# Compact stellar model in higher torsion gravitational theory

G.G.L. Nashed<EMAIL_ADDRESS>Centre for Theoretical Physics, The British
University, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt  Laboratory for
Theoretical Cosmology, Tomsk State University of Control Systems and
Radioelectronics (TUSUR), 634050 Tomsk, Russia  Kazuharu Bamba
<EMAIL_ADDRESS>Faculty of Symbiotic Systems Science, Fukushima
University, Fukushima 960-1296, Japan

(August 28, 2024)

###### Abstract

In this study, we address the issue of a spherically symmetrical interior
solution to the quadratic form of
$f\mathcal{(T)}=\mathcal{T}+\epsilon\mathcal{T}^{2}$ gravitational theory
using a physical tetrad that provides vanishing components of the off-
components of the field equation, in contrast to what exists in the current
literature. To be able to formulate the resulting differential equation in a
closed form, we employ the Krori-Barua (KB) ansatz. Using the KB spacetime
form, we derive the analytic form of the energy-density, radial, and
tangential pressures and the anisotropic form. All of these quantities are
affected by the dimensional parameter $\epsilon$, which causes them to have a
noted difference from those given in the frame of Einstein general relativity.
The derived model of this study exhibits a non-trivial form of torsion scalar,
and it also contains three constants that we drew from the matching of the
boundary condition with a line element that also features a non-trivial form
of torsion scalar. Having established the physical conditions that are needed
for any real stellar, we check our model and show in detail that it bypasses
all of these. Finally, we analyze the model’s stability utilizing the Tolman-
Oppenheimer-Volkoff equation and adiabatic index and show that our model
satisfies these.

$\mathbf{F(T)}$ gravitational theory, interior solution, compactness, energy
conditions, Tolman-Oppenheimer-Volkoff equation, adiabatic index.

###### pacs:

04.50.Kd, 04.25.Nx, 04.40.Nr

##  I Introduction

Shortly after the development of general relativity (GR), different
gravitational theories were still being followed. Among these was one given by
H. Weyl that sought to unify gravitation with electromagnetism in (1918) Weyl
(1918); an endeavor that was unsuccessful. Einstein in (1928) Einstein (2006)
tried to achieve the same goal as Weyl and he drew on the geometry of
Weitzenböck. In this geometry, one must to introduce a tetrad field as the
dynamical variable in the theory, unlike GR, whose dynamical variable is the
metric. The tetrad field consists of 16 components, which means that Einstein
thought that the additional six components from GR could prescribe the
parameters of the electromagnetic field. Nonetheless, researchers showed that
these extra six components were linked to local Lorentz invariance in the
theory Mueller-Hoissen and Nitsch (1983).

Despite the failure of Weyl and Einstein’s efforts they still supplied the
notion of gauge theories, and so the search for a gauge theory of gravitation
began in earnest O’Raifeartaigh (1997); Blagojevic and Hehl (2013). In (1979),
Hayashi and Shirafuji Hayashi and Shirafuji (1979) proposed a gravitational
theory that they termed “new GR” that was a gauge theory of the translation
group. This theory entailed three free parameters to be experimentally-
determined.

Another theory that was built on Weitzenböck geometry is the teleparallel
equivalent of GR (TEGR). The theories of TEGR and GR are commensurate with the
field equations however, at the level of actions, they differ in their total
divergence term Maluf (2013); Wu and Geng (2012). Despite this conceptual
difference, TEGR and GR do not differ experimentally. In the TEGR theory,
gravity is encoded in the torsion, with a vanishing curvature, unlike GR in
which gravity is encoded in the metric with vanishing torsion Li  _et al._
(2011); Shirafuji and Nashed (1997); Krssak (2017); Nashed (2003); Bahamonde
_et al._ (2015).

Compact objects are formed when stars exhaust all of their nuclear fuel.
Nowadays, such compact stars are of interest to scientists, especially the
class of neutron stars that are boosted by neutron degeneracy pressure against
the attraction of gravity. Meanwhile another type of compact star exists
namely the white dwarf which is boosted by electron degeneracy pressures
against gravity. It is well known that the first precise vacuum solution of GR
is that by Schwarzschild Schwarzschild (1916). Thereafter, many solutions to
investigate static compact stellar formulations were derived. The search for
interior GR solutions that describe realistic compact stellar objects became a
fertile domain to scientists. Einsteinian GR became a highly important theory
that influenced the study of compact stellar objects following the discovery
of quasars in (1960). Using the equation of state (EoS), one can analyze the
stability structure of a compact stellar object. By means of EoS, one can also
investigate the physical stellar behavior using the Tolman-Oppenheimer-Volkov
(TOV) equation, which is the general relativistic equation of stellar
constructions.

In order to study stellar configurations in GR the distribution of matter must
be assumed to be isotropic, which means that the radial and tangential
pressures are equal. However, in reality, such an assumption is not held and
one can find that the two components of radial and tangential pressures are
not equal and that the difference between them results in the anisotropy. Such
stellar configurations exhibit unequal radial and tangential pressures. In
1933 Leimatre was the first to predict anisotropic model Lemaitre (1997).
Moreover, researchers showed that to reach the maximum on the surface of a
star then, the radial pressure decreased and ultimately vanished at the center
Mak and Harko (2004a, b). Many factors must be taken into account in assuming
that a star is anisotropic; among these is a high-density regime in which
nuclear interactions are relativistically treated Ruderman (1972); Canuto
(1975). Furthermore, the existence of a solid core or 3A-type superfluid may
cause a star to become anisotropic Kippenhahn  _et al._ (2013). Another source
exists that causes stellar anisotropy namely, the strong magnetic field Tamta
and Fuloria (2017). The slow rotation can be considered a source of anisotropy
Herrera and Santos (1994). Letelier showed that the combination between
perfect and null fluids can be prescribed as anisotropic in nature Letelier
(1980). Many reasons can be taken into account that can yield the anisotropic-
like pion condensation Sawyer (1972), including the strong electromagnetic
field Usov (2004) and phase transition Sokolov (1980). Dev and Gleiser Dev and
Gleiser (2003, 2002) and Gleiser and Dev Gleiser and Dev (2004) investigated
operators that affect the pressure to be anisotropic. Researchers showed that
the effect of shear, electromagnetic field, etc $\cdots$ on self-bound systems
can be neglected if the system is anisotropic Ivanov (2010). Systems that
consists of scalar fields such as boson stars possess anisotropic qualities
Schunck and Mielke (2003). Gravastars and wormholes are considered anisotropic
models Morris and Thorne (1988); Cattoen  _et al._ (2005); DeBenedictis  _et
al._ (2006). An application of an anisotropic model to the stable
configuration of neutron stars has also been discussed Bowers and Liang (1974)
and they show that anisotropy could exhibit non-negligible effects on
equilibrium mass and surface red-shift. A fine study that describes the origin
and influence of anisotropy can be found in Chan  _et al._ (2003); Herrera and
Santos (1997). Super dense anisotropic neutron stars have also been
considered, and the conclusion that no limiting mass of such stars exits was
discussed Heintzmann and Hillebrandt (1975). The issue of the stability of
such stars was analyzed in etd  and a conclusion was drawn that such stability
is similar to that of isotropic stars. Many anisotropic models deal with
anisotropic pressure in the energy-momentum tensor in the compositions of
matter. Several exactly spherically symmetrical solutions of interior stellar
have also been developed Bayin (1982); Krori  _et al._ (1984); Bondi (1993,
1999); El Hanafy and Nashed (2016); Barreto (1993); Barreto  _et al._ (2007);
Coley and Tupper (1994); Martinez  _et al._ (1994); Singh  _et al._ (1995);
Hern??ndez  _et al._ (1999); Harko and Mak (2000); Nashed and Bamba (2018);
Shirafuji and Nashed (1997); Awad and Nashed (2017); Patel and Mehta (1995);
Lake (2004); Bohmer2006; Nashed (2011); Boehmer and Harko (2007); Esculpi  _et
al._ (2007); Shirafuji  _et al._ (1996); Khadekar and Tade (2007); Karmakar
_et al._ (2007); Abreu  _et al._ (2007a); IVANOV (2010); Herrera  _et al._
(2009); Nashed (2008); Mak and Harko (2003); Sharma and Mukherjee (2002);
Maharaj and Maartens (1989); Chaisi and Maharaj (2006); Herrera  _et al._
(2008); Chaisi and Maharaj (2005); Gokhroo and Mehra (1994); Lake (2009); V O
_et al._ (2012); Thomas and Ratanpal (2007); Elizalde  _et al._ (2020);
Tikekar and Thomas (2005); Thirukkanesh and Maharaj (2008a, b); Sharma and
Ratanpal (2013); Pandya  _et al._ (2015); Bhar  _et al._ (2015).

