Gibbs measures for SOS models with external field on a Cayley tree

M.M.Rahmatullaev, O.Sh.Karshiboev

###### Abstract

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin
values $0,1,\ldots,m,$ $m\geq 2,$ and nonzero external field, on a Cayley tree
of degree $k$ (with $k+1$ neighbors). We are aiming to extend the results of
[6] where the SOS model is studied with (mainly) three spin values and zero
external field. The SOS model can be treated as a natural generalization of
the Ising model (obtained for $m=1$). We mainly assume that $m=2$ (three spin
values) and study translation-invariant (TI) and splitting (S) Gibbs measures
(GMs).(Splitting GMs have a particular Markov-type property specific for a
tree.) For $m=2$, in the antiferromagnit (AFM) case, a TISGM is unique for all
temperatures with an external field. In the ferromagnetic (FM) case, for
$m=2,$ the number of TISGMs varies with the temperature and the external
field: this gives an interesting example of phase transition.

Our second result gives a classification of all TISGMs of the Three-State SOS-
Model on the Cayley tree of degree two with the presence of an external field.
We show uniqueness in the case of antiferromagnetic interactions and existence
of up to seven TISGMs in the case of ferromagnetic interactions, where the
number of phases depends on the interaction strength and external field.

Key words: Cayley tree, Gibbs measure, SOS model, external field,
configuration.

##  1 Introduction

One of the central problems in the theory of Gibbs measures is to describe
infinite-volume (or) limiting Gibbs measures corresponding to a given
Hamiltonian. The existence of such measures for a wide class of Hamiltonians
was established in the ground-breaking work of Dobrushin (see [9]). However, a
complete analysis of the set of limiting Gibbs measures for a specific
Hamiltonian is often a difficult problem. On a cubic lattice, for small values
of $\beta=\frac{1}{T},$ where $T>0$ is the temperature, a Gibbs measure is
unique (Refs. [1],[9]) which reflects a physical fact that at high
temperatures there is no phase transition. The analysis for low temperatures
requires specific assumptions on the form of the Hamiltonian.

In this paper we consider models with a nearest neighbour interaction on a
Cayley tree. Models on a Cayley tree were discussed in [1], [14], [22]-[26]. A
classical example of such a model is the Ising model, with two values of spin,
$\pm 1.$ It was considered in [9], [23] and became a focus of active research
in the first half of the ’90s and afterwards; Models considered in this paper
are generalisations of the Ising model and can be described as SOS (solid-on-
solid) models with constraints. In the case of a cubic lattice they were
analysed in [5] where an analogue of the so-called Dinaburg-Mazel-Sinai theory
was developed. Besides interesting phase transitions in these models, the
attention to them is motivated by applications, in particular in the theory of
communication networks (see, for example, [20], [21]). In [30] SOS model with
spins 0, 1, 2, 3 on a Cayley tree of degree $k\geq 1$ is analysed and
translation-invariant, periodic splitting Gibbs measures for this model are
constructed. In [27] the SOS model with magnetic external field on
$Z^{d},\leavevmode\nobreak\ d\geq 2$ is studied. In [7] a classification of
all translation-invariant splitting Gibbs measures of SOS model on the Binary
tree with spin values $0,\leavevmode\nobreak\ 1\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ 2$ is given and whether these measures are
extremal or non-extremal in the set of all Gibbs measures is investigated. In
[18] Potts and SOS models with $q\geq 2$ states on the Cayley tree of order
$k\geq 1$ are considered. For any values of the parameter $q$ in the Potts
model and $q\leq 6$ in the SOS model, sets containing all translation-
invariant Gibbs measures are found. In [15] the authors studied translation-
invariant and periodic ground states for the SOS model with external fields.
In [16] the SOS model for $m=2$ on a Cayley tree of degree $k\geq 3$ is
considered and using translation-invariant Gibbs measures on a Cayley tree of
degree two, some non-translation-invariant Gibbs measures are constructed. In
[17] weakly periodic ground state for the SOS model with competing
interactions on the Cayley tree of degree two are studied. In [28], [31]
translation-invariant and periodic Gibbs measures for the Potts-SOS model on
the Cayley tree are studied. In [8] the $q$-state Potts model (i.e., with spin
values in $\\{1,\ldots,q\\}$) on a Cayley tree of degree $k\geq 2$ in an
external (possibly random) field is considered. Recently, in [29] a systematic
review of the theory of Gibbs measures of Potts model on Cayley trees
(developed since 2013) is given and discussed many applications of the Potts
model to real world situations.

##  2  Set-up

Following [8] we start by summarizing the basic consepts for Gibbs measures on
a Cayley tree, and also fix some notation.

_Cayley tree._ Let $\mathbb{T}^{k}$ be a (homogeneous) Cayley tree of degree
$k\geq 2$, that is, an infinite connected cycle-free (undirected) regular
graph with each vertex incident to $k+1$ edges. For example,
$\mathbb{T}^{1}=\mathbb{Z}.$ Denote by $V=\\{x\\}$ the set of the vertices of
the tree and by $E=\\{\langle x,y\rangle\\}$ the set of its (non-oriented)
edges connecting pairs of neighbouring vertices. The natural distance $d(x,y)$
on $\mathbb{T}^{k}$ is defined as the number of edges on the unique path
connecting vertices $x,y\in V.$ In particular, $\langle x,y\rangle\in E$
whenever $d(x,y)=1.$ A (non-empty) set $\Lambda\subset V$ is called
_connected_ if for any $x,y\in\Lambda$ the path connecting $x$ and $y$ lies in
$\Lambda.$ We denote the complement of $\Lambda$ by
$\Lambda^{c}:=V\backslash\Lambda$ and its _boundary_ by
$\partial\Lambda:=\\{x\in\Lambda^{c}:\exists y,d(x,y)=1\\},$ and we write
$\overline{\Lambda}=\Lambda\bigcup\partial\Lambda.$ The subset of edges in
$\Lambda$ is denoted $E_{\Lambda}:=\\{\langle x,y\rangle\in
E:x,y\in\Lambda\\}.$

Fix a vertex $x_{0}\in V$, interpreted as the _root_ of the tree. We say that
$y\in V$ is a _direct successor_ of $x\in V$ if $x$ is penultimate vertex on
the unique path leading from the root $x_{0}$ to the vertex $y$; that is,
$d(x_{0},y)=d(x_{0},x)+1$ and $d(x,y)=1$. The set of all direct successors of
$x\in V$ is denoted by $S(x).$ It is convenient to work with the family of the
radial subsets centred at $x_{0}$, defined for
$n\in\mathbb{N}_{0}:=\\{0\\}\bigcup\mathbb{N}$ by

$V_{n}:=\\{x\in V:d(x_{0},x)\leq n\\},\quad W_{n}:=\\{x\in V:d(x_{0},x)=n\\},$

interpreted as the ’ball’ and ’sphere’, respectively, of radius $n$ centred at
the root $x_{0}$. Clearly, $\partial V_{n}=W_{n+1}.$ Note that if $x\in W_{n}$
then $S(x)=\\{y\in W_{n+1}:d(x,y)=1\\}$. In the special case $x=x_{0}$ we have
$S(x_{0})=W_{1}.$ For short, we set $E_{n}:=E_{V_{n}}.$

Remark 1 Note that the sequence of balls ($V_{n}$) ($n\in\mathbb{N}_{0}$) is
_cofinal_ (see [14], section 1.2, page 17, see also [8]), that is, any finite
subset $\Lambda\subset V$ is contained in some $V_{n}.$

_The SOS model and Gibbs measures._ In the SOS model with external field, the
spin at each vertex $x\in V$ can take values in the set
$\Phi:=\\{0,1,...,m\\}.$ Thus, the spin configuration on $V$ is a function
$\sigma:V\rightarrow\Phi$ and the set of all configurations is $\Phi^{V}$. For
a subset $\Lambda\subset V$, we denote by
$\sigma_{\Lambda}:\Lambda\rightarrow\Phi$ the restriction of configuration
$\sigma$ to $\Lambda,$

$\sigma_{\Lambda}:=\sigma(x),\quad x\in\Lambda.$

The SOS model with external field is defined by the formal Hamiltonian

$H(\sigma)=-J\sum_{\langle x,y\rangle\in E}|\sigma(x)-\sigma(y)|-\sum_{x\in
V}\alpha_{\sigma(x),x},$ (2.1)

where $\sigma\in\Phi^{V},\leavevmode\nobreak\ $$J\in\mathbb{R}$ and
${\textbf{\textalpha}}_{x}=(\alpha_{0,x},\alpha_{1,x},...,\alpha_{m,x})\in\mathbb{R}^{m+1}$
is the external (possible random) field.

Here, $J<0$ gives a ferromagnetic and $J>0$ an anti-ferromagnetic model.

For each finite subset $\Lambda\subset V$ ($\Lambda\neq\emptyset$) and any
fixed subconfiguration $\eta\in\Phi^{\Lambda^{c}}$ (called the
_configurational boundary condition_), the _Gibbs distribution_
$\gamma^{\eta}_{\Lambda}$ is a probability measure in $\Phi^{\Lambda}$ defined
by the formula

$\gamma^{\eta}_{\Lambda}(\varsigma)=\frac{1}{Z^{\eta}_{\Lambda}(\beta)}\exp\Big{\\{}-\beta
H_{\Lambda}(\varsigma)+\beta
J\sum_{x\in\Lambda}\sum_{y\in\Lambda^{c}}|\varsigma(x)-\eta(y)|\Big{\\}},\quad\varsigma\in\Phi^{\Lambda},$
(2.2)

where $\beta\in(0,\infty)$ is a parameter (having the meaning of _inverse
temperature_), $H_{\Lambda}$ is the restriction of the Hamiltonian (2.1) to
subconfigurations in $\Lambda,$

$H_{\Lambda}(\varsigma)=-J\sum_{\langle x,y\rangle\in
E_{\Lambda}}|\varsigma(x)-\varsigma(y)|-\sum_{x\in\Lambda}\alpha_{\varsigma(x),x},\quad\varsigma\in\Phi^{\Lambda},$
(2.3)

and $Z^{\eta}_{\Lambda}$ is the normalizing constant (often called the
_canonical partition function_),

$Z^{\eta}_{\Lambda}=\sum_{\varsigma\in\Phi^{\Lambda}}\exp\Big{\\{}-\beta
H_{\Lambda}(\varsigma)+\beta
J\sum_{x\in\Lambda}\sum_{y\in\Lambda^{c}}|\varsigma(x)-\eta(y)|\Big{\\}}.$

Due to the nearest-neighbour interaction, formula (2.2) can be rewritten as

$\gamma_{\Lambda}(\varsigma)=\frac{1}{Z^{\eta}_{\Lambda}}\exp\Big{\\{}-\beta
H_{\Lambda}(\varsigma)+\beta
J\sum_{x\in\Lambda}\sum_{y\in\partial\Lambda}|\varsigma(x)-\eta(y)|\Big{\\}}.$
(2.4)

