YB}$ problem without correspondence,” _IEEE Robotics and Automation Letters_ , vol. 1, no. 1, pp. 145–152, 2015.
  * [30] K. Pachtrachai, F. Vasconcelos, G. Dwyer, V. Pawar, S. Hailes, and D. Stoyanov, “Chess—$\mathrm{C}$alibrating the hand-eye matrix with screw constraints and synchronization,” _IEEE Robotics and Automation Letters_ , vol. 3, no. 3, pp. 2000–2007, 2018.
  * [31] P. Furgale, T. D. Barfoot, and G. Sibley, “Continuous-time batch estimation using temporal basis functions,” in _2012 IEEE International Conference on Robotics and Automation_. IEEE, 2012, pp. 2088–2095.
  * [32] J. Rehder, R. Siegwart, and P. Furgale, “A general approach to spatiotemporal calibration in multisensor systems,” _IEEE Transactions on Robotics_ , vol. 32, no. 2, pp. 383–398, 2016.
  * [33] H. Sommer, J. R. Forbes, R. Siegwart, and P. Furgale, “Continuous-time estimation of attitude using $\mathrm{B}$-splines on $\mathrm{L}$ie groups,” _Journal of Guidance, Control, and Dynamics_ , vol. 39, no. 2, pp. 242–261, 2016.
  * [34] P. Geneva, K. Eckenhoff, W. Lee, Y. Yang, and G. Huang, “Open$\mathrm{VINS}$: A research platform for visual-inertial estimation,” in _2020 IEEE International Conference on Robotics and Automation_. IEEE, 2020, pp. 4666–4672.