This paper is structured as follows: In Section II, we present the ingredient
bases of TEGR theory. In Section III, we apply the non-vacuum charged field
equations of $f\mathcal{(T)}$ theory to a non-diagonal tetrad, and we derive
the non-vanishing components of the differential equations. Then we show that
the number of the nonlinear differential equations is less than the number of
unknowns, and, therefore, we make two different assumptions, one related to
the $g_{rr}$ potential metric and the other to the anisotropy, and derive a
new solution in Section III. In Section IV, we use the matching condition to
match the constants of integration involving in the interior solution with an
exterior spherically symmetric spacetime that its torsion scalar depends on
the radial coordinate. In Section V, we discuss the physical content of the
interior solution and show that it possesses many merits that make it
physically acceptable. Among these is that it satisfies the energy and
causality conditions,. In Section VI, we study the stability using the TOV
equation and the adiabatic index and show that our interior solution exhibits
static stability. In Section VII we conclude and discuss the main results of
this study.

##  II Ingredient tools of $f(\mathcal{T})$ gravitational theories

In this section, we briefly present the $f(\mathcal{T})$ gravitational theory.
In torsional formulations of gravity using the tetrad fields $b^{i}{{}_{\mu}}$
as dynamical variables proves to be convenient, which form an orthonormal
basis for tangent space at each point of spacetime. These are related to the
metric through the following:

${\textit{g}_{\mu\nu}:=\eta_{ij}{b^{i}}_{\mu}{b^{j}}_{\nu},}$ (1)

with $\eta_{ij}=(-,+,+,+)$ being the $4$-dimensional Minkowskian metric of the
tangent space (The Greek indices are used for the coordinate space and the
Latin indices for the tangent one). The curvatureless Weitzenböck connection
is introduced as
$\overset{{\textbf{w}}}{\Gamma}^{\lambda}_{\mu\nu}:={b_{i}}^{\lambda}\leavevmode\nobreak\
\partial_{\nu}b^{i}_{\mu}$ Weitzenböck (1923), and thus the torsion tensor is
defined as the following:

${\mathcal{T}^{\alpha}}_{\mu\nu}:=\overset{{\textbf{w}}}{\Gamma}^{\alpha}_{\nu\mu}-\overset{{\textbf{w}}}{\Gamma}^{\alpha}_{\mu\nu}={b_{i}}^{\alpha}\left(\partial_{\mu}{b^{i}}_{\nu}-\partial_{\nu}{b^{i}}_{\mu}\right),$
(2)

which carries all of the information on the gravitational field. Finally, in
contracting the torsion tensor, we obtain the torsion scalar as the following:

$\mathcal{T}\equiv\frac{1}{4}\mathcal{T}^{\rho\mu\nu}\mathcal{T}_{\rho\mu\nu}+\frac{1}{2}\mathcal{T}^{\rho\mu\nu}\mathcal{T}_{\nu\mu\rho}-\mathcal{T}_{\rho\mu}^{\
\ \rho}\mathcal{T}_{\ \ \ \nu}^{\nu\mu}.$ (3)

When $\mathcal{T}$ is used as the Lagrangian in the action of teleparallel
gravity the obtained theory is the TEGR, as variation with respect to the
tetrads lead to exactly the same equations using GR.

Inspired by the $f(R)$ extensions of GR, one can extend $\mathcal{T}$ to
$f\mathcal{(T)}$, obtaining $f\mathcal{(T)}$ gravity, as determined by the
following action Cai  _et al._ (2016):

${\mathcal{L}}=\frac{1}{2\kappa}\int|b|f\mathcal{(T)}\leavevmode\nobreak\
d^{4}x,$ (4)

where $|b|=\sqrt{-g}=\det\left({b^{a}}_{\mu}\right)$ is the determinant of the
metric and $\kappa$ is a dimensional constant defined as $\kappa=8\pi$.

In this work, we seek to study an interior solution within the framework of
$f(\mathcal{T})$ gravity. Hence, in action (4) we add the Lagrangian matter
too. Therefore, the considered action is the following:

${\mathcal{L}}=\frac{1}{2\kappa}\int|b|f\mathcal{(T)}\leavevmode\nobreak\
d^{4}x+\int|b|{\mathcal{L}}_{matter}\leavevmode\nobreak\ d^{4}x,$ (5)

where ${\mathcal{L}}_{matter}$, is the Lagrangian matter.

Variation of action (5) with respect to the tetrads leads to Cai  _et al._
(2016):

$\displaystyle{\mathop{\mathcal{E}}}^{\nu}{}_{\mu}={S_{\mu}}^{\rho\nu}\partial_{\rho}\mathcal{T}f_{\mathcal{T}\mathcal{T}}+\left[b^{-1}{b^{i}}_{\mu}\partial_{\rho}\left(b{b_{i}}^{\alpha}{S_{\alpha}}^{\rho\nu}\right)-{\mathcal{T}^{\alpha}}_{\lambda\mu}{S_{\alpha}}^{\nu\lambda}\right]f_{\mathcal{T}}-\frac{f}{4}\delta^{\nu}_{\mu}+\frac{1}{2}\kappa{{{\mathfrak{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{matter}}}}}}}^{\nu}_{\mu}\equiv
0,$ (6)

with $f:=f(\mathcal{T})$, $f_{\mathcal{T}}:=\frac{\partial
f(\mathcal{T})}{\partial\mathcal{T}}$,
$f_{\mathcal{T}\mathcal{T}}:=\frac{\partial^{2}f\mathcal{(T)}}{\partial\mathcal{T}^{2}}$.
Here
${{{\mathfrak{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{matter}}}}}}}^{\nu}_{\mu}$
is the energy-momentum tensor of an anisotropic fluid defined as the
following:

${{{\mathfrak{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{matter}}}}}}}^{\mu}_{\nu}=\left(\frac{p_{t}}{c^{2}}+\rho\right)u^{\mu}u_{a}+p_{t}\delta_{a}{}^{\mu}+(p_{r}-p_{t})\xi_{a}\xi^{\mu}\,,$
(7)

where $c$ is the speed of light. Here, $u_{\mu}$ the time-like vector is
defined as $u^{\mu}=[1,0,0,0]$ and $\xi_{\mu}$ is a unit space-like vector in
the radial direction, defined as $\xi^{\mu}=[0,1,0,0]$ such that
$u^{\mu}u_{\mu}=-1$ and $\xi^{\mu}\xi_{\mu}=1$. In this paper, $\rho$ is the
energy-density and $p_{r}$ and $p_{t}$ are the radial and tangential
pressures, respectively. The above equations determine $f(\mathcal{T})$
gravity in 4-dimension space.

##  III Stellar equations in f(R) gravitational theory

We are using the following spherically symmetrical spacetime, which
demonstrates two unknown functions as111We do not discuss the issue of spin
connection in the frame of $f(T)$ gravity because its value is vanishing
identically for the tetrad (9) DeBenedictis and Ilijic (2016); Ilijic and
Sossich (2018); Bahamonde  _et al._ (2019) which we use as the main input in
this study.:

$\displaystyle
ds^{2}=-e^{\alpha(r)}dt^{2}+e^{\beta(r)}dr^{2}+r^{2}d\Omega\,,\qquad{\textrm{w}here}\qquad
d\Omega=(d\theta^{2}+\sin^{2}d\phi^{2})\,,$ (8)

where $\alpha(r)$ and $\beta(r)$ are unknown functions. The above metric of
Eq. (8) can be reproduced from the following covariant tetrad field Bahamonde
_et al._ (2019)

$b^{a}{}_{\mu}=\left(\begin{array}[]{cccc}e^{\alpha(r)/2}&0&0&0\\\
0&e^{\beta(r)/2}\cos(\phi)\sin(\theta)&r\cos(\phi)\cos(\theta)&-r\sin(\phi)\sin(\theta)\\\
0&e^{\beta(r)/2}\sin(\phi)\sin(\theta)&r\sin(\phi)\cos(\theta)&r\cos(\phi)\sin(\theta)\\\
0&e^{\beta(r)/2}\cos(\theta)&-r\sin(\theta)&0\\\ \end{array}\right)\,.$ (9)

The torsion scalar of Eq. (9) takes the form:

$\displaystyle{\mathcal{T}(r)}=\frac{2(1-e^{\beta/2})(e^{\beta/2}-1-r\alpha^{\prime})}{r^{2}e^{\beta}}\,,$
(10)

where $\alpha\equiv\alpha(r)$, $\beta\equiv\beta(r)$,
$\alpha^{\prime}=\frac{d\alpha}{dr}$,
$\alpha^{\prime\prime}=\frac{d^{2}\alpha}{dr^{2}}$ and
$\beta^{\prime}=\frac{d\beta}{dr}$. In this study, we consider the model of as

$\displaystyle f(\mathcal{T})=\mathcal{T}+1/2\epsilon\mathcal{T}^{2}\,,$ (11)

where $\epsilon$ is a dimensional parameter which has a unit of distance
squared Bahamonde  _et al._ (2019). For the tetrad (9), the non-vanishing
components of the field equations (6) when
$f(\mathcal{T})=\mathcal{T}+1/2\epsilon\mathcal{T}^{2}$ take the following
form Cai  _et al._ (2016):