Finally, a measure $\mu=\mu_{\beta,\alpha}$ on $\Phi^{V}$ is called a _Gibbs
measure_ if, for any non-empty finite set $\Lambda\subset V$ and any
$\eta\in\Phi^{\Lambda^{c}},$

$\mu(\sigma_{\Lambda}=\varsigma|\sigma_{\Lambda^{c}}=\eta)\equiv\gamma^{\eta}_{\Lambda}(\varsigma),\quad\varsigma\in\Phi^{V}.$
(2.5)

_SGM construction._ It is convenient to construct Gibbs measures on the Cayley
tree $\mathbb{T}^{k}$ using a version of Gibbs distributions on the balls
($V_{n}$) defined via auxiliary fields encapsulating the interaction with the
exterior of the balls. More precisely, for a vector field $V\ni x\mapsto
h_{x}=(h_{0,x},...,h_{m,x})\in\mathbb{R}^{m+1}$ and each $n\in\mathbb{N}_{0}$,
define a probability measure in $V_{n}$ by the formula

$\mu^{h}_{n}(\sigma_{n})=\frac{1}{Z_{n}}\exp\Big{\\{}-\beta
H_{n}(\sigma_{n})+\beta\sum_{x\in
W_{n}}h_{\sigma_{n}(x),x}\Big{\\}},\quad\sigma_{n}\in\Phi^{V_{n}},$ (2.6)

where $Z_{n}=Z_{n}(\beta,h)$ is the normalizing factor and $H_{n}:=H_{V_{n}}$,
that is (see (2.3))

$H_{n}(\sigma_{n})=-J\sum_{\langle x,y\rangle\in
E_{n}}|\sigma_{n}(x)-\sigma_{n}(y)|-\sum_{x\in
V_{n}}\alpha_{\sigma_{n}(x),x},\quad\sigma_{n}\in\Phi^{V_{n}}.$ (2.7)

The vector field $\\{h_{x}\\}_{x\in V}$ in (2.6) is called _generalized
boundary condition (GBC)_.
We say that the probability distributions (2.6) are compatible (and the
intristic GBC $\\{h(x)\\}$ are _permissible_) if for each $n\in\mathbb{N}_{0}$
the following identity holds,

$\sum_{\omega\in\Phi^{W_{n+1}}}\mu^{h}_{n+1}(\sigma\vee\omega)\equiv\mu^{h}_{n}(\sigma),\quad\sigma_{n}\in\Phi^{V_{n}},$
(2.8)

where the symbol $\vee$ stands for concatenation of subconfigurations. A
criterion for permissibility of GBC is provided by Theorem 1 (see below). By
Kolmogorov’s extension theorem (see [19], chapter II, section 3, theorem 4,
page 167), the compatibility condition (2.8) ensures that there exists a
unique measure $\mu^{h}=\mu^{h}_{\beta,\alpha}$ on $\Phi^{V}$ such that, for
all $n\in\mathbb{N}_{0}$,

$\mu^{h}(\sigma_{V_{n}}=\sigma_{n})\equiv\mu^{h}_{n}(\sigma_{n}),\quad\sigma_{n}\in\Phi^{V_{n}},$
(2.9)

or more explicitly (substituting (2.6)),

$\mu^{h}_{n}(\sigma_{V_{n}}=\sigma_{n})=\frac{1}{Z_{n}}\exp\Big{\\{}-\beta
H_{n}(\sigma_{n})+\beta\sum_{x\in
W_{n}}h_{\sigma_{n}(x),x}\Big{\\}},\quad\sigma_{n}\in\Phi^{V_{n}}.$ (2.10)

Definition. Measure $\mu^{h}$ satisfying (2.9) is called a _splitting Gibbs
measure (SGM)_.
Remark 2. Note that adding a constant $c=c_{x}$ to all coordinates $h_{i,x}$
of the vector $\textbf{h}_{x}$ does not change the probability measure (2.6)
due to the normalization $Z_{n}$. The same is true for the external field
$\textbf{\textalpha}_{x}$ in the Hamiltonian (2.1). Therefore, without loss of
generality we can consider _reduced GBC_ $\tilde{\textbf{h}}_{x}$, for example
defined as

$\tilde{h}_{i,x}=h_{i,x}-h_{m,x},\quad i=0,...,m-1.$

The same remark also applies to the external field  and its reduced version
$\tilde{\textbf{\textalpha}}(x)$ defined by

$\tilde{\alpha}_{i,x}:=\alpha_{i,x}-\alpha_{m,x}\quad i=0,...,m-1.$

##  3  Compatibility criterion

In the view of Remark 2, when working with vectors and vector-valued functions
and fields it will often be convenient to pass from a generic vector
$\textbf{u}=(u_{0},...,u_{m})\in\mathbb{R}^{m+1}$ to a ’reduced vector’
$\tilde{\textbf{u}}=(\tilde{u}_{0},...,\tilde{u}_{m-1})\in\mathbb{R}^{m}$ by
setting $\tilde{u}_{i}=u_{i}-u_{m}$ $(i=0,...,m-1).$

The following general statement describes a criterion for GBC
$\\{h_{x}\\}_{x\in V}$ to guarantee compatibility of the measure
$\\{\mu^{h}_{n}\\}_{n\in\mathbb{N}_{0}}.$
Theorem 1. The probability distributions
$\\{\mu^{h}_{n}\\}_{n\in\mathbb{N}_{0}}$ defined in (2.6) are compatible (and
the underlying GBC $\\{\textbf{h}_{x}\\}_{x\in V}$ are permissible) if and
only if the following vector identity holds

$\tilde{\textbf{h}}_{x}=\tilde{\textbf{\textalpha}}_{x}+\sum_{y\in
S(x)}\textbf{F}(\tilde{\textbf{h}}_{y};\theta),\quad x\in V,$ (3.1)

here and below

$\theta=e^{\beta J}$ (3.2)

and

$\begin{split}\tilde{\textbf{h}}_{x}=(\tilde{h}_{i,x},...,\tilde{h}_{m-1,x}),\leavevmode\nobreak\
\tilde{\textbf{\textalpha}}_{x}=(\tilde{\alpha}_{i,x},...,\tilde{\alpha}_{m-1,x}),\\\
\tilde{h}_{i,x}=\beta\Big{(}h_{i,x}-h_{m,x}\Big{)}+\beta\Big{(}\alpha_{i,x}-\alpha_{m,x}\Big{)},\\\
\tilde{\alpha}_{i,x}=\beta\Big{(}\alpha_{i,x}-\alpha_{m,x}\Big{)},\quad
i=0,...,m-1,\end{split}$ (3.3)

the map
$\textbf{F}(\textbf{u};\theta)=(F_{0}(\textbf{u};\theta),...,F_{m-1}(\textbf{u};\theta))$
is defined for $\textbf{u}=(u_{0},...,u_{m-1})\in\mathbb{R}^{m}$ and
$\theta>0$ by the formulas

$F_{i}(\textbf{u};\theta):=\ln\frac{\sum_{j=0}^{m-1}\theta^{|i-j|}e^{u_{j}}+\theta^{m-i}}{\sum_{j=0}^{m-1}\theta^{m-j}e^{u_{j}}+1},\quad
i=0,...,m-1.$ (3.4)

Proof. For shorthand, denote temporarily
$\textbf{\textzeta}_{x}:=\textbf{h}_{x}+\textbf{\textalpha}_{x}.$ Suppose that
the compatibility condition (2.8) holds. On substituting (2.6), it is easy to
see that (2.8) simplifies to

$\prod_{x\in W_{n}}\prod_{y\in
S(x)}\sum_{\omega(y)\in\Phi}\exp\Big{\\{}\beta\Big{(}J|\sigma_{n}(x)-\omega(y)|+\zeta_{\omega(y),y}\Big{)}\Big{\\}}=\frac{Z_{n+1}}{Z_{n}}\prod_{x\in
W_{n}}\exp\\{\beta h_{\sigma_{n}(x),x}\\},$ (3.5)

for any $\sigma_{n}\in\Phi^{V_{n}}.$ Consider the equality (3.5) on
configurations $\sigma^{1}_{n}$, $\sigma^{2}_{n}\in\Phi^{V_{n}}$ that coincide
everywhere in $V_{n}$ except at vertex $x\in W_{n}$, where
$\sigma_{n}^{1}(x)=i\leq m-1$ and $\sigma^{2}_{n}(x)=m.$ Taking the log-ratio
of the two resulting relations, we obtain

$\sum_{y\in
S(x)}\ln\frac{\sum_{j\in\Phi}\exp\\{\beta(J|i-j|+\zeta_{j,y})\\}}{\sum_{j\in\Phi}\exp\\{\beta(J|m-j|+\zeta_{j,y})\\}}=\beta(h_{i,x}-h_{m,x})$
(3.6)

which is readily (3.1) in view of the notations (3.3) and (3.4). Conversely,
again using (3.3) and (3.4), equation (3.1) can be rewritten in the coordinate
form as follows,

$\prod_{y\in
S(x)}\sum_{j=0}^{m}\exp\\{\beta(J|i-j|+\zeta_{j,y})\\}=a(x)\exp\\{\beta
h_{i,x}\\},\quad i=0,...,m-1,$ (3.7)

where (omitting the immaterial dependence on $\beta,$ h and ) we denote

$a(x):=\exp\\{\beta h_{m,x}\\}\prod_{y\in
S(x)}\sum_{j=0}^{m}\exp\\{\beta(J|m-j|+\zeta_{j,y})\\},\quad x\in V.$

Hence, using (3.7) and setting $A_{n}:=\prod_{x\in W_{n}}a(x),$ we get

$\begin{split}&\sum_{\omega\in\Phi^{W_{n+1}}}\mu^{h}_{n+1}(\sigma\vee\omega)=\frac{\exp\\{-\beta
H_{n}(\sigma_{n})\\}}{Z_{n+1}}\prod_{y\in W_{n}}\prod_{y\in
S(x)}\sum_{j=0}^{m}\exp\\{\beta(J|\sigma_{n}(x)-j|+\zeta_{j,y})\\}=\\\
&=\frac{A_{n}}{Z_{n+1}}\exp\Big{\\{}-\beta H_{n}(\sigma_{n})+\beta\sum_{x\in
W_{n}}h_{\sigma_{n(x),x}}\Big{\\}}=\frac{A_{n}Z_{n}}{Z_{n+1}}\mu^{h}_{n}(\sigma_{n}).\end{split}$
(3.8)

Finally, observe that

$\sum_{\sigma_{n}\in\Phi^{V_{n}}}\sum_{\omega\in\Phi^{W_{n+1}}}\mu^{h}_{n+1}(\sigma_{n}\vee\omega)=$
$=\sum_{\sigma_{n+1}\in\Phi^{V_{n+1}}}\mu^{h}_{n+1}(\sigma_{n+1})=1$

whereas from the RHS of (3.8) the same sum is given by

$\sum_{\sigma_{n}\in\Phi^{V_{n}}}\frac{A_{n}Z_{n}}{Z_{n+1}}\mu^{h}_{n}(\sigma_{n})=\frac{A_{n}Z_{n}}{Z_{n+1}}.$

Hence, $A_{n}Z_{n}/Z_{n+1}=1$ and formula (3.8) yields (2.8), as required.
This completes the proof of Theorem 1.

Proposition 1. Any measure $\mu$ with local distributions $\mu^{(h)}_{n}$
satisfying (2.6),(2.8) is an SGM.

Proof. Straightforward.

##  4  Translation-Invariant SGMs.

From Proposition 1 it follows that for any
$\tilde{\textbf{h}}=\\{\tilde{\textbf{h}}_{x},x\in V\\}$ satisfying (3.1)
there exists a unique GM $\mu$ (with restriction $\mu^{(h)}_{n}$ as in (2.6))
and vice versa. We suppose that the number spin values $m+1$ is 3, i.e. $m=2$
and $\Phi=\\{0,1,2\\}.$

For $m=2$, it follows from (3.1) that

$\left\\{\begin{array}[]{ll}\tilde{h}_{0,x}=\tilde{\alpha}_{0,x}+\sum\limits_{y\in
S(x)}\ln\frac{\exp{(\tilde{h}_{0,y})}+\theta\exp{(\tilde{h}_{1,y})}+\theta^{2}}{\theta^{2}\exp{(\tilde{h}_{0,y})}+\theta\exp{(\tilde{\tilde{h}}_{1,y})}+1},\\\\[14.22636pt]
\tilde{h}_{1,x}=\tilde{\alpha}_{1,x}+\sum\limits_{y\in
S(x)}\ln\frac{\theta\exp{(\tilde{h}_{0,y})}+\exp{(\tilde{h}_{1,y})}+\theta}{\theta^{2}\exp{(\tilde{h}_{0,y})}+\theta\exp{(\tilde{h}_{1,y})}+1}.\\\
\end{array}\right.$ (4.1)

It is natural to begin with translation-invariant solutions (4.1), i.e. to
assume that
$\tilde{\textbf{h}}_{x}=\tilde{\textbf{h}}\in\mathbb{R}^{2},\leavevmode\nobreak\
\forall x\in V.$ We suppose that the external field
$\tilde{\textbf{\textalpha}}_{x}$ is also translation-invariant, i.e.
$\tilde{\textbf{\textalpha}}_{x}=\tilde{\textbf{\textalpha}}=(\tilde{\alpha}_{0},\tilde{\alpha}_{1}),\leavevmode\nobreak\
\forall x\in V$ where $\tilde{\alpha}_{i}\in\mathbb{R},\leavevmode\nobreak\
i=0,1$. For the sake of simplicity, we restrict ourselves to the case
$\tilde{\alpha}_{0}=0.$ Set $z_{0}=\exp{(\tilde{h}_{0,x})},$
$z_{1}=\exp(\tilde{h}_{1,x})$ and $\lambda=\exp(\tilde{\alpha}_{1,x}),$ $x\in
V.$ From (4.1) we have

$z_{0}=\Big{(}\frac{z_{0}+\theta z_{1}+\theta^{2}}{\theta^{2}z_{0}+\theta
z_{1}+1}\Big{)}^{k},$ (4.2) $z_{1}=\lambda\Big{(}\frac{\theta
z_{0}+z_{1}+\theta}{\theta^{2}z_{0}+\theta z_{1}+1}\Big{)}^{k}.$ (4.3)

Remark 3. The case $\lambda=1$ is studied in [6].

Observe that $z_{0}=1$ satisfies the equation (4.2) independently of
$k,\theta,z_{1}$ and $\lambda.$ Putting $z_{0}=1$ into (4.3), we obtain

$z_{1}=\lambda\Big{(}\frac{2\theta+z_{1}}{\theta^{2}+\theta
z_{1}+1}\Big{)}^{k}.$ (4.4)

Set

$a=a(\theta):=\frac{2\theta^{k+1}}{\lambda},\quad
b=b(\theta):=\frac{1+\theta^{2}}{2\theta^{2}},\quad x=\frac{z_{1}}{2\theta}.$
(4.5)

From (4.4) we get:

$ax=\left(\frac{1+x}{b+x}\right)^{k}.$ (4.6)

The Eq. (4.6) is well known in the theory of Markov chains on the Cayley tree
(see, e.g., [1], Proposition 10.7), and it is easy to analyse the number of
its positive solutions. According to Proposition 10.7 of [1] we have the
following

Lemma 1. Denote $b_{0}=b_{0}(k):=\left(\frac{k+1}{k-1}\right)^{2}$ and let
$\nu\equiv\nu(a,b,k)$ be the number of solutions of the Eq. (4.6) with $a,b>0$
and $k\geq 2.$

(i) If $b\leq b_{0}$ then for all $a>0$ the Eq. (4.6) has a unique solution,
i.e. $\nu=1.$

(ii) For $b>b_{0}$ set

$a_{i}=a_{i}(b,\theta):=\frac{1}{x_{i}}\Big{(}\frac{1+x_{i}}{b+x_{i}}\Big{)}^{k},\quad
i=1,2,$ (4.7)

with $0<a_{1}<a_{2}$ where $0<x_{1}(b,\theta)<x_{2}(b,\theta)$ are the
distinct solutions of the following quadratic equation

$x^{2}+[2-(b-1)(k-1)]x+b=0,$ (4.8)

such that

$x_{1}=\frac{(b-1)(k-1)-2-\sqrt{D(b,k)}}{2},$ (4.9)

and

$x_{2}=\frac{(b-1)(k-1)-2+\sqrt{D(b,k)}}{2},$ (4.10)

where $D(b,k):=\Big{(}2-(b-1)(k-1)\Big{)}^{2}-4b=(b-1)(k-1)^{2}(b-b_{0}).$
Then the number of solution $\nu$ is defined as follows

$\nu=\left\\{\begin{array}[]{lll}1\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{if}\leavevmode\nobreak\ \leavevmode\nobreak\
0<a<a_{1}\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\ a>a_{2},\\\
2\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
\leavevmode\nobreak\ a=a_{1}\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
a=a_{2},\\\ 3\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{if}\leavevmode\nobreak\ \leavevmode\nobreak\ a_{1}<a<a_{2}.\\\
\end{array}\right.$ (4.11)

It is easily seen from (4.5) that the condition $b>b_{0}$ implies that

$\theta<\theta_{c}\equiv\theta_{c}(k)=\frac{k-1}{\sqrt{k^{2}+6k+1}}.$ (4.12)

If $b>b_{0}$ then there are one, two or three solutions according as
$\lambda\notin[\lambda_{1}^{*},\lambda_{2}^{*}],\leavevmode\nobreak\
\lambda\in\\{\lambda_{1}^{*},\lambda_{2}^{*}\\}$ or
$\lambda\in(\lambda_{1}^{*},\lambda_{2}^{*}),$ respectively, where

$\lambda_{1}^{*}=\lambda_{1}^{*}(\theta):=\frac{2\theta^{k+1}}{a_{2}}\quad\mbox{and}\quad\lambda_{2}^{*}=\lambda_{2}^{*}(\theta):=\frac{2\theta^{k+1}}{a_{1}}.$
(4.13)

By $card(GM)_{TI}$ we denote the number of translation-invariant splitting
Gibbs measures (TISGMs) for the SOS model with external field.

Theorem 2. Let $k\geq 2.$ For the system (4.2),(4.3) the following assertions
hold

$card(GM)_{TI}\geq\left\\{\begin{array}[]{lll}1\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
\theta\geq\theta_{c}\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
\theta<\theta_{c}\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\lambda\notin[\lambda_{1}^{*},\lambda_{2}^{*}],\\\ 2\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
\theta<\theta_{c}\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\lambda\in\\{\lambda_{1}^{*},\lambda_{2}^{*}\\},\\\ 3\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\
\theta<\theta_{c}\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\lambda\in(\lambda_{1}^{*},\lambda_{2}^{*}),\\\ \end{array}\right.$ (4.14)

where $\theta_{c}$ is given in (4.12) and
$\lambda^{*}_{1,2}=\lambda^{*}_{1,2}(\theta)$ and defined by (4.13).

Proposition 2. If $J\geq 0\leavevmode\nobreak\ (\theta\geq
1),\leavevmode\nobreak\ \lambda>0$ then the system of equations (4.2),(4.3)
has a unique solution.

Proof. Let $E=z_{0}+\theta z_{1}+\theta^{2},\leavevmode\nobreak\
F=\theta^{2}z_{0}+\theta z_{1}+1,$ then from (4.2) we have:

$(z_{0}-1)\Big{[}F^{k}+(\theta^{2}-1)(E^{k-1}+\ldots+F^{k-1})\Big{]}=0.$
(4.15)

Since $\theta\geq 1(J\geq 1),$ we deduce from (4.15) that $z_{0}=1$ is the
only solution. Then $b=\frac{1+\theta^{2}}{2\theta^{2}}\leq
1<\left(\frac{k+1}{k-1}\right)^{2}.$ By Lemma 1, the Eq. (4.4) has a unique
solution. Thus we have proved that the system (4.2), (4.3) has a unique
solution.

By Proposition 2 we have

Corollary 1. If $\theta\geq 1$ then there is a unique TISGM.

##  5  Full analysis of solutions of the system (4.2), (4.3) in the case $k=2$

Assuming $k=2$ the two-dimensional fixed point of the system (4.2), (4.3) for
the two components of the boundary law can be written in terms of the
convenient variables $x=\sqrt{z_{0}}$, $y=\sqrt{z_{1}}$ and
$\sqrt{\lambda}=\zeta$ in the form

$x=\frac{x^{2}+\theta y^{2}+\theta^{2}}{\theta^{2}x^{2}+\theta y^{2}+1},$
(5.1) $y=\zeta\frac{\theta x^{2}+y^{2}+\theta}{\theta^{2}x^{2}+\theta
y^{2}+1}.$ (5.2)

Remark 4. The case $\lambda=1$, that is $\zeta=1$, is analyzed in [7].