$\displaystyle\frac{8\pi
G}{c^{2}}\rho=\frac{e^{-2\beta}}{r^{4}}\Big{[}2\epsilon\,e^{\beta/2}\\{2r\beta^{\prime}[2r\alpha^{\prime}+3]-4r^{2}\alpha^{\prime\prime}-r^{2}\alpha^{\prime
2}+8\\}+8\epsilon e^{3\beta/2}+4\epsilon
r^{2}\alpha^{\prime\prime}(1+e^{\beta})+e^{2\beta}(r^{2}-\epsilon)+r\beta^{\prime}\Big{\\{}e^{\beta}(r^{2}-6\epsilon)$
$\displaystyle-2r\epsilon(3+e^{\beta})\alpha^{\prime}-6\epsilon\Big{\\}}+\epsilon
r^{2}\alpha^{\prime
2}(1+e^{\beta})-e^{\beta}(r^{2}+18\epsilon)-5\epsilon\Big{]}\,,$
$\displaystyle\frac{8\pi
G}{c^{4}}p_{r}=\frac{e^{-2\beta}}{r^{4}}\Big{[}\epsilon\,e^{\beta/2}\\{12r\alpha^{\prime}+4r^{2}\alpha^{\prime
2}+8\\}+e^{2\beta}\\{\epsilon-r^{2}\\}-r^{2}\epsilon\,\alpha^{\prime
2}\\{3+e^{\beta}\\}+r\,\alpha^{\prime}\\{(r^{2}-6\epsilon)e^{\beta}-6\epsilon\\}+e^{\beta}(r^{2}-6\epsilon)-3\epsilon\Big{]}\,,$
$\displaystyle\frac{8\pi
G}{c^{4}}p_{t}=\frac{e^{-2\beta}}{4r^{4}}\Big{[}2\epsilon
e^{\beta/2}\\{4r^{2}(3+r\alpha^{\prime})\alpha^{\prime\prime}+r^{3}\alpha^{\prime
3}-2\alpha^{\prime
2}\,r^{2}(r\beta^{\prime}-3)-12r^{2}\alpha^{\prime}\beta^{\prime}-12r\beta^{\prime}-16\\}+8r\epsilon\alpha^{\prime}e^{3/2\beta}$
$\displaystyle+2r^{2}\alpha^{\prime\prime}(e^{\beta}[r^{2}-6\epsilon]-4\epsilon
r\alpha^{\prime}-6\epsilon)-4\epsilon e^{2\beta}-2\epsilon r^{3}\alpha^{\prime
3}+r^{2}\alpha^{\prime 2}(6\epsilon
r\beta^{\prime}+e^{\beta}(r^{2}-6\epsilon)-6\epsilon)-r\alpha^{\prime}\Big{(}[e^{\beta}(r^{2}-6\epsilon)-18\epsilon]r\beta^{\prime}$
$\displaystyle+2e^{\beta}(6\epsilon-r^{2})-4\epsilon\Big{)}-2r\beta^{\prime}[e^{\beta}(r^{2}-6\epsilon)-6\epsilon]+12\epsilon(e^{\beta}+1)\Big{]}\,,$
(12)

where we use the form of anisotropic energy momentum-tensor that exhibits the
form $T_{\mu}{}^{\nu}=[\rho\,c^{2},\,p_{r},\,p_{t},and\,\,\,p_{t}]$. It is
noteworthy to mention that if the dimensional parameter is $\epsilon=0$ and we
will return to the differential equations given in Roupas and Nashed (2020);
Nashed  _et al._ (2020). We define the anisotropic parameter of the stellar as
the following:

$\displaystyle\Delta(r)=\frac{8\pi
G}{c^{4}}\Big{[}p_{{}_{{}_{t}}}-p_{r}\Big{]}=\frac{e^{-2\beta}}{4r^{4}}\Big{[}2\epsilon
e^{\beta/2}\\{4r^{2}(3+r\alpha^{\prime})\alpha^{\prime\prime}+r^{3}\alpha^{\prime
3}-2\alpha^{\prime
2}\,r^{2}(r\beta^{\prime}+1)-12r\alpha^{\prime}[2+r\beta^{\prime}]-4[3r\beta^{\prime}+8]\\}$
$\displaystyle+8r\epsilon\alpha^{\prime}e^{3/2\beta}+2r^{2}\alpha^{\prime\prime}(e^{\beta}[r^{2}-6\epsilon]-4\epsilon
r\alpha^{\prime}-6\epsilon)+4(r^{2}-2\epsilon)e^{2\beta}-2\epsilon
r^{3}\alpha^{\prime 3}+r^{2}\alpha^{\prime 2}(6\epsilon
r\beta^{\prime}+e^{\beta}(r^{2}-2\epsilon)+6\epsilon)$
$\displaystyle-r\alpha^{\prime}\Big{(}[e^{\beta}(r^{2}-6\epsilon)-18\epsilon]r\beta^{\prime}+2e^{\beta}(r^{2}-6\epsilon)-28\epsilon\Big{)}-2r\beta^{\prime}[e^{\beta}(r^{2}-6\epsilon)-6\epsilon]+24\epsilon(2e^{\beta}+1)-4r^{2}e^{\beta}\Big{]}\,.$
(13)

Moreover, we introduce the characteristic density thus

$\rho_{\star}\equiv\frac{c^{2}}{4\pi GR^{4}}\,,$ (14)

which we will apply to re-scale the density, radial and tangential pressures,
to obtain the dimensionless variables:

$\tilde{\rho}=\frac{\rho}{\rho_{\star}}\,,\;\qquad\tilde{p}_{r}=\frac{p}{\rho_{\star}c^{2}}\,,\;\qquad\tilde{p}_{t}=\frac{p_{t}}{\rho_{\star}c^{2}}\,,\;\qquad\tilde{\Delta}=\frac{\Delta}{\rho_{\star}c^{2}}\,,\;$
(15)

Moreover, we re-scale the dimension parameter $\epsilon$ to put it in a
dimensionless form as:

$\varepsilon=\frac{\epsilon}{\chi^{2}R^{2}}\,,\;$ (16)

where $\chi$ is a dimensionless variable defined as

$\displaystyle\chi\equiv\frac{r}{R}\in[0,1]\,,$ (17)

where $R$ is the radius of the stellar.

Equations (III) are three nonlinear differentials in five unknowns, $\alpha$,
$\beta$, $\rho$, $p_{r}$ ,and $p_{{}_{{}_{t}}}$. Therefore, to adjust the
unknowns with the differential equations, we require two additional
conditions. These two constrains are the Krori-Barua (KB) ansatz which takes
the following form:

$\displaystyle\alpha(\chi)=f_{0}\chi^{2}+f_{1}\,,\quad\beta(\chi)=f_{2}\chi^{2}\,.$
(18)

The parameters $f_{0}$, $f_{1}$, $f_{2}$ in the ansatz (18) are dimensionless
and their value will be determined from the matching conditions on the
boundary. The KB (18) is a physical ansatz as the gravitational potentials and
their derivatives at the center are finite.

Using Eq. (18) in Eq. (III) we obtain the following:

$\displaystyle\tilde{\rho}(\chi)=\frac{e^{-2f_{2}\chi^{2}}}{\chi^{4}}\Big{[}8\varepsilon
e^{f_{2}\chi^{2}/2}[2-f_{0}\chi^{4}(f_{0}-4f_{2})-\chi^{2}(2f_{0}-3f_{2})]+8\varepsilon
e^{3f_{2}\chi^{2}/2}+e^{2f_{2}\chi^{2}}(\chi^{2}-\varepsilon)+e^{f_{2}\chi^{2}}\Big{[}2\varepsilon\Big{(}2\chi^{2}\\{2f_{0}-3f_{2}\\}$
$\displaystyle+2f_{0}\chi^{4}(f_{0}-2f_{2})-9\Big{)}+(2f_{2}\chi^{2}-1)\chi^{2}\Big{]}+\varepsilon\\{4\chi^{2}(2f_{0}-3f_{2})+4\chi^{4}f_{0}(f_{0}-6f_{2})-5\\}\Big{]}\,,$
$\displaystyle\tilde{p}_{r}(\chi)=\frac{e^{-2f_{2}\chi^{2}}}{\chi^{4}}\Big{[}8\varepsilon
e^{f_{2}\chi^{2}/2}(1+f_{0}\chi^{2}[3+2f_{0}\chi^{2}])+e^{2f_{2}\chi^{2}}(\varepsilon-\chi^{2})+e^{f_{2}\chi^{2}}(2f_{0}\chi^{4}+\chi^{2}-2\varepsilon[3+2f_{0}\chi^{2}\\{3+\chi^{2}\\}])$
$\displaystyle-3\varepsilon(2f_{0}\chi^{2}+1)^{2}\Big{]}\,,$
$\displaystyle\tilde{p}_{{}_{{}_{t}}}(\chi)=\frac{e^{-2f_{2}\chi^{2}}}{\chi^{4}}\Big{[}4\varepsilon
e^{f_{2}\chi^{2}/2}[3(f_{0}-f_{2})\chi^{2}+f_{0}(5f_{0}-6f_{2})\chi^{4}+f_{0}{}^{2}(f_{0}-2f_{2})\chi^{6}-2]+4e^{3f_{2}\chi^{2}/2}\varepsilon
f_{0}\chi^{2}-\,\varepsilon e^{2f_{2}\chi^{2}}$
$\displaystyle+e^{f_{2}\chi^{2}}[f_{0}\chi^{6}(f_{0}-f_{2})+\chi^{4}(2f_{0}-f_{2}-6\varepsilon
f_{0}[f_{0}-f_{2}])-6\varepsilon\chi^{2}(2f_{0}-f_{2})+6\varepsilon]+\varepsilon[3-4f_{0}{}^{2}\chi^{6}(f_{0}-3f_{2})-2f_{0}(7f_{0}-9f_{2})\chi^{4}$
$\displaystyle-2(2f_{0}-3f_{2})\chi^{2}]\Big{]}\,,$
$\displaystyle\Delta(\chi)=\frac{e^{-2f_{2}\chi^{2}}}{\chi^{4}}\Big{[}4\varepsilon
e^{f_{2}\chi^{2}/2}[f_{0}{}^{2}(f_{0}-2f_{2})\chi^{6}+f_{0}(f_{0}-6f_{2})\chi^{4}-3(f_{0}-f_{2})\chi^{2}-4]+4\varepsilon
f_{0}\chi^{2}e^{3f_{2}\chi^{2}/2}+(\chi^{2}-2\varepsilon)e^{2f_{2}\chi^{2}}$
$\displaystyle+e^{f_{2}\chi^{2}}\Big{(}f_{0}[f_{0}-f_{2}]\chi^{6}-\chi^{4}[f_{2}+2f_{0}\varepsilon(f_{0}-3f_{2})]-\chi^{2}(1-6\varepsilon
f_{2})+6\varepsilon\Big{)}+2\varepsilon(1+f_{0}\chi^{2})(3-2f_{0}\chi^{4}[f_{0}-3f_{2}]+\chi^{2}[3f_{2}+f_{0}])\Big{]}\,.$