From the Eq. (5.1) we have $x=1$ or

$\theta y^{2}=(1-\theta^{2})x-\theta^{2}(x^{2}+1).$ (5.3)

Remark 5. Rearranging the RHS of (5.3) we obtain that $\theta
y^{2}=x(1-\theta^{2}(x+\frac{1}{x}+1))$. Since $x>0$ and $x+\frac{1}{x}\geq 2$
we have that the equality (5.3) can hold if
$\theta<\theta^{\prime}_{c}=\frac{1}{\sqrt{3}}.$

5.1 Case: $x=1$

In this case from the Eq. (5.2) we get

$\theta y^{3}-\zeta y^{2}+(\theta^{2}+1)y-2\zeta\theta=0.$ (5.4)

According to the Descartes’ Rule of Signs, the Eq. (5.4) has not any negative
root, has at least one positive root and has at most three positive roots.
Denote

$a=-\frac{\zeta}{\theta},\leavevmode\nobreak\
b=\frac{\theta^{2}+1}{\theta},\leavevmode\nobreak\ c=-2\zeta.$

We calculate the discriminant of (5.4) as in [32] taking into account
$\zeta=\sqrt{\lambda}$

$\begin{split}\Delta^{\prime}(\theta,\lambda):=-\Delta(\theta,\lambda)=4a^{3}c-a^{2}b^{2}-18abc+4b^{3}+27c^{2}=\\\
=\frac{1}{\theta^{4}}\Big{(}8\theta\lambda^{2}+(71\theta^{4}-38\theta^{2}-1)\lambda+4\theta^{7}+12\theta^{5}+12\theta^{3}+4\theta\Big{)}.\end{split}$
(5.5)

It is known (see [32], Theorem 4.3.8) that if $\Delta^{\prime}>0$ then the Eq.
(5.4) has one real root and two imaginary roots. If $\Delta^{\prime}=0$ then
the Eq. (5.4) has three real roots, at least two of which are equal. If
$\Delta^{\prime}<0$ then the Eq. (5.4) has three distinct real roots. Note
that $\Delta^{\prime}(\cdot)$ is a quadratic function with respect to
$\lambda.$ Denote

$A=A(\theta):=\frac{8\theta}{\theta^{4}},\leavevmode\nobreak\
B=B(\theta):=\frac{71\theta^{4}-38\theta^{2}-1}{\theta^{4}},\leavevmode\nobreak\
C=C(\theta):=\frac{4\theta^{7}+12\theta^{5}+12\theta^{3}+4\theta}{\theta^{4}},$

and $\theta_{1}\approx 0.7486$. Note that $A>0,\leavevmode\nobreak\ C>0$ for
all $\theta>0$ and $B\geq 0$ if $\theta\geq\theta_{1}$ and $B<0$ if
$\theta<\theta_{1}$. It is easy to see that if $B\geq 0$ then
$\Delta^{\prime}>0$ for all $\lambda>0.$ Therefore, we consider only the case
$B<0$, that is $\theta<\theta_{1}.$ We calculate the discriminant of quadratic
function (5.5)

$D=D(\theta):=B^{2}-4AC=\frac{(1-\theta)(1+\theta)(1-17\theta^{2})^{3}}{\theta^{8}}.$
(5.6)

Denote $\theta_{2}=\sqrt{\frac{1}{17}}\approx 0.2425$. It is easy to verify
that $D\geq 0$ if $\theta\leq\theta_{2}$ and $D<0$ if
$\theta_{2}<\theta<\theta_{1}.$

Consider the following cases

Case $0<\theta<\theta_{2}.$ In the case $D>0$ and $B<0$ then we solve (5.5)
with respect to $\lambda$

$\lambda_{1}(\theta)=\frac{-71\theta^{4}+38\theta^{2}+1-\sqrt{(1-\theta)(1+\theta)(1-17\theta^{2})^{3}}}{16\theta},$
(5.7)
$\lambda_{2}(\theta)=\frac{-71\theta^{4}+38\theta^{2}+1+\sqrt{(1-\theta)(1+\theta)(1-17\theta^{2})^{3}}}{16\theta}.$
(5.8)

Note that $\lambda_{i}(\theta)>0,\leavevmode\nobreak\ i=1,2.$ It is easy to
see that if
$\lambda\in(0,\lambda_{1}(\theta))\bigcup(\lambda_{2}(\theta),\infty)$ then
$\Delta^{\prime}>0,$ if $\lambda=\lambda_{1}(\theta)$ or
$\lambda=\lambda_{2}(\theta)$ then $\Delta^{\prime}=0,$ if
$\lambda\in(\lambda_{1}(\theta),\lambda_{2}(\theta))$ then
$\Delta^{\prime}<0.$

Case $\theta=\theta_{2}$. In this case $D=0$ and $B<0$, then from (5.5) we
have

$\tilde{\lambda}:=\lambda_{1}(\theta_{2})=\frac{-71\theta_{2}^{4}+38\theta_{2}^{2}+1}{16\theta_{2}}=\frac{54\sqrt{17}}{289}\approx
0.7704.$ (5.9)

It follows that if $\theta=\theta_{2}$ and $\lambda=\tilde{\lambda}$ then
$\Delta^{\prime}=0,$ if $\theta=\theta_{2}$ and $\lambda\neq\tilde{\lambda}$
then $\Delta^{\prime}>0.$ Moreover, substituting $\sqrt{\tilde{\lambda}}$ and
$\theta_{2}$ into the Eq. (5.4) we obtain that the Eq. (5.4) has one three-
fold solution.

Case $\theta_{2}<\theta<\theta_{1}$. In this case $D<0$ and $B<0$ then we get
$\Delta^{\prime}>0.$

Combining above results we summarise

Lemma 2. There exists a unique $\theta_{2}=\frac{1}{\sqrt{17}}$ such that

  * •

If $\theta\geq\theta_{2}$ or $\theta<\theta_{2}$ and
$\lambda\notin[\lambda_{1}(\theta),\lambda_{2}(\theta))]$ then Eq. (5.4) has
one positive solution $y_{1}>0$

  * •

If $\theta<\theta_{2}$ and
$\lambda\in\\{\lambda_{1}(\theta),\lambda_{2}(\theta)\\}$ then Eq. (5.4) has
two positive solutions $y_{2}<y_{1}$

  * •

If $\theta<\theta_{2}$ and
$\lambda\in(\lambda_{1}(\theta),\lambda_{2}(\theta))$ then Eq. (5.4) has three
positive solutions $y_{3}<y_{2}<y_{1}$.

where $\lambda_{1}(\theta)$ and $\lambda_{2}(\theta)$ are given by (5.7) and
(5.8), respectively.

Remark 6. Note that Lemma 2 is a particular case of Theorem 2.

5.2 Case: (5.3) is satisfied

By Remark 5 we should only consider the case
$\theta<\theta^{\prime}_{c}=\frac{1}{\sqrt{3}}.$ The Eq. (5.2) can be written
as

$y^{2}=\zeta^{2}\Big{(}\frac{\theta x^{2}+y^{2}+\theta}{\theta^{2}x^{2}+\theta
y^{2}+1}\Big{)}^{2}.$ (5.10)

In the case when the equality (5.3) is satisfied then from the Eq. (5.10) we
get

$((1-\theta^{2})x-\theta^{2}(x^{2}+1))\theta=\zeta^{2}\Big{(}\frac{x}{x+1}\Big{)}^{2}$

which is equivalent to

$\theta^{3}x^{4}+\theta(3\theta^{2}-1)x^{3}+(4\theta^{3}+\zeta^{2}-2\theta)x^{2}+\theta(3\theta^{2}-1)x+\theta^{3}=0.$
(5.11)

Denoting $\xi=x+1/x$ from (5.11) we have

$\theta^{3}\xi^{2}+\theta(3\theta^{2}-1)\xi+2\theta^{3}-2\theta+\zeta^{2}=0.$
(5.12)

Note that the Eq. (5.12) is a quadratic equation with respect to $\xi.$ We
calculate its discriminant taking into account that $\zeta=\sqrt{\lambda}$

$D^{\prime}=D^{\prime}(\theta,\lambda):=\theta^{2}(\theta^{4}+2\theta^{2}+1-4\theta\lambda).$
(5.13)

The Eq. (5.12) has no solution if $D^{\prime}<0$; it has a unique solution if
$D^{\prime}=0$ and two solutions if $D^{\prime}>0.$ We note that
$D^{\prime}=0$ has a unique solution with respect to $\lambda$

$\lambda_{3}(\theta):=\frac{\theta^{4}+2\theta^{2}+1}{4\theta}.$ (5.14)

Thus we have the following

  * •

if $\lambda<\lambda_{3}(\theta)$ then the Eq. (5.12) has two solutions
$\xi_{1}<\xi_{2}$ with

$\xi_{1}=\xi_{1}(\theta,\lambda):=\frac{1-3\theta^{2}-\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}},$
(5.15)
$\xi_{2}=\xi_{2}(\theta,\lambda):=\frac{1-3\theta^{2}+\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}};$
(5.16)

  * •

if $\lambda=\lambda_{3}(\theta)$ then the Eq. (5.12) has a unique solution
$\xi_{1}=\xi_{2}=\frac{1-3\theta^{2}}{2\theta^{2}};$

  * •

if $\lambda>\lambda_{3}(\theta)$ then the Eq. (5.12) has no solution.

In accordance with $\xi=x+1/x$, the condition $\xi\geq 2$ must be satisfied.
Let $\lambda=\lambda_{3}(\theta)$. In the case we should solve the following
inequality

$\xi_{1}=\xi_{2}=\frac{1-3\theta^{2}}{2\theta^{2}}\geq 2.$ (5.17)

Denote $\theta_{3}=\sqrt{\frac{1}{7}}\approx 0.3780$. It follows from (5.17)
that if $\theta=\theta_{3}$ then $\xi_{1}=\xi_{2}=2$ and if
$\theta<\theta_{3}$ then $\xi_{1}=\xi_{2}>2.$

Let $\lambda<\lambda_{3}(\theta).$ Then the following cases arise

Case $\xi_{1}<2,\leavevmode\nobreak\ \xi_{2}=2$. In this case we should solve

$\left\\{\begin{array}[]{ll}\xi_{1}=\frac{1-3\theta^{2}-\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}<2,\\\\[14.22636pt]
\xi_{2}=\frac{1-3\theta^{2}+\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}=2.\\\
\end{array}\right.$ (5.18)

Simplifying (5.18) we have

$\left\\{\begin{array}[]{ll}\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}>1-7\theta^{2},\\\\[17.07182pt]
\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}=7\theta^{2}-1.\\\
\end{array}\right.$ (5.19)

It follows that the second part of (5.19) has not any solution if
$\theta\leq\theta_{3}.$ For $\theta>\theta_{3}$, solving the second part of
(5.19) we get

$\lambda_{4}(\theta):=4\theta(1-3\theta^{2}).$ (5.20)

This solution also satisfies the first part of (5.19). Consider the following
difference

$\lambda_{3}(\theta)-\lambda_{4}(\theta)=\frac{\theta^{4}+2\theta^{2}+1}{4\theta}-4\theta(1-3\theta^{2})=\frac{(1-7\theta^{2})^{2}}{4\theta}.$
(5.21)

It is easily verified that if $\theta=\theta_{3}$ then
$\lambda_{3}(\theta)=\lambda_{4}(\theta)$, and if $\theta\neq\theta_{3}$ then
$\lambda_{3}(\theta)>\lambda_{4}(\theta).$

Case $\xi_{1}=2,\leavevmode\nobreak\ \xi_{2}>2$. In this case we should solve

$\left\\{\begin{array}[]{ll}\xi_{1}=\frac{1-3\theta^{2}-\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}=2\\\\[17.07182pt]
\xi_{2}=\frac{1-3\theta^{2}+\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}>2\\\
\end{array}\right.$ (5.22)

From (5.22) we obtain

$\left\\{\begin{array}[]{ll}\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}=1-7\theta^{2},\\\\[11.38092pt]
\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}>7\theta^{2}-1.\\\
\end{array}\right.$ (5.23)

It is obvious that the first part of (5.23) has not any solution if
$\theta\geq\theta_{3}.$ For $\theta<\theta_{3}$, solving the first part of
(5.23) we get $\lambda=\lambda_{4}(\theta).$ This solution also satisfies the
second part of (5.23).

Case $\xi_{1}>2,\leavevmode\nobreak\ \xi_{2}>2.$ Since $\xi_{1}<\xi_{2}$ it
suffices to solve the following inequality

$\xi_{1}=\frac{1-3\theta^{2}-\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}>2.$
(5.24)

After simple algebra we have

$\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}<1-7\theta^{2}.$ (5.25)

Let $\theta<\theta_{3}$, then from (5.25) we get
$\lambda>\lambda_{4}(\theta).$ For $\theta\geq\theta_{3}$ the inequality
(5.25) has no solution.

Case $\xi_{1}<2,\leavevmode\nobreak\ \xi_{2}>2.$ In this case we should solve

$\left\\{\begin{array}[]{ll}\xi_{1}=\frac{1-3\theta^{2}-\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}<2,\\\\[19.91684pt]
\xi_{2}=\frac{1-3\theta^{2}+\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}}{2\theta^{2}}>2.\\\
\end{array}\right.$ (5.26)

Simplifying (5.26) we get

$\left\\{\begin{array}[]{ll}\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}>1-7\theta^{2},\\\\[11.38092pt]
\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}>7\theta^{2}-1.\\\
\end{array}\right.$ (5.27)

Let $\theta<\theta_{3}$, then from the first inequality of (5.27) we get
$\lambda<\lambda_{4}(\theta).$ For $\theta<\theta_{3}$ the second inequality
is true for all $\lambda<\lambda_{3}(\theta).$ For $\theta=\theta_{3}$ the
second inequality of (5.27) holds for
$\lambda<\lambda_{3}(\theta_{3})=\frac{16\sqrt{7}}{49}\approx 0.8639.$ For
$\theta>\theta_{3}$ the first inequality of (5.27) holds for all
$\lambda<\lambda_{3}(\theta)$ and from the second inequality of (5.27) we have
$\lambda<\lambda_{4}(\theta).$

If $\xi_{1}\geq 2$ then we can find positive solutions to the Eq. (5.11)
explicitly, i.e. we have

$x_{4}=\frac{1}{2}\Big{(}\xi_{1}-\sqrt{\xi_{1}^{2}-4}\Big{)},\leavevmode\nobreak\
\leavevmode\nobreak\
x_{5}=\frac{1}{2}\Big{(}\xi_{1}+\sqrt{\xi_{1}^{2}-4}\Big{)}.$ (5.28)

Similarly, if $\xi_{2}\geq 2$ then we can find positive solutions to the Eq.
(5.11) explicitly, i.e. we have

$x_{6}=\frac{1}{2}\Big{(}\xi_{2}-\sqrt{\xi_{2}^{2}-4}\Big{)},\leavevmode\nobreak\
\leavevmode\nobreak\
x_{7}=\frac{1}{2}\Big{(}\xi_{2}+\sqrt{\xi_{2}^{2}-4}\Big{)}.$ (5.29)

Now to find corresponding $y$ we need the following

Lemma 3. For each $x\in\\{x_{4},x_{5},x_{6},x_{7}\\}$, if
$\theta<\theta^{\prime}_{c}=\frac{1}{\sqrt{3}}$ and
$\lambda\leq\lambda_{3}(\theta)$ then the RHS of (5.3) is positive, i.e.

$(1-\theta^{2})x-\theta^{2}(x^{2}+1)>0.$ (5.30)

where $\lambda_{3}(\theta)$ is given by (5.14).

_Proof._ We shall use that $x\in\\{x_{4},x_{5},x_{6},x_{7}\\}$:

$(1-\theta^{2})x-\theta^{2}(x^{2}+1)=x-\theta^{2}(x^{2}+x+1)=$
$=x\Big{(}1-\theta^{2}(x+\frac{1}{x}+1\Big{)}\Big{)}=x(1-\theta^{2}(\xi_{i}+1)),\quad
i=1,2.$

In case $i=1$ we have

$1-\theta^{2}(\xi_{1}+1)=\frac{1}{2}\Big{(}1+\theta^{2}+\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}\Big{)}$

which is positive if $\theta<\theta^{\prime}_{c}$ and
$\lambda\leq\lambda_{3}(\theta)$

For $i=2$ we get

$1-\theta^{2}(\xi_{2}+1)=\frac{1}{2}\Big{(}1+\theta^{2}-\sqrt{\theta^{4}+2\theta^{2}+1-4\theta\lambda}\Big{)}$

which is positive if $\theta<\theta^{\prime}_{c}$ and
$\lambda\leq\lambda_{3}(\theta)$. Using this lemma we can define

$y_{i}=\frac{1}{\sqrt{\theta}}\sqrt{(1-\theta^{2})x_{i}-\theta^{2}(x_{i}^{2}+1)},\quad
i=4,5,6,7.$ (5.31)

Combining above results we summarise

Lemma 4. The following assertions hold

  1. 1.

Let $\theta<\theta_{3}.$ The system (5.2), (5.3) has

    1. (a)

4 positive solutions $(x_{i},y_{i}),\leavevmode\nobreak\ i=4,5,6,7$ if
$\lambda\in(\lambda_{4}(\theta),\lambda_{3}(\theta))$

    2. (b)

3 positive solutions $(1,y_{5})\leavevmode\nobreak\
(x_{i},y_{i}),\leavevmode\nobreak\ i=6,7$ if $\lambda=\lambda_{4}(\theta)$

    3. (c)

2 positive solutions $(x_{6},y_{6}),\leavevmode\nobreak\ (x_{7},y_{7})$ if
$\lambda=\lambda_{3}(\theta)$ or
$\lambda\in\Big{(}0,\lambda_{4}(\theta)\Big{)}$

    4. (d)

no positive solution otherwise

  2. 2.

Let $\theta_{3}\leq\theta<\theta^{\prime}_{c}.$ The system (5.2), (5.3) has

    1. (a)

2 positive solutions $(x_{6},y_{6}),\leavevmode\nobreak\ (x_{7},y_{7})$ if
$\lambda\in(0,\lambda_{4}(\theta))$

    2. (b)

1 positive solution $(1,y_{6})$ if $\lambda=\lambda_{4}(\theta)$

    3. (c)

no positive solution otherwise

  3. 3.

Let $\theta\geq\theta^{\prime}_{c}.$ The system (5.2), (5.3) has not any
positive solution,

where $\theta_{3}=\frac{1}{\sqrt{7}}$,
$\theta^{\prime}_{c}=\frac{1}{\sqrt{3}},$ $x_{i}$ and
$y_{i},\leavevmode\nobreak\ i=4,5,6,7$ are given by (5.29) and (5.31),
respectively.

Remark 7. Let $x=1$ be a solution of (5.3). Then it is easy to verify that the
equations (5.3) and (5.4) have a common solution $y$ if
$\theta<\theta^{\prime}_{c}$ and $\lambda=\lambda_{4}(\theta)$.

To give an overall view of the solutions of the system (5.1), (5.2), we need
to compare the obtained external fields. The external fields
$\lambda_{1}(\theta),\leavevmode\nobreak\ \lambda_{2}(\theta)$ exist as long
as $\theta\leq\theta_{2},$ hence we consider only the case
$\theta\leq\theta_{2}.$ Denote $\theta_{4}=\sqrt{\frac{1}{19}}\approx 0.2294.$
Elementary, but cumbersome calculation shows that

  * •

If $0<\theta<\theta_{4}$ then
$\lambda_{1}(\theta)<\lambda_{4}(\theta)<\lambda_{2}(\theta)<\lambda_{3}(\theta).$

  * •

If $\theta=\theta_{4}$ then
$\lambda_{1}(\theta)<\lambda_{2}(\theta)=\lambda_{4}(\theta)<\lambda_{3}(\theta).$

  * •

If $\theta_{4}<\theta\leq\theta_{2}$ then
$\lambda_{1}(\theta)\leq\lambda_{2}(\theta)<\lambda_{4}(\theta)<\lambda_{3}(\theta).$

Summarizing we get the full characterization of solutions:

Lemma 5. The set of solutions to the system (5.1), (5.2) changes under
variations of the parameters $\theta$ and $\lambda$. There are
$\theta_{4}(\approx 0.2294)$, $\theta_{2}(\approx 0.2425)$,
$\theta_{3}(\approx 0.3780)$, $\theta^{\prime}_{c}(\approx 0.5773),$ and
$\lambda_{j}(\theta)$, $j=\overline{1,4}$ are defined by (5.7), (5.8),(5.14)
and (5.20), respectively. We have the following assertions

  1. 1.

Let $0<\theta<\theta_{4}$ then the system (5.1), (5.2) has

    1. (a)

7 solutions $(x_{i},y_{i})$, $i=\overline{1,7}$ if
$\lambda\in(\lambda_{4}(\theta),\lambda_{2}(\theta))$

    2. (b)

6 solutions $(1,y_{1})$, $(1,y_{2}),$ $(x_{i},y_{i}),$ $i=4,5,6,7$ if
$\lambda=\lambda_{2}(\theta)$

    3. (c)

5 solutions $(1,y_{i}),$ $i=1,2,3$ and $(x_{6},y_{6})$, $(x_{7},y_{7})$ if
$\lambda\in(\lambda_{1}(\theta),\lambda_{4}(\theta)]$

    4. (d)

5 solutions $(1,y_{1}),$ $(x_{i},y_{i})$ $i=4,5,6,7$ if
$\lambda\in(\lambda_{2}(\theta),\lambda_{3}(\theta))$

    5. (e)

4 solution $(1,y_{1}),\leavevmode\nobreak\ (1,y_{2}),\leavevmode\nobreak\
(x_{6},y_{6}),\leavevmode\nobreak\ (x_{7},y_{7})$ if
$\lambda=\lambda_{1}(\theta)$

    6. (f)

3 solutions $(1,y_{1}),$ $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda\in(0,\lambda_{1}(\theta))$ or $\lambda=\lambda_{3}(\theta)$

    7. (g)

otherwise 1 solution $(1,y_{1})$ (see Fig. 1)

Figure 1. The distribution of the solutions when $\theta<\theta_{4}$.

  2. 2.