Using Eq. (18) in (10) we get

$\displaystyle{\mathcal{T}(\chi)}=\frac{2e^{-f_{2}\chi^{2}}[1-e^{f_{2}\chi^{2}/2}+2f_{0}\chi^{2}][e^{f_{2}\chi^{2}/2}-1]}{\chi^{2}}\,,$
(20)

which shows that the torsion scalar has a non-trivial value and its leading
order is $O\left(\frac{1}{\chi^{2}}\right)$.

##  IV Matching conditions

Due to the fact that solution (III) exhibits a non-trivial torsion scalar, we
should therefore match it with an spherically symmetrical exterior solution
that exhibits a non-constant torsion scalar. For this goal, we are going to
match solution (III) with the neutral one presented in Bahamonde  _et al._
(2019). This spherically symmetrical solution is presented in Bahamonde  _et
al._ (2019) takes the following form:

$\displaystyle
ds^{2}=-A(\chi)dt^{2}+B(\chi)dr^{2}+r^{2}d\Omega_{2}{}^{2},\quad
A(\chi)=\mu^{2}+\varepsilon\frac{13-99\mu^{2}+128\mu^{3}-45\mu^{4}+3\mu^{6}+12(1-3\mu^{2})\ln(\mu)}{6\chi^{2}(1-\mu^{2})^{2}},$
$\displaystyle
B(\chi)=\mu^{-2}-\varepsilon\frac{1+24\mu-12\mu^{2}-64\mu^{3}+75\mu^{4}-24\mu^{5}+12\ln(\mu)}{6\chi^{2}(1-\mu^{2})},\quad\textrm{where}\quad\mu=\sqrt{1-\frac{2u}{\chi}}\,,$
(21)

where $2u<\chi$ and $u=M/R$ and $d\Omega_{2}{}^{2}$ is the line element on the
unit 2- sphere. We must match the interior spacetime metric (8) with the
exterior one given by Eq. (IV) at the boundary of the star, $r=R$ in addition
to the vanishing of the radial pressure at the surface of the star. These
boundary conditions yield the following:

$\displaystyle\alpha(r=R)=\ln\,A,\qquad\qquad\beta(r=R)=ln\,B\,,\qquad\qquad\tilde{p}_{r}=0.$
(22)

One the basis of the above conditions, we obtain a complicated form of the
parameters $f_{0}$, $f_{1}$ and $f_{2}$. The form that these parameters take
is the following numerical one when $\varepsilon=0.5$ for the white dwarf
companion pulsar $PRSJ0437-4715$ whose mass is
$1.44^{+0.07}_{-0.07}M_{\odot}$, where $1.44$ is the mean value and $\pm 0.07$
is the error and its radius $13.6^{+0.9}_{-0.8}$ km, respectively222 Here the
unit of the mass of the stellar is the mass sun and the unit of the radius is
a Kilometer.:

$\displaystyle f_{0}=0.2232587486,\qquad\qquad
f_{1}=-0.5918023605,\qquad\qquad f_{2}=.3685944011\,.$ (23)

We will use the above values of parameters $f_{0}$, $f_{1}$ and $f_{2}$ for
the reminder of this study.

##  V Terms of physical viability of the solution (III)

Now, we are going to check solution (III) to determine if it describes a real
physical stellar space. To do this, we are going to state the following
necessary conditions which are important for any actual star.

###  V.1 The energy–momentum tensor

It is known that for a true interior solution the components of the energy-
momentum tensor, energy-density, radial and transverse pressures must
demonstrate positive values. Moreover, the components of the energy-momentum
tensor must be finite at the center of the star after that they decrease
toward the surface of the stellar space and the radial pressure must be
greater than the tangential one. Figure 1 indicates the behavior of the
energy-momentum components, density, radial and transverse pressures. It is
clear from this figure that $\rho(\chi=0)_{\varepsilon=0}=0.5976904312$,
$\tilde{\rho}(\chi=0)_{\varepsilon=0.5}=1.195668990$,
$\tilde{p}_{r}(\chi=0)_{\varepsilon=0}=\tilde{p}_{t}(\chi=0)_{\varepsilon=0}=0.04553459825$,
and
$p_{r}(\chi=0)_{\varepsilon=0.5}=p_{t}(\chi=0)_{\varepsilon=0.5}=0.0922628584$.
Also, Fig. 1, shows that the density, radial and tangential pressures are
decreasing toward the stellar surface. Moreover Fig. 1 shows that the values
of the components of the energy momentum at the center in case $\varepsilon=0$
are smaller than those when $\varepsilon\neq 0$. We can also deduce from Fig.
1 that the density, radial and tangential pressures move to the surface of the
stellar more rapidly in case $\varepsilon=0$ than in the case of
$\varepsilon=0.5$. Finally, it can be seen that when $\varepsilon=0$, one
obtains $\tilde{p}=\tilde{p}_{t}$ at the center however, as we reach the
surface of the stellar space we can see that $\tilde{p}_{t}\geq\tilde{p}$.

In Fig. 2, we plot the behavior of the anisotropy and anisotropic force which
are defined as $\Delta(\chi)=p_{t}-p_{r}$, $F=\Delta(\chi)/\chi$,
respectively, for different values of $\varepsilon$. As Fig. 2 1(a) shows the
anisotropic is positive when $\varepsilon=0.5$ which means that it possesses a
repulsive, gravitational, outward force, due to the fact that $p_{t}\geq
p_{r}$ which means that it enables the formation of supermassive stars. When
$\varepsilon=0$ ,the anisotropic force becomes negative, which means that it
possesses an inward gravitational force due to the fact that $p_{r}\geq
p_{t}$.

(a) The density of solution (III)

(b) The radial pressure of solution (III)

(c) The transverse pressure of solution (III)

Figure 1: Density, radial and transverse pressures when $\varepsilon=0$ and
$\varepsilon=0.5$.

(a) The anisotropy $\Delta$ of solution (III)

(b) The anisotropic force $\Delta/\chi$ of solution (III)

Figure 2: Anisotropy $\Delta(\chi)$ and anisotropy force when $\varepsilon=0$
and $\varepsilon=0.5$.

(a) The gradient of energy-density, radial and tangential pressures of
solution (III) when $\varepsilon=0$

(b) The gradient of energy-density, radial and tangential pressures of
solution (III) when $\varepsilon=0.5$

Figure 3: Gradients of the density, radial, and transverse pressures when
$\varepsilon=0$ and $\varepsilon=0.5$

###  V.2 Causality

To show the behavior of the sound velocities we must calculate the gradient of
the energy–density, radial and transverse pressures which take the following
form:

$\displaystyle\tilde{\rho}^{\prime}=-\frac{2e^{-2f_{2}\chi^{2}}}{\chi^{5}}\Bigg{[}4\varepsilon
e^{f_{2}\chi^{2}/2}\Big{\\{}2-3f_{0}f_{2}\chi^{6}(2f_{0}-5f_{2})-12f_{2}\chi^{4}(f_{0}-f_{2})-\chi^{2}(8f_{0}-11f_{2})\Big{\\}}+e^{3f_{2}\chi^{2}/2}\Big{(}2\varepsilon[4+f_{2}\chi^{2}]$
$\displaystyle+e^{f_{2}\chi^{2}/2}[\chi^{2}-2\varepsilon]\Big{)}+e^{f_{2}\chi^{2}}\Big{\\{}\varepsilon(8f_{2}\chi^{4}[f_{0}-f_{2}]-4[3-f_{0}f_{2}(f_{0}-f_{2})\chi^{6}]+2\chi^{2}[4f_{0}-7f_{2}])-f_{2}\chi^{4}(1-2f_{2}\chi^{2})-\chi^{2}\Big{\\}}$
$\displaystyle+2\varepsilon\Big{\\{}4f_{0}f_{2}(f_{0}-4f_{2})\chi^{6}+8f_{2}\chi^{4}(f_{0}-f_{2})+\chi^{2}(4f_{0}-5f_{2})-1\Big{\\}}\Big{]}\,,$
$\displaystyle\tilde{p}^{\prime}_{r}=-\frac{2e^{-2f_{2}\chi^{2}}}{\chi^{5}}\Bigg{[}4\varepsilon
e^{f_{2}\chi^{2}/2}\Big{\\{}4+6f_{0}{}^{2}f_{2}\chi^{6}+9f_{0}f_{2}\chi^{4}+3(2f_{0}+f_{2})\chi^{2}\Big{\\}}-4e^{2f_{2}\chi^{2}}(\chi^{2}-2\varepsilon)-e^{f_{2}\chi^{2}}\Big{[}12\varepsilon+\chi^{2}\Big{\\{}2f_{0}f_{2}\chi^{4}\Big{(}2f_{0}\varepsilon$
$\displaystyle-1\Big{)}+\chi^{4}(12f_{0}f_{2}\varepsilon-
f_{2})+6\varepsilon(2f_{0}+f_{2})\Big{\\}}\Big{]}-6\varepsilon(1+2f_{0}\chi^{2})(1+f_{2}\chi^{2}[1+2f_{0}\chi^{2}])\Big{]}\,,$
$\displaystyle\tilde{p}^{\prime}_{t}=-\frac{2e^{-2f_{2}\chi^{2}}}{\chi^{5}}\Bigg{[}\varepsilon
e^{f_{2}\chi^{2}/2}\Big{\\{}3f_{0}{}^{2}f_{2}\chi^{8}(2f_{0}-3f_{2})+4f_{0}\chi^{6}(9f_{0}f_{2}-6f_{2}{}^{2}-2f_{0}{}^{2})-12f_{2}\chi^{4}(f_{2}-2f_{0})+\chi^{2}(16f_{0}-11f_{2})-4\Big{\\}}$
$\displaystyle+\varepsilon
e^{3f_{2}\chi^{2}/2}(4+2f_{0}f_{2}\chi^{4}+\chi^{2}[4f_{0}+f_{2}])-2\varepsilon
e^{2f_{2}\chi^{2}}+\chi^{2}e^{f_{2}\chi^{2}}[f_{0}f_{2}\chi^{6}(f_{0}-f_{2})+\chi^{4}\\{2f_{0}f_{2}\varepsilon(2f_{2}-3f_{0})+f_{2}(3f_{0}-f_{2})-f_{0}{}^{2}\\}$
$\displaystyle-2\varepsilon\,(f_{2}\chi^{2}+1)(7f_{0}-2f_{2})]+2\varepsilon\\{1-4f_{0}{}^{2}f_{2}\chi^{8}(f_{0}-2f_{2})+2f_{0}\chi^{6}(f_{0}{}^{2}+6f_{2}{}^{2}-9f_{0}f_{2})+2f_{2}\chi^{4}(3f_{0}-f_{2})+3\chi^{2}(f_{0}-f_{2})\\}\Bigg{]}\,,$

where $\tilde{\rho}^{\prime}=\frac{d\tilde{\rho}}{d\chi}$,
$\tilde{p}^{\prime}_{r}=\frac{d\tilde{p_{r}}}{d\chi}$ and
$\tilde{p}^{\prime}_{t}=\frac{d\tilde{p}_{t}}{dx}$. Before graphically
analyzing the gradient of the energy-momentum components given by Eq. (V.2) we
will write the asymptote form of Eq. (V.2) which yields the following:

$\displaystyle\tilde{\rho}^{\prime}\approx-
f_{2}{}^{2}\chi(4\varepsilon[f_{2}-4f_{0}]+5)+\cdots\,,\qquad\tilde{p}^{\prime}\approx-
f_{2}\chi(\varepsilon[6f_{0}f_{2}-8f_{0}{}^{2}-f_{2}{}^{2}]+4f_{0}-f_{2})+\cdots\,,$
$\displaystyle\tilde{p}^{\prime}_{t}\approx-\frac{\chi(3f_{2}\varepsilon[6f_{0}f_{2}-8f_{0}{}^{2}-f_{2}{}^{2}]-4f_{0}{}^{2}+12f_{0}f_{2}-4f_{2}{}^{2})}{2}+\cdots\,,\,,$
(25)

which gives a negative gradient of the energy-momentum components provided
that we use the values of $f_{0}$ and $f_{2}$ given by Eq. (23). The behavior
of the gradient of density, radial and tangential pressures are shown in Fig.
3 2(a) and 2(b) for $\varepsilon=0$ and $\varepsilon=0.5$.

In order to investigate the causality condition for either the radial or
transverse sound speeds $v_{r}{}^{2}$ and $v_{\bot}{}^{2}$, we must show that
the values of both of them are less than the speed of light. Using Eq. (V.2)
we obtain the asymptotic form of the radial and transverse speeds as the
following:

$\displaystyle
v_{r}{}^{2}=\frac{\tilde{p}^{\prime}_{r}}{\tilde{\rho}^{\prime}}\approx\frac{1}{12f_{2}(5-4[4f_{0}-f_{2}]\varepsilon)^{2}}\Big{[}3\varepsilon^{2}[11f_{2}{}^{3}\chi^{2}+2f_{2}{}^{2}(8-19f_{0}\chi^{2})-32f_{0}f_{2}(3+f_{0}\chi^{2})-32f_{0}{}^{2}(f_{0}\chi^{2}-4)](4f_{0}-f_{2})$
$\displaystyle-\varepsilon[30f_{0}{}^{3}\chi^{2}+f_{2}{}^{2}(108-196f_{0}\chi^{2})-f_{2}(744f_{0}-592f_{0}{}^{2}\chi^{2})-96f_{0}{}^{2}(2f_{0}\chi^{2}-13)]+60(4f_{0}-f_{2})-16(f_{0}+f_{2})f_{2}\chi^{2}\Big{]}\,,$
$\displaystyle
v_{t}{}^{2}=\frac{\tilde{p}^{\prime}_{t}}{\tilde{\rho}^{\prime}}\approx\frac{1}{24f_{2}{}^{2}(5-4[4f_{0}-f_{2}]\varepsilon)^{2}}\Big{[}57\varepsilon^{2}f_{2}{}^{5}\chi^{2}+6f_{2}{}^{4}\varepsilon[10\chi^{2}-3\varepsilon(8-47f_{0}\chi^{2})]+4f_{2}{}^{3}[4\chi^{2}-\varepsilon(93-123f_{0}\chi^{2})]$
$\displaystyle-
f_{2}{}^{2}(96\varepsilon^{2}f_{0}{}^{2}[48-73f_{0}\chi^{2}]+12\varepsilon
f_{0}[202+37f_{0}\chi^{2}]-48[6f_{0}\chi^{2}+5])+f_{2}(384\varepsilon^{2}f_{0}{}^{3}[12+5f_{0}\chi^{2}]$
$\displaystyle-96\varepsilon
f_{0}{}^{2}[41-9f_{0}\chi^{2}]+16f_{0}[45+16f_{0}\chi^{2}])+6f_{0}{}^{3}\varepsilon[131-32f_{0}\chi^{2}]-240f_{0}{}^{2}\Big{]}\,.$
(26)

Equation (V.2) shows that both $c^{2}>v_{r}{}^{2}>0$ and $c^{2}>v_{t}{}^{2}>0$
utilize the data presented in Eq. (23). The behavior of the radial and
tangential speeds of sound are shown in Fig. 4 3(a) and 3(b).

The appearance of the non–vanishing total radial force with different signs in
different regions of the fluid is called gravitational cracking when this
radial force is directed inward to the inner part of the sphere for all values
of the radial coordinate $\chi$ between the center and some value beyond which
the force reverses its direction Herrera (1994). In Abreu  _et al._ (2007b) it
is stated that a simple requirement for avoiding gravitational cracking is the
following $0<v_{r}{}^{2}-v_{t}{}^{2}<c^{2}$. In Fig. 4 3(c) we demonstrate
that solution (III) is locally stable against cracking for $\varepsilon=0.5$
but not for $\varepsilon=0$.