Let $\theta=\theta_{4}$ then the system (5.1), (5.2) has

    1. (a)

5 solutions $(1,y_{i}),$ $i=1,2,3$ and $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda\in(\lambda_{1}(\theta),\lambda_{2}(\theta))$

    2. (b)

5 solutions $(1,y_{1}),$ $(x_{i},y_{i}),$ $i=4,5,6,7$ if
$\lambda\in(\lambda_{2}(\theta),\lambda_{3}(\theta))$

    3. (c)

4 solutions $(1,y_{1})$, $(1,y_{2}),$ $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda=\lambda_{1}(\theta)$ or $\lambda=\lambda_{2}(\theta)$

    4. (d)

3 solutions $(1,y_{1}),$ $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda=\lambda_{3}(\theta)$ or $\lambda\in(0,\lambda_{1}(\theta))$

    5. (e)

otherwise 1 solution $(1,y_{1})$ (see Fig. 2)

Figure 2. The distribution of the solutions when $\theta=\theta_{4}$.

  3. 3.

Let $\theta_{4}<\theta<\theta_{2}$ then the system (5.1), (5.2) has

    1. (a)

5 solutions $(1,y_{i}),$ $i=1,2,3$ and $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda\in(\lambda_{1}(\theta),\lambda_{2}(\theta))$

    2. (b)

5 solutions $(1,y_{1}),$ $(x_{i},y_{i}),$ $i=4,5,6,7$ if
$\lambda\in(\lambda_{4}(\theta),\lambda_{3}(\theta))$

    3. (c)

4 solutions $(1,y_{1})$, $(1,y_{2}),$ $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda=\lambda_{1}(\theta)$ or $\lambda=\lambda_{2}(\theta)$

    4. (d)

3 solutions $(1,y_{1}),$ $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda=\lambda_{3}(\theta)$ or $\lambda=(0,\lambda_{1}(\theta))$ or
$\lambda\in(\lambda_{2}(\theta),\lambda_{4}(\theta)]$

    5. (e)

otherwise 1 solution $(1,y_{1})$ (see Fig. 3)

Figure 3. The distribution of the solutions when
$\theta_{4}<\theta<\theta_{2}$.

  4. 4.

Let $\theta_{2}\leq\theta<\theta_{3}$ then the system (5.1), (5.2) has

    1. (a)

5 solutions $(1,y_{1}),$ $(x_{i},y_{i}),$ $i=4,5,6,7$ if
$\lambda\in(\lambda_{4}(\theta),\lambda_{3}(\theta))$

    2. (b)

3 solutions $(1,y_{1},)$ $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda=\lambda_{3}(\theta)$ or $\lambda\in(0,\lambda_{4}(\theta)]$

    3. (c)

otherwise 1 solution $(1,y_{1})$ (see Fig. 4)

Figure 4. The distribution of the solutions when
$\theta_{2}\leq\theta<\theta_{4}$.

  5. 5.

Let $\theta_{3}\leq\theta<\theta^{\prime}_{c}$ then the system (5.1), (5.2)
has

    1. (a)

3 solutions $(1,y_{1})$, $(x_{6},y_{6}),$ $(x_{7},y_{7})$ if
$\lambda\in(0,\lambda_{4}(\theta))$

    2. (b)

otherwise 1 solution $(1,y_{1})$ (see Fig. 5)

Figure 5. The distribution of the solutions when
$\theta_{3}\leq\theta<\theta^{\prime}_{c}$.

  6. 6.

Let $\theta\geq\theta^{\prime}_{c}$ then the system (5.1), (5.2) has 1
solution $(1,y_{1})$ for all $\lambda>0$

where $y_{i},$ $i=1,2,3$ are solutions of the Eq. (5.4) which can be given by
Cardano’s formula, $x_{1}=x_{2}=x_{3}=1,$ $x_{i}$ and $y_{i}$ for $i=4,5,6,7$
are given by formulas (5.28), (5.29) and (5.31).

We denote by $\mu_{i},\leavevmode\nobreak\ i=1,\ldots,7$ the TISGMs which
correspond to the solutions of the system (4.2), (4.3) for $k=2.$ We have the
following

Theorem 3. For the Three-State SOS model with external field on the Cayley
tree of order two the following assertions hold: there exist
$\theta_{4}(\approx 0.2294)$, $\theta_{2}(\approx 0.2425)$,
$\theta_{3}(\approx 0.3780)$, $\theta^{\prime}_{c}(\approx 0.5773),$ and
$\lambda_{j}(\theta)$, $j=\overline{1,4}$ are defined by (5.7), (5.8),(5.14)
and (5.20), respectively, such that

  1. 1.

There are exactly seven TISGMs $\mu_{i},\leavevmode\nobreak\ i=1,2,3,4,5,6,7$

    1. (a)

if $\theta\in(0,\theta_{4})$ and
$\lambda\in(\lambda_{4}(\theta),\lambda_{2}(\theta))$

  2. 2.

There are exactly six TISGMs $\mu_{i},\leavevmode\nobreak\ i=1,2,4,5,6,7$

    1. (a)

if $\theta\in(0,\theta_{4})$ and $\lambda=\lambda_{2}(\theta)$

  3. 3.

There are exactly five TISGMs $\mu_{i},\leavevmode\nobreak\ i=1,4,5,6,7$

    1. (a)

if $\theta\in(0,\theta_{4}]$ and
$\lambda\in(\lambda_{2}(\theta),\lambda_{3}(\theta))$

    2. (b)

if $\theta\in(\theta_{4},\theta_{3})$ and
$\lambda\in(\lambda_{4}(\theta),\lambda_{3}(\theta))$

  4. 4.

There are exactly another five such measures $\mu_{i},\leavevmode\nobreak\
i=1,2,3,6,7$

    1. (a)

if $\theta\in(0,\theta_{4})$ and
$\lambda\in(\lambda_{1}(\theta),\lambda_{4}(\theta)]$

    2. (b)

if $\theta\in[\theta_{4},\theta_{2})$ and
$\lambda\in(\lambda_{1}(\theta),\lambda_{2}(\theta))$

  5. 5.

There are exactly four such measures $\mu_{i},\leavevmode\nobreak\ i=1,2,6,7$

    1. (a)

if $\theta\in(0,\theta_{2})$ and $\lambda=\lambda_{1}(\theta)$

    2. (b)

if $\theta\in[\theta_{4},\theta_{2})$ and $\lambda=\lambda_{2}(\theta)$

  6. 6.

There are exactly three such measures $\mu_{i},\leavevmode\nobreak\ i=1,6,7$

    1. (a)

if $\theta\in(0,\theta_{2})$ and $\lambda\in(0,\lambda_{1}(\theta))$

    2. (b)

if $\theta\in(0,\theta_{3})$ and $\lambda=\lambda_{3}(\theta)$

    3. (c)

if $\theta\in(\theta_{4},\theta_{2})$ and
$\lambda\in(\lambda_{2}(\theta),\lambda_{4}(\theta)]$

    4. (d)

if $\theta\in[\theta_{2},\theta^{\prime}_{c})$ and
$\lambda\in(0,\lambda_{4}(\theta))$

    5. (e)

if $\theta\in[\theta_{2},\theta_{3})$ and $\lambda=\lambda_{4}(\theta)$

  7. 7.

Otherwise there exists a unique such measure $\mu_{1}$

Remark 8. If $\lambda=1$ then Theorem 3 coincides with Part 1 of the Theorem 1
in [7].

Acknowledgments. The authors express their deep gratitude to Professor U. A.
Rozikov for the useful advice.

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# Task-grasping from human demonstration

Daichi Saito1,2, Kazuhiro Sasabuchi1, Naoki Wake1, Jun Takamatsu1, Hideki
Koike2, and Katsushi Ikeuchi1 1Authors are with Applied Robotics Research,
Microsoft, Redmond, WA, USA<EMAIL_ADDRESS>are with
Tokyo Institute of Technology<EMAIL_ADDRESS>
###### Abstract

A challenge in robot grasping is to achieve task-grasping which is to select a
grasp that is advantageous to the success of tasks before and after grasps.
One of the frameworks to address this difficulty is Learning-from-Observation
(LfO), which obtains various hints from human demonstrations. This paper
solves three issues in the grasping skills in the LfO framework: 1) how to
functionally mimic human-demonstrated grasps to robots with limited grasp
capability, 2) how to coordinate grasp skills with reaching body mimicking, 3)
how to robustly perform grasps under object pose and shape uncertainty. A deep
reinforcement learning using contact-web based rewards and domain
randomization of approach directions is proposed to achieve such robust
mimicked grasping skills. Experiment results show that the trained grasping
skills can be applied in an LfO system and executed on a real robot. In
addition, it is shown that the trained skill is robust to errors in the object
pose and to the uncertainty of the object shape, and can be combined with
various reach-coordination.

##  I Introduction

The study of robot grasps has evolved in the industrial field. Yet, a
challenge is to achieve context-optimal grasping. The problem of selecting a
grasp that is advantageous to the success of tasks before and after grasping
is called task-grasping. The parameters that determine the optimality of a
grasp often involve a higher-level understanding of the tasks to be performed.
For example, Figure 1 (A)-left shows a case where a pick acceptable for bin-
picking is achieved with a dexterous hand, but does not fulfill the grasp in
terms of task-grasping: grasp in a way so that is easy to place the object
neatly in a household environment. The gap between conventional bin-picking
and task-grasping lies in the difficulty of understanding the higher-level
knowledge about the adjacent tasks.

One of the frameworks to address this difficulty is Learning-from-Observation
(LfO), in which a human provides manipulation instruction to a robot through a
one-shot demonstration [1, 2]. We have previously shown that the LfO framework
can be used to teach the humanoid robot implicit high-level knowledge for the
task, such as the semantic and physical constraints that exist between the
environment and the grasping object, and the reaching body posture that is
less likely to conflict with the environment [3]. Planning grasps based on
these implicit knowledge has the potential to realize task-grasping.

The LfO-based task-grasping problem can be divided into two sub-problems: 1)
developing grasping skills which is capable of leveraging the implicit
knowledge that is obtained from human-demonstrated grasps but is also robust
against stochasticity such as recognition errors, and 2) the actual obtaining
of the implicit knowledge through real time recognition of the one-shot human
demonstration.

For sub-problem 1, recent reinforcement learning techniques have been shown to
be effective for improving grasping skill robustness, as simulators can
generate various error patterns which imitate recognition errors, and the
robot can be trained under the different error conditions [4, 5, 6]. However,
while previous works have achieved robust picks, the implicit knowledge about
adjacent tasks obtained from human-demonstrated grasps cannot be leveraged for
two reasons: First, previous reward designs do not guarantee-to-mimic human
grasps especially when the robot’s fingers have limited capability. Second,
previous works have not considered the coordination between the grasp and the
reaching body (the approach direction of the grasp toward an object), due to
the assumption that the robot arm can reach from the front of the object. This
is not always the case in a complex task sequence as shown in Figure 1 (B) and
[3]. Thus, this paper focuses on solving sub-problem 1 by proposing a reward
design which guarantees to functionally mimic human grasps using a so-called
contact-web expression, and achieve the coordination with the reaching body by
training with domain randomization of approach directions.