(a) The radial and transverse speeds of solution (III) when $\varepsilon=0$

(b) The radial and transverse speeds of solution (III) when $\varepsilon=0.5$

(c) The cracking of (III) when $\varepsilon=0$ and $\varepsilon=0.5$

Figure 4: Radial, transverse speeds of sound when $\varepsilon=0$ and
$\varepsilon=0.5$

###  V.3 Energy conditions

The energy conditions are an important test for the non–vacuum solution. The
dominant energy condition (DEC) leads to the fact that the speed of the energy
must be less than that of light. The asymptotic form of the DEC, strong energy
condition and weak energy condition take the following forms:

$\displaystyle
DEC\quad\left\\{\begin{array}[]{l}\tilde{\rho}-\tilde{p}_{r}\approx
2(2f_{2}-f_{0})-1/2\chi^{2}(2f_{2}{}^{2}+4f_{0}f_{2}-4f_{2}[2f_{0}-f_{2}]+\varepsilon
f_{2}[5f_{2}{}^{2}-22f_{0}f_{2}+8f_{0}f_{2}])+\cdots\,,\\\ \\\
\tilde{\rho}-\tilde{p}_{t}\approx
2(2f_{2}-f_{0})-1/2\chi^{2}(3f_{2}{}^{2}+2f_{0}(f_{0}+f_{2})-4f_{2}[2f_{0}-f_{2}]+\varepsilon
f_{2}[11f_{2}{}^{2}-25f_{0}f_{2}+12f_{0}f_{2}])+\cdots\,,\\\
\end{array}\right.$ (30) $\displaystyle
WEC\quad\left\\{\begin{array}[]{l}\tilde{\rho}+\tilde{p}_{r}\approx
2(f_{2}+f_{0})-1/2\chi^{2}(8f_{2}{}^{2}-4f_{0}f_{2}+4f_{2}[2f_{0}-f_{2}]+\varepsilon
f_{2}[3f_{2}{}^{2}-10f_{0}f_{2}-8f_{0}f_{2}])+\cdots\,,\\\ \\\
\tilde{\rho}-\tilde{p}_{t}\approx
2(f_{2}+f_{0})-1/4\chi^{2}(14f_{2}{}^{2}-4f_{0}(f_{0}+f_{2})+8f_{2}[2f_{0}+f_{2}]+\varepsilon
f_{2}[5f_{2}{}^{2}-14f_{0}f_{2}-24f_{0}f_{2}])+\cdots\,,\\\
\end{array}\right.$ (34) $\displaystyle
SEC\quad\tilde{\rho}-\tilde{p}_{r}-2\tilde{p}_{t}\approx
6(f_{2}-f_{0})+\chi^{2}(f_{2}{}^{2}+6(2f_{0}-f_{2})f_{2}-2f_{0}[2f_{2}-f_{0}]+4\varepsilon
f_{2}[5f_{0}f_{2}-f_{2}{}^{2}-4f_{0}f_{2}])+\cdots\,.$

All of the above energy conditions are analytically and graphically satisfied
using Eq. (23). Figure 5 shows the behavior of the energy conditions for
vanishing and non-vanishing values of the dimensional parameter $\rho$.

(a) The DEC, $\tilde{\rho}-\tilde{p}_{r}$, of solution (III)

(b) The DEC, $\tilde{\rho}-\tilde{p}_{t}$, of solution (III)

(c) The WEC, $\tilde{\rho}+\tilde{p}_{r}$ of solution (III)

(d) The WEC $\tilde{\rho}+\tilde{p}_{t}$ of solution (III)

(e) The SEC $\tilde{\rho}-\tilde{p}_{r}-2\tilde{p}_{t}$ of solution (III)

Figure 5:  Energy conditions of solution (III) when $\varepsilon=0$ and
$\varepsilon=0.5$

###  V.4 Mass–radius–relation

The compactification factor $u(r)$ of a compact stellar is defined as the
ratio between the mass and radius and is considered as an ingredient role to
use to understand the physical properties of the star. From solution (III) we
derived the gravitational mass as the following:

$\displaystyle M(\chi)={\int_{0}}^{\chi}\tilde{\rho(\xi)}\xi^{2}d\xi$
$\displaystyle=\frac{1}{36f_{0}{}^{3/2}\chi}\Bigg{[}8\chi\varepsilon\sqrt{6\pi}\,erf(\sqrt{6f_{2}}\,\chi/2)(3f_{2}{}^{2}-2f_{2}{}^{2}-7f_{0}f_{2})+48\varepsilon
e^{-3/2f_{2}\chi^{2}}\,\sqrt{f_{2}}(2f_{0}{}^{2}\chi^{2}-f_{2}[3+5f_{0}\chi^{2}])-144\varepsilon
f_{2}{}^{3/2}e^{-f_{2}\chi^{2}/2}$ $\displaystyle-72\chi\varepsilon
f_{2}{}^{2}\sqrt{2\pi}\,erf(\sqrt{2f_{2}}\,\chi/2)-36e^{-f_{2}\chi^{2}}\sqrt{f_{2}}[2\varepsilon
f_{0}{}^{2}\chi^{2}+f_{2}(\chi^{2}-2\varepsilon\\{3-f_{0}\chi^{2}\\})-\varepsilon
e^{-f_{2}\chi^{2}}(f_{0}{}^{2}\chi^{2}+f_{2}\\{1-4\chi^{2}f_{0}\\})]$
$\displaystyle+\varepsilon\chi\sqrt{2\pi}\,erf(\sqrt{2f_{2}}\,\chi)[f_{0}{}^{2}+4f_{0}f_{2}-4f_{2}{}^{2}]+36\varepsilon\chi\sqrt{\pi}(f_{0}+2f_{2})(f_{0}+f_{2})\,erf(\sqrt{f_{2}}\,\chi)+36f_{2}{}^{3/2}(\chi^{2}+\varepsilon)\Bigg{]}\,,$
$\displaystyle\approx\frac{f_{2}\chi^{3}}{10}(5[2-f_{2}\chi^{2}]-4\varepsilon
f_{2}\chi^{2}[f_{2}-4f_{0}])\,,$ (36)

where $erf(\chi)$ is the error function which is defined as the following:

$\displaystyle erf(\chi)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-y^{2}}dy\,.$ (37)

The compactification factor $u(\chi)$ which is defined as the following:

$\displaystyle
u(\chi)=\frac{M(\chi)}{R}\approx\frac{f_{2}\chi^{3}}{10R}(5[2-f_{2}\chi^{2}]-4\varepsilon
f_{2}\chi^{2}[f_{2}-4f_{0}])\,,$ (38)

where $R$ is the radius of the star and we use Eq. (V.4) in (38). The behavior
of the gravitational mass and the compactification factor are plotted in Fig.
6 for $\varepsilon=0$ and $\varepsilon=0.5$.

(a) The gravitational mass of (III) when $\varepsilon=0$

(b) The compactification factor of (III) when $\varepsilon=0.5$

Figure 6: Gravitational mass and compactification factor of solution (III)
when $\varepsilon=0$ and $\varepsilon=0.5$

As Fig. 6 5(a) shows the gravitational mass increases as the radial coordinate
does whereas Fig. 6 5(b) displays the compactification factor in direct
proportion to the radial coordinate.

###  V.5 Equation of state

Das et al. Das  _et al._ (2019) showed that the EoS of neutral compact stars
in the frame of GR behaves in a linear form however, this study shows that the
EoS of solution (III) is not a linear one either for the TEGR or higher order
torsion tensor. To rectify this, we define the radial and transverse EoS as
the following:

$\displaystyle\omega_{r}=\frac{\tilde{p}_{r}}{\tilde{\rho}}\approx\frac{2f_{0}-f_{2}}{3f_{2}}-\frac{2f_{2}[f_{0}+f_{2}]-\varepsilon
f_{2}[6f_{0}f_{2}-f_{2}{}^{2}-8f_{0}{}^{2}]}{18f_{2}}\chi^{2}+\cdots\,,$
$\displaystyle\omega_{t}=\frac{\tilde{p}_{t}}{\tilde{\rho}}\approx\frac{4f_{0}-3f_{2}}{6f_{2}}+\frac{2[f_{2}{}^{2}-8f_{0}f_{2}+6f_{0}{}^{2}]+\varepsilon
f_{2}[f_{2}{}^{2}-6f_{0}f_{2}++8f_{0}{}^{2}]}{36f_{2}}\chi^{2}+\cdots\,,$ (39)

where $\omega_{r}$ and $\omega_{t}$ are the radial and transverse. The
behavior of the EoS is shown in Fig 7.

(a) Radial and tangential EoS when $\varepsilon=0$

(b) Radial and tangential EoS when $\varepsilon\neq 0$

Figure 7: Radial and transverse EoS’s of solution (III) for $\varepsilon=0$
and $\varepsilon=-0.01$

As Fig 7 6(a) and 6(b) show that the EoS are also not linear for the case of
TEGR, $\varepsilon=0$ or for the higher order torsion. Figure 7 6(a) and 6(b)
indicate that for the TEGR case the radial EoS moves to the surface of the
star rapidly in comparison to the tangential EoS however for the higher order
torsion scalar case the inverse is occurs.

##  VI Stability of the model

The model’s stability is an important aspect that any physical model must
consider. For this we are going to use two different methods, the first being
the TOV equation and the second being the adiabatic index.

###  VI.1 Tolman-Oppenheimer-Volkoff equation

We assume a hydrostatic equilibrium in order to study the stability of
solution (III). The TOV equation Tolman (1939); Oppenheimer and Volkoff (1939)
which was presented in Ponce de Leon (1993) is as follows:

$\displaystyle\frac{2[\tilde{p}_{t}-\tilde{p}_{r}]}{\chi}-\frac{M_{g}(\chi)[\tilde{\rho}(\chi)+\tilde{p}_{t}(\chi)]e^{[\alpha(\chi)-\beta(\chi)]/2}}{\chi}-\frac{d\tilde{p}(\chi)}{\chi}=0.$
(40)

The quantity h $M_{g}(\chi)$ is the gravitational mass of radius $\chi$ which
is defined as the following:

$\displaystyle
M_{g}(\chi)=4\pi{\int_{0}}^{\chi}\Big{(}{{\mathcal{T}}_{t}}^{t}-{{\mathcal{T}}_{r}}^{r}-{{\mathcal{T}}_{\theta}}^{\theta}-{{\mathcal{T}}_{\phi}}^{\phi}\Big{)}\chi^{2}e^{[\alpha(\chi)+\beta(\chi)]/2}d\chi=\frac{\chi^{2}\alpha^{\prime}e^{[\beta(\chi)-\alpha(\chi)]/2}}{2}\,,$
(41)

Using Eqs. (41) in (40) we get

$\displaystyle\frac{2(\tilde{p}_{t}-\tilde{p}_{r})}{\chi}-\frac{d\tilde{p}_{r}}{d\chi}-\frac{\alpha^{\prime}[\tilde{\varepsilon}(\chi)+\tilde{p}_{r}(\chi)]}{2}=F_{g}+F_{a}+F_{h}=0\,.$
(42)

The quantities $F_{g}=-\frac{\alpha^{\prime}[\tilde{\rho}+\tilde{p}]}{2}$,
$F_{a}=\frac{2(\tilde{p}_{t}-\tilde{p}_{r})}{\chi}$ and
$F_{h}=-\frac{d\tilde{p}_{r}(\chi)}{d\chi}$ are the gravitational, anisotropic
and hydrostatic forces, respectively. The behavior of the TOV equation of
model (III) is shown in Fig. 8 which

(a) The TOV Equation when $\varepsilon=0$

(b) The TOV Equation when $\varepsilon=0.5$

Figure 8: TOV of solution (III) when $\varepsilon=0$ and $\varepsilon=0.5$

indicates that the anisotropic and gravitational forces are negative and
dominated by the hydrostatics force which is positive and keeps the system in
static equilibrium in the case of TEGR whereas in the higher order torsion the
anisotropic and hydrostatic forces are positive and dominated by the
gravitational force which is negative and keeps the system in static
equilibrium.

###  VI.2 The adiabatic index

It is well-known that the adiabatic index is $\gamma$ and defined as Chan  _et
al._ (1994); Herrera and Santos (1997)

$\displaystyle\gamma=\Big{(}1+\frac{\tilde{\rho}}{\tilde{p}}\Big{)}\frac{d\tilde{p}}{d\tilde{\varepsilon}}\,,$
(43)

which allows us to link the structure of a symmetrically spherical static
object and the EoS of the interior solution so that we can investigate its
stability Moustakidis (2017). For any interior solution to exhibit stability,
it is important that its adiabatic index be greater than $4/3$ Heintzmann and
Hillebrandt (1975) and if $\gamma=\frac{4}{3}$ then the isotropic sphere will
be in neutral equilibrium. In accordance with the work of Chan et al. Chan
_et al._ (1993), the condition of the stability of a relativistic anisotropic
sphere $\gamma>\Gamma$ must be satisfied such that

$\displaystyle\Gamma=\frac{4}{3}-\left\\{\frac{4(\tilde{p}-\tilde{p}_{t})}{3\lvert\tilde{p}^{\prime}\lvert}\right\\}_{max}\,.$
(44)

The behavior of the adiabatic index is shown in Figure 9 which ensures the
stability condition of (III).

(a) The adiabatic index of (III) when $\varepsilon=0$

(b) The adiabatic index of (III) when $\varepsilon=0.5$

Figure 9: Adiabatic index of solution (III) when $\varepsilon=0$ and $\varepsilon=0.5$ Table 1: Values of model parameters when $\varepsilon=0.5$ Pulsar | Mass ($M_{\odot}$) | Radius (km) | $f_{0}$ | $f_{1}$ | $f_{2}$ |
---|---|---|---|---|---|---
J0437-4715 | $1.44^{+0.7}_{-0.07}$ |  $13.6^{+0.9}_{-0.8}$ Reardon  _et al._ (2016); González-Caniulef  _et al._ (2019) | 0.2454095942 | -0.6438925458 | 0.39855633 |
J0030+0451 | $1.44^{+0.15}_{-0.16}$ |  $13.02^{+1.25}_{-1.06}$ Miller  _et al._ (2019) | 0.2841372756 | -0.7327677384 | 0.44874578 |
J0030+0451 | $1.34^{+0.15}_{-0.16}$ |  $12.71^{+1.14}_{-1.19}$Riley  _et al._ (2019) | 0.2665730489 | -0.6927871908 | 0.4263174699 |
EXO 1785 - 248 | $1.3^{+0.2}_{-0.2}$ | $8.849^{+0.45}_{-0.45}$ | 0.003403660033 | -0.01018791514 | 0.006784254813 |
Cen X-3 | $1.49^{+0.08}_{-0.08}$ | $9.178^{+0.13}_{-0.13}$ | 0.5216844162 | -1.227938771 | 0.7072044229 |
RX J 1856 -37 | $0.9^{+0.2}_{-0.2}$ | $\simeq 6$ | 0.6103362288 | -1.393834058 | 0.7865648502 |
Her X-1 | $0.85^{+0.15}_{-0.15}$ | $8.1^{+0.41}_{-0.41}$ | 0.2892875212 | -0.7441973693 | 0.4552206652 |

Table 2: Values of physical quantities when $\varepsilon=0$ Pulsar | $\tilde{\rho}\lvert_{{}_{{}_{\chi=0}}}$ | $\tilde{\rho}\lvert_{{}_{{}_{\chi=1}}}$ | $\frac{d\tilde{p}_{r}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=0}}}$ | $\frac{d\tilde{p}_{r}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=1}}}$ | $\frac{d\tilde{p}_{t}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=0}}}$ | $\frac{d\tilde{p}_{t}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=1}}}$ | $(\tilde{\rho}-\tilde{p}_{r}-2\tilde{p}_{t})\lvert_{{}_{{}_{\chi=0}}}$ | $(\tilde{\rho}-\tilde{p}_{r}-2\tilde{p}_{t})\lvert_{{}_{{}_{\chi=1}}}$ | $z\lvert_{{}_{{}_{\chi=1}}}$ |
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |
|  |  |  |  | Millisecond Pulsars |  |  |  |  |
|  |  |  |  | with White Dwarf |  |  |  |  |
|  |  |  |  | Companions |  |  |  |  |
J0437-4715 | 0.598 | 0.27$\times 10^{-2}$ | 0 | -9$\times 10^{-28}$ | -1 | -1 | 0.461 | 0.81$\times 10^{-2}$ | 0.039 |
J0030+0451 | 0.67 | 0.3$\times 10^{-2}$ | 0 | -9$\times 10^{-29}$ | -1 | -1 | 0.497 | 0.88$\times 10^{-2}$ | 0.041 |
J0030+0451 | 0.64 | 0.31$\times 10^{-2}$ | 0 | -9$\times 10^{-26}$ | -1 | -1 | 0.48 | 0.01 | 0.042 |
|  |  |  |  | Pulsars Presenting |  |  |  |  |
|  |  |  |  | Thermonuclear |  |  |  |  |
|  |  |  |  | Bursts |  |  |  |  |
EXO 1785 - 248 | 1.1 | 0.0064 | 0 | -6$\times 10^{-20}$ | -1 | -1 | 0.58 | 0.02 | 0.062 |
Cen X-3 | 1.06 | 006 | 0 | -6$\times 10^{-22}$ | -1 | 01 | 0.58 | 0.018 | 0.069 |
RX J 1856 -37 | 1.17 | 0.014 | 0 | -2.1$\times 10^{-9}$ | -1 | -0.99 | 0.567 | 0.0417 | 0.095 |
Her X-1 | 0.64 | 0.007 | 0 | -2$\times 10^{-10}$ | -1 | -1 | 0.48 | 0.02 | 0.07 |
Table 3: Values of physical quantities when $\varepsilon=0.5$ Pulsar | $\tilde{\rho}\lvert_{{}_{{}_{\chi=0}}}$ | $\tilde{\rho}\lvert_{{}_{{}_{\chi=1}}}$ | $\frac{d\tilde{p}_{r}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=0}}}$ | $\frac{d\tilde{p}_{r}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=1}}}$ | $\frac{d\tilde{p}_{t}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=0}}}$ | $\frac{d\tilde{p}_{t}}{d\tilde{\rho}}\lvert_{{}_{{}_{\chi=1}}}$ | $(\tilde{\rho}-\tilde{p}_{r}-2\tilde{p}_{t})\lvert_{{}_{{}_{\chi=0}}}$ | $(\tilde{\rho}-\tilde{p}_{r}-2\tilde{p}_{t})\lvert_{{}_{{}_{\chi=1}}}$ | $z\lvert_{{}_{{}_{\chi=1}}}$ |
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |
|  |  |  |  | Millisecond Pulsars |  |  |  |  |
|  |  |  |  | with White Dwarf |  |  |  |  |
|  |  |  |  | Companions |  |  |  |  |
J0437-4715 | 1.2 | 0.9 | -1.14 | 0.28 | -2.4$\times 10^{-7}$ | 0.16 | 0.92 | 0.86 | 0.77 |
J0030+0451 | 1.3 | 0.97 | 1.4 | 0.29 | 0.32$\times 10^{-5}$ | 0.17 | 0.98 | 0.93 | 0.97 |
J0030+0451 | 1.3 | 0.94 | 9.3 | 0.28 | 0.16$\times 10^{-5}$ | 0.16 | 0.96 | 0.9 | 0.89 |
|  |  |  |  | Pulsars Presenting |  |  |  |  |
|  |  |  |  | Thermonuclear |  |  |  |  |
|  |  |  |  | Bursts |  |  |  |  |
EXO 1785 - 248 | 1.7 | 1.2 | 0 | 0.31 | 3.9$\times 10^{-7}$ | 0.19 | 1.1 | 1.2 | 2.8 |
Cen X-3 | 1.8 | 1.3 | 0 | 0.31 | -0.15$\times 10^{-3}$ | 0.2 | 1.1 | 1.2 | 4.4 |
RX J 1856 -37 | 1.8 | 1.3 | 0 | 0.31 | 1.7$\times 10^{5}$ | 0.2 | 1.1 | 1.2 | 3.8 |
Her X-1 | 1.4 | 1 | 0 | 0.3 | 0.1 | 0.2 | 1 | 0.95 | 0.8 |