Figure 1: (A) An example of a grasp acceptable for bin-picking but not for
task-grasping (left) versus a grasp acceptable for task-grasping (right). The
left is difficult to control the placing task of the object. (B) An example of
a grasp constrained by the reaching body. In the dual-arm operation, the left
arm is in front of the drawer to perform a pulling task, and the body cannot
grasp from the front of the object. Therefore, the grasp is constrained to
approach the object from the right.

For sub-problem 2, Kang et al. proposed a system to determine the grasp
strategy from a human demonstration [7]. However, due to the lack of solving
sub-problem 1, the grasping was vulnerable to errors in the position and
orientation of the object at runtime.

The contributions of this paper are as follows:

  * •

Contact-web based reward: Grasping skills necessary for task-grasping are
achieved by using contact-web expressions, which can distinguish grasps by
defining the desired contact locations and force directions onto an object. A
contact-web based reward in reinforcement learning is proposed to achieve
these skills with robustness.

  * •

Domain randomization of approach directions: Coordination between the grasp
and the reaching body is achieved through training-in-simulation by
randomizing the ”approach directions of the grasp toward an object.”

  * •

Task-grasping from human demonstration: The proposed grasping skills are
trained in simulation and then applied in an LfO system and executed on a real
robot. Execution results are shown with three representative grasps which
functionally mimic human grasps.

##  II Related Works

In the early years, grasping was more of an analysis research of how to
formulate stable grasps [8]. After the early years, grasping became more of a
kinematic planning problem of how to obtain stable grasps with a specific
robot hardware as well as how to reach such stable grasp states from a pre-
grasp state [9]. In these approaches, mostly, the models of the objects to
grasp were assumed to be known for calculating the kinematics. With the
development of the deep neural network techniques, several grasp detection
methods for unknown objects using convolutional neural network have been
proposed [10, 11, 12]. However, these methods were vulnerable to errors in the
estimated object pose because of the open-loop control. To solve this problem,
several end-to-end systems have been proposed to stably control gripper
directly from visual input [13, 14, 15, 16]. In addition, multi-modal grasping
methods combining vision and tactile have been proposed [17, 18]. These
methods are very promising in terms of providing a stable grasp. However, they
ignore the context of tasks and aim for the so-call bin-picking instead of
task-grasping. Thus, they cannot achieve the task-grasp along with the task
situation and environment, which can be obtained and hinted through human
demonstration for optimizing a given task sequence [3].

In parallel with the effort on stable grasping, researchers also explored the
methods to determine the optimal grasping strategy for a given task.
Cutkosky’s pioneering work proposed a taxonomy of grasping and a rule-based
determination method for a given task [19]. Kang et al. proposed a system to
determine grasping strategy from a human demonstration; they reorganize
Cutkosky’s taxonomy and derived a learning-from-observation system to
determine the grasp type [7]. However, even if the grasping is obtained, it is
vulnerable to errors in the position and orientation of the object at runtime,
and stochastic processes must be introduced. To this end, most of the recent
studies have applied reinforcement learning to train grasping skills robust
against recognition errors [6, 20, 21, 22, 4, 5]. Especially, [22, 4, 5] can
train grasping skills which mimic the human grasp shape. Yet, it is
impractical to mimic human grasp shapes when the robot’s fingers have limited
capability. In addition, the prior taxonomies are redundant for robots with
limited capability.

##  III Method

This section will first explain the representative grasps required in the LfO-
based task-grasping problem, and how they can be expressed using the contact-
web expression. Then, using these expressions, we explain how we can train and
develop robust grasping skills based on each grasp.

###  III-A defining grasps and contact-webs

Figure 2: Three representative grasps performed with a one-degree-of-freedom
gripper (top) and the corresponding contact-webs (bottom). Black circles and
arrows indicate the contact points and the desired force directions,
respectively. The black arcs illustrate the gripper.

Reproducing human grasps is essential in task-grasping, and especially when
human knowledge must be integrated to overcome the high-level decision making.
Since the human can only grasp within the possible shape producible by his or
her fingers, the number of grasps must be finite, therefore, we can define a
skill set beforehand, and the high-level decision making can decide which
skill to use.

Yet, the difficulty in defining the different grasps, is that, the definition
of the grasp must make sense to the robot. Defining grasps by the shape of the
human grasp will not make sense to robotic fingers which capabilities are
usually limited compared to the human fingers; in the worst case with only
two-opposing fingers. Due to the limited capability of the fingers, it is
impractical or even impossible to generate a finger configuration that mimics
exactly like a human grasp shape. Instead of looking at the grasps by shape,
we propose looking at grasps by functions using the closure theory
perspective. The theory generalizes to a more variety of finger
configurations, and can mimic human grasps in terms of the function of the
grasp, i.e., the requirements between the fingers and object/tools to achieve
tasks after grasping.

When we categorize grasps by closure, we are able to define three
representative grasps [23]. That is, passive-form closure, passive-force
closure, and active-force closure. The difference between form and force
closure is whether some extra force is required to be produced by the fingers
in order to resist external force such as gravity. Taking this definition
broadly, we can also say a loose-hooking against a handle (Figure 2(A)) is
also a form closure in that, the handle cannot get away, and gravity is self
compensated by the environment and does not require extra force. Some may
argue this is not a grasp, however, in terms of initiating connection with a
tool, this is common and in-fact the second most type of connections done by a
human [24], and thus cannot be ignored. The difference between passive-force
and active-force closure is whether fingertips can actively produce the force
or not. The passive closure allows a more firm grasp, while the active closure
can produce precise control against external force.

Figure 2 shows that even for a simple one-degree-of-freedom gripper, the grasp
can be distinguished in terms of the mimicked functions. Even just three,
distinguishing these is critical for task-grasping.

These grasps by closure can be defined by the relation between the grasping
object and the contact locations of the virtual fingers [25] (fingers that
coordinate in movement are counted as one virtual finger; for example, index-
middle-ring-little are four fingers but can be counted as one virtual finger
for the grasps in this paper), as well as the contact force direction produced
(which should be perpendicular to the object’s surface). The active-force
closure can be expressed as appropriately locating $N(\geq 2)$ contacts on the
object using the fingertips of the virtual-fingers. The passive-force closure
requires appropriately locating $N(\geq 3)$ contact points using the virtual
fingers and optionally also the base (e.g., palm) to form a firm closure. The
passive-form closure wraps around the grasped object by the shape of the grasp
and can add any $N(\geq 1)$ number of contacts between the object and the
form-closed grasp shape.

We call these appropriate virtual finger/base contact locations (with also
information on the desired contact force direction on each location) as
contact-webs.

###  III-B train and developing skills

Our goal is to generate robust task-grasping skills by functionally mimicking
human grasps. For the robustness, we use reinforcement learning with domain
randomization. Functional mimicking can be guaranteed if the contact-web is
fulfilled with the robot’s fingers. Therefore, we design rewards that can
guide the robot’s fingers to become close to the desired contact-web. Note
that a unified reward design about contact-webs is used for all grasps, and
each grasp only differ in the desired contact-web to become close against.

The training is done in simulation. After the training, the trained grasping
skill policy is directly applied to the real robot without any extra training.
This is achieved with the domain randomization by adding random noises to the
initial state of each training episode during the training. These random
noises are generated to be similar to the uncertainties the real robot
encounters (e.g., recognition errors). In order to apply the trained policy on
the real robot, the state space must contain information that can be acquired
by both the simulator and the real robot. In contrast, a reward is only used
during the training, so it uses information obtained in the simulator but not
necessarily obtainable directly with the real robot.

Although this is a grasp training problem, in order to coordinate with the
reaching, we define the grasping task as a two-phase problem consisting an
approach phase (getting the robot finger contacts closer to the desired
contact-web contact location from 15 cm away along the approach direction
explained in the next section) and a contact-to-grasp phase (getting the
finger contacts close to the desired contact-web contact directions). The
details of the training and reward designs are described below.

####  III-B1 state and action space

Figure 3: Illustration of the approach direction defined with zenith and
azimuth angles respective to the contact-web.

In the state space, we include the actual robot finger positions, the robot
wrist orientation, and the approach direction of the grasp (represented as
zenith, azimuth angle as in Figure 3) which is the information used to
coordinate with the reach. These states are defined respective to the contact-
web, which is known in simulation during training, but is recognized when
using the trained policy after training with the real robot (see Section IV
for details about after training). The approach direction is a fixed value
during one episode, but is initiated with different values. All other states
are updated at every control step (e.g., 100 ms) to the latest state.

In addition, we also add any available sensor feedback with the real robot
hardware, such as tactile feedback (e.g., binary values of contacted or not on
each finger). If the real hardware does not have any available sensors, states
such as the desired finger positions can be used to infer about the contact
through desired-actual deviation. Other information obtainable from simulation
(e.g., a vector-formulated contact direction) may be used for the reward
calculation, but are not added to the state as is difficult to obtain
precisely with a real robot.

For the action space, we include the finger joints of the virtual fingers, as
well as changing the wrist pose along the approach direction, i.e., the wrist
is allowed to slightly move its position but the wrist orientation is
constrained straight around the approach direction.

####  III-B2 fundamentals for the reward design

For each contact $i$ in the contact-web, a desired contact position
$\bm{c}_{i}$ and a desired contact direction $\bm{n}_{i}$ is defined. We
minimize the distance $d_{p}$ between $\bm{c}_{i}$ and the current position
$\bm{p}_{i}$ of the robot’s finger contact, but also minimize the cosine
distance $d_{f}$ so that the finger contact direction $\bm{f}_{i}$ faces
$\bm{n}_{i}$. More precisely, $d_{p}$ should be minimized by approaching the
robot’s finger contact along a line which goes through $\bm{c}_{i}$ and is
parallel to $\bm{n}_{i}$. Thus, for each time step $t$, the finger contact
should be close enough by $\delta_{t}$ to some guiding point, which point is
at a proportional distance $k_{t}$ along $\bm{n}_{i}$ from $\bm{c}_{i}$
(Figure 4):

$d_{p}(\bm{p}_{i},t)=\max(\|\bm{p}_{i}-(\bm{c}_{i}+k_{t}\bm{n}_{i})\|-\delta_{t},0).$
(1) $d_{f}(\bm{f}_{i})=\arccos(\bm{f}_{i}\cdot\bm{n}_{i}).$ (2)

Note that $\bm{c}_{i}$ and $\bm{f}_{i}$ is subject to the object’s location
and pose. During training, the exact state of the object is known in
simulation and therefore the above distances can be used for reward
calculation.

####  III-B3 the grasping reward

For each phase (approach and contact-to-grasp), a reward can be designed using
the $d_{p}$, $d_{f}$ distance metric on each virtual finger. In addition, we
can design two type of rewards: a driving reward, and an avoiding penalty.
Driving rewards encourage the agent (in this case, the robot) to either
achieve or maintain some problem goal. Avoiding penalties discourage the agent
from entering an inappropriate state such as failing to maintain a goal. We
terminate an episode if any avoiding penalties are met [26].

The approach phase can be expressed as a driving reward per finger and is
formulated as below using an upper-bound baseline $b_{p}$:

$\log(b_{p}-d_{p}(\bm{p}_{i},t)).$ (3)

The approach phase switches to a contact-to-grasp phase once a contact is
detected in simulation. The contact-to-grasp phase should achieve the target
contact direction, i.e., the finger must adjust its contact direction to face
a desired direction, while maintaining the contact position. After contacting
at iteration $T$, the finger should not deviate by more than $\delta_{T}$.
Thus, the contact reward per virtual finger is formulated as below using an
upper-bound baseline $b_{f}$:

$\begin{cases}\log(b_{f}-d_{f}(\bm{f}_{i}))+\log(b_{p}-d_{p}(\bm{p}_{i},T))&\text{if
}d_{p}<\delta_{T}\\\ -r_{1}&\text{otherwise}.\end{cases}$ (4)

The contact-to-grasp phase finishes once all finger contact directions have
converged within $\theta$.

In order to ensure that the grasp/contact phase switches, an avoiding penalty
of $-r_{2}(<-r_{1})$ is applied if the phase does not switch from an approach
phase within a certain number of iterations. Also, an avoiding penalty is
added if the switch condition is violated (e.g., loosing contact between the
finger and the object during the contact-to-grasp phase).

Figure 4: Illustration of the contact-web based reward design (for simplicity,
figure shows only one contact finger). (A) Approach phase. (B) Contact-to-
grasp phase.

####  III-B4 other reward constraints

During the training, we ensure that the object does not change much in
position by adding a penalty if the object moves too much (the object can be
slightly oriented however). In addition, the passive-form closure assumes that
the grasp-object’s (e.g., handle’s) gravity is compensated by the environment.
This means that the object is usually connected closely to some sort of
environment (e.g., door). Therefore, we train the passive-form closure in a
condition where an environmental plane is close to the grasp-object, and the
fingers must go between the plane and the grasp-object.

In addition, the active/passive-force closure must evaluate whether enough
contact force is being produced to resist against an external force. After
finishing the contact-to-grasp phase, we evaluate the resistance against the
external force that the grasp must resist against, and provide a
$-r_{3}(>-r_{1})$ penalty on failure. The external force to resist against
depends on the task sequence. For the experiment, we assume a pick which
external force is gravity.

####  III-B5 domain randomization

Domain randomization is done in simulation, and consists two stages. At every
episode in the first stage, we simulate a ”recognized contact-web” by adding
random noise less than $5mm$ translation and $1.5^{\circ}$ rotation to the
actual location of the contact-web in the simulator. Then, on a spherical
surface which center matches the center of the ”recognized contact-web,” we
randomly place the robot hand. This placing on the spherical surface is
equivalent to randomizing the approach direction and can be represented with
the azimuth and zenith angles. The state is represented respective to the
”recognized contact-web” while the rewards are calculated using the contact-
web’s actual location in the simulator.

Once the agent no longer gets penalties, we do a second-stage randomization.
At every episode in the second stage, we further randomize the object shapes
using superquadrics parameters [27] so that the policy can become robust
against object shape uncertainty. The object scale and the shape of the
object’s upper surface are randomized within the range of the YCB object
dataset [28].

##  IV Skill application in an LfO system

Figure 5: The proposed LfO system overview during a grasping demonstration.
Note that a human can demonstrate different tasks before and after the grasp,
but the figure shows only the grasping part for the scope of the paper.

In this section, we explain how to achieve task-grasping by using the trained
skills with the leveraging of the human demonstration.

The LfO system has two phases (Figure 5). In the online grasp demonstration
phase, the grasping skill appropriate for the task situation is recognized
from the human-demonstration [29, 30]. The approach direction is also
recognized [31, 2] and is passed to the trained policy as a state111We expect
that the object locations in the grasp execution phase to be roughly similar
as in the grasp demonstration phase (we are mainly targeting a task-grasping
in a human environment where we can assume the human stores/puts-back an
object mostly in similar locations: a shelf, drawer, kitchen table etc.),
therefore this approach direction is a reusable information for execution..

In the online grasp execution phase, the trained skill is applied to an
arbitrary scenery by recognizing the contact-web locations within the depth
image, and then the initial states are transformed respect to the recognized
contact-web. From the initial state, the trained skill policy decides its
first action, and then returns a new action from an updated state every 100
ms.

##  V Experiments

###  V-A Training settings

We trained the skills of three representative grasps (active-force closure,
passive-force closure, and passive-form closure) using a robotic hand with
necessary complexity. That is, simple-structured enough to experiment our
concept on functionality mimicking grasps, but complex-enough to show that the
training converges with a certain degrees-of-high-freedom.

We used the Shadow Dexterous Hand Lite which has four fingers, but we bind the
index-middle-ring finger to bend together to produce simply-structured
functional movement (two opposing virtual fingers), but still has $3$ degrees-
of-freedom for the bind fingers, and $4$ degrees-of-freedom for the thumb.

We train with $N=2$ contact points for the active-force closure, with $N=3$
including the virtual fingertips and the palm for the passive-force closure,
with $N=1$ (the virtual fingertip of the bind fingers) for the passive-form
closure.

We used the proximal policy optimization (PPO) [32] to learn the policy and
used the Microsoft Bonsai Platform222https://preview.bons.ai/ to run multiple
simulators in parallel for the training. Each finger had a binary tactile
feedback available in the states.

###  V-B Training results of the three grasping skills

Figure 6: Successful execution of the trained three grasping skills on the
Shadow Dexterous Hand Lite. (A) grasp a box using active-force closure grasp.
(B) grasp a cup using passive-force closure grasp. (C) hook on a door handle
using passive-form closure grasp.

Figure 6 shows a successful execution of the trained three grasping skills on
the real Shadow Hand Lite using the LfO pipeline described in Section IV. The
contact-web was recognized using a trained convolutional neural network (CNN).
The CNN was trained using simulated depth images with ground truth labels
using camera parameters from the real camera. The recognition results had pose
errors within the range of noise used in the policy training, and the robot
did not have the scales and shapes of the object used in the human
demonstration. The results indicate that, using the training design and domain
randomization in simulation, a robust task-grasping skill can be executed with
the real robot under recognition/object-model uncertainty.

###  V-C Robustness evaluation of the trained grasping skill

Figure 7: Successful (yellow) and failed (green) grasps in each object shape
for the active-force closure skill. $\epsilon_{2}$ represents the curvature of
the top surface of the object. (A) Tested without object pose uncertainty. (B)
Tested with 16 sampled pose uncertainty patterns.

We evaluate how well the policy was trained in terms of robustness against
object shape variance, which is critical if we would like to handle
demonstrations using objects which the robot has no prior knowledge about
(knowledge as in the precise model of the object). We focus on the policy on
the active-force closure grasp, which is considered to be most difficult of
the three representative grasps.

The experiment was carried out in the simulator to investigate the robustness
in detail. The approach direction was fixed to $(0^{\circ},0^{\circ})$. For
the scale and shape parameter, we evaluate whether the trained policy is
capable of handling the range that we targeted for the skill (same as the
range in the domain randomization in which we trained the skill). Depth was
changed by $1cm$ within $[2cm,6cm]$ and width was changed by $1cm$ within
$[6cm,10cm]$, and $\epsilon_{2}$ was set to $0$ or $1$ or $2$. Whether the
grasp was successful or not was evaluated by lifting the object after
finishing the grasp. Figure 7 (A) shows the successful scales and shapes in
yellow and failed in green. Figure 7 (B) shows the robustness when recognition
errors of the contact-webs are added on top, in which each object shape was
grasped with 16 error patterns sampled from the range $5mm$ and $1.5^{\circ}$
noise range.

The results indicate that the policy was well-trained to grasp against a large
range of objects (even under noisy conditions) despite being only a single
trained policy. When recognition errors are added, the policy fails the thick
and high-curvature shaped objects. The errors cause the fingers to easily
collide with thick objects, and slipping-at-contact with high-curvature
shapes. This indicates the areas where grasping skills may require to train a
sub-category based on the shape created by the fingertips (an encompassing
shape instead of a parallel shape) to overcome ”weak” geometries. However,
such a strategy is only applicable in the case where the robot hand has more
than two opposing fingers.

###  V-D Reach-coordination evaluation of the trained skill

We show how well our active-force closure grasp was trained against approach
variance. The experiment was done in a controlled simulation environment, and
we chose three successful objects from Section V-C but with different degrees
of difficulty. The first object (A) was the average size/shape object of the
successful grasp (depth=4, width=10, $\epsilon_{2}\simeq 0$). The second
object (B) was selected as a challenging-scale object with a small width
(depth=4, width=6, $\epsilon_{2}\simeq 0$). The third object (C) was selected
as a challenging-shape object with the upper surface shaped in a diamond
(depth=4, width=10, $\epsilon_{2}$=2), The approach zenith was changed by
$10^{\circ}$ within $[0^{\circ},60^{\circ}]$ and azimuth was change by
$15^{\circ}$ within $[-45^{\circ},45^{\circ}]$. The recognition errors were
not added.

Figure 8 shows the successful approach direction in yellow and failed in
green. The object (A) was successfully grasped with all tested approach
directions. The object (B) and (C) were successfully grasped with almost all
tested approach, but were failed when the azimuth was largely negative. The
asymmetric result on the azimuth is due to the slight asymmetrical structure
of the robot hand used in the experiment, where the root of the thumb is
located towards the robot’s index-finger; the hand is specifically designed
for the ”right arm.” The hand structure makes the approaching of the thumb to
the contact positions difficult for the largely negative azimuth, yet, this
reaching direction is rare to happen during a task if executing with the
”right arm,” which the hand was designed for. The success against approach
directions is sensitive to the design of the hand.

Figure 8: Successful (yellow) or failed (green) grasps with each approach
direction in each tested object. (A) is the object which has (depth, width,
$\epsilon_{2}$) $=$ ($4,10,\simeq 0$), (B) is the object which has
($4,6,\simeq 0$) and (C) is the object which has ($4,10,2$).

##  VI Conclusions

We proposed a method for training robust task-grasping skills using contact-
web based rewards and domain randomization of approach directions. The skills
can functionally mimic human-demonstrated grasping, and can coordinate with
the reaching body to solve the LfO-based task-grasping problem. Our results
show that our trained policies are able to execute three representative grasps
(active-force closure, passive-force closure, passive-form closure) on the
real robot. We confirmed that the trained policy is able to grasp an object
without the precise model of the object, under recognition error, and is able
to grasp from various directions.

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