##  VII Discussion and conclusion

In the context of quadratic
$f\mathcal{(T)}=\mathcal{T}+\epsilon\mathcal{T}^{2}$ modified gravity and for
a KB metric potentials we gave a precise interior solution that has the
ability to describe real compact star configurations. The regularity
conditions at the origin as well as at the surface of the stellar show that
our solution has a good attitude over the stellar structure using the white
dwarf companion $PRSJ0437-4715$ . Moreover, we show that the anisotropy of
this model has a positive value when the dimensional parameter
$\varepsilon\neq 0$ which can be interpreted as a repulsive force. This fact
is because the tangential pressure is greater than the radial pressure, i.e.,
$p_{t}>p_{r}$ Sunzu  _et al._ (2019). When the dimensional parameter
$\varepsilon=0$, which is the GR case, we got a negative anisotropy parameter.
Furthermore, we study the issue of stability and showed that the derived model
is stable against the different forces (gravitational, hydrostatic,
anisotropic, and electromagnetic) acting on it. We also calculated the sound
of speed and showed that it is consistent with realistic compact stars.
Finally, we calculated the adiabatic index of our model and showed that it is
also consistent with a realistic physical star. It is worth mention that the
adiabatic index presented in Singh  _et al._ (2019) has a negative value that
is not consistent with realistic stellar models. The main reason for getting
negative adiabatic index Singh  _et al._ (2019) is due to the use of vanishing
radial pressure. This indicates, in a clear way, that our assumption of the KB
metric potentials (18) are the physical assumption that makes the resulting
stellar model in the frame of quadratic $f(T)$ gravitational theory is
consistent with real stellar objects.

In this study and for the first time we show the effect of the higher order
torsion in modified teleparallel gravitational theories by assuming the form
of $f\mathcal{(T)}=\mathcal{T}+\epsilon\mathcal{T}^{2}$. We showed that the
effect of the dimensional parameter $\epsilon$ made all the physical parameter
consistent with a real stellar pulsar as we discussed in the context of this
study. Moreover, we tested our model over a wide range of reported observed
values of masses and radii of pulsars are reported in (Tables I, II and III).
We show that the fit is good also in these cases.

The present approach can be summarized as follows: we have used a non-diagonal
form of the tetrad field that gives a null value of the torsion tensor as soon
as the metric potentials approach $1$. This is a crucial condition for any
physical tetrad field as reported in various studies DeBenedictis and Ilijic
(2016); Abbas  _et al._ (2015); Momeni  _et al._ (2018); Abbas  _et al._
(2015); Chanda  _et al._ (2019); Debnath (2019); Ilijic and Sossich (2018).
Moreover, we stress the fact that the tetrad field (9) is a necessary tetrad
in the application of any spherically symmetrical interior/exterior solution
in the frame of $f(T)$ because it makes the off-diagonal components of the
field equations vanishing identically.

Bhatti et al. Bhatti  _et al._ (2018) have discussed the physical features of
compact stars using the ansatz of Krori and Barua in the frame of
$f(G,{\mathcal{T}})$, where $G$ is the Gauss-Bonnet term and ${\mathcal{T}}$
is the trace of the energy-momentum tensor, showing the behavior of material
variables through plots and discussed the physical viability of compact stars
through energy conditions. Moreover, in the frame of $f(R)$, discussions of
the behavior of different forces, equation of state parameter, measure of
anisotropy and TOV equation in the modeling of stellar structures have been
carried out. The comparison from their graphical representations provided
clear evidence for the realistic and viable $f(R)$ gravity models at both
theoretical and the astrophysical scale Yousaf  _et al._ (2017).

We can conclude that the comparison of our exact interior solution with the
physical parameters of pulsars gives indications that the model can
realistically represent observed systems. Furthermore, the approach can be
extended to a large class of metrics and anisotropy if the above physical
requirements are satisfied. However, a detailed confrontation with
observational data is needed. This will be our forthcoming study.

## Acknowledgement

The work of KB was supported in part by the JSPS KAKENHI Grant Number
JP21K03547.

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# Robust asymptotic observer of motion states with nonlinear friction

Michael Ruderman  University of Agder
Department of Engineering Sciences
P.B. 422, Kristiansand, 4604, Norway
email<EMAIL_ADDRESS>
###### Abstract

This paper revisits the previously proposed linear asymptotic observer of the
motion state variables with nonlinear friction and provides a robust design
suitable for both, transient presliding and steady-state sliding phases of the
relative motion. The class of motion systems with the only measurable output
displacement is considered. The reduced-order Luenberger-type observer is
designed based on the obtained simplified state-space representation with a
time-varying system matrix. The resulted observation error dynamics proves to
be robust and appropriate for all variations of the system matrix, which are
due to the nonlinear spatially-varying friction. A specially designed
tribological setup to accurately monitor the relative motion between two
contacting friction surfaces is used to collect the experimental data of the
deceleration trajectories when excited by a series of impulses. The
performance of the state estimation using the proposed observer is shown based
on the collected experimental data.

###### keywords:

Friction observer state estimation robust observation motion dynamics

††thanks: Author accepted manuscript (IFAC World Congress 2023)

##  1 Introduction

The observation of dynamic states in the motion systems is a long-term problem
within (motion) control communities. Often, only the relative displacement of
an actuated system is measured by the sensing elements, such as e.g. encoders,
while the relative velocity as well as some internal dynamic states of the
entire system need also to be known in real time. The classical observation
strategies, like Luenberger observer (see Luenberger (1964)), disturbance
observer (see e.g. Ohishi et al. (1987); Shim et al. (2016)), or sliding-mode
observers (see e.g. Shtessel et al. (2014) and references therein), are well
known. Also a variety of related and analogous approaches have been developed
over the past three decades; several more specific references are omitted here
for reasons of space and too large a number published works. One of the
fundamental problems related to dynamic states of relative motion and their
observation is nonlinear friction, which is present in almost all driven
mechanical systems. As well known, the kinetic friction can (i) lead to the
undesired and long-period stick-slip cycles in case of a feedback-controlled
motion (see e.g. Ruderman (2022)), (ii) affect the motion breakaway and
reversals phases (see e.g. Ruderman (2017) and Ruderman and Rachinskii
(2017)), and (iii) result in unacceptable steady-state control errors (see
e.g. Ruderman and Iwasaki (2016)) if it is not properly compensated. Moreover,
the friction forces are well known to be not directly measurable (see e.g.
Harnoy et al. (2008)) and, in addition, they are mostly uncertain due various
system-internal and environment-external factors (see e.g. Ruderman and
Iwasaki (2015)). A robust, like discontinuity-based, compensation of nonlinear
friction (see e.g. Ruderman and Fridman (2022)) can partially solve the
problem, but can have a limited acceptance in several applications due to a
high-frequent relay-type control action. Therefore, it is often understood
that an accurate on-line estimation of friction force values can be of great
benefit to the performance of motion control systems with friction, cf. e.g.
Olsson et al. (1998) and Chen et al. (2000). Since the dynamics of frictional
forces are far from trivial, see e.g. Al-Bender and Swevers (2008) for a
detailed tutorial, designing a reliable friction observer remains a
significant challenge for the motion control and its applications. Some recent
experimentally confirmed scenarios revealing the nature of frictional
perturbations for motion control can be found in the literature, for instance
in Beerens et al. (2019); Kim et al. (2019) just to name a few applications
here. An observer-based strategy for compensating the nonlinear friction was
proposed in Ruderman and Iwasaki (2015), which allowed for an accurate
positioning control up to the level of encoder resolution. However, a
systematic analysis of the error dynamics and, hence, observer synthesis
remained partially undisclosed and less explained as methodology. The aim of
this paper is to close this gap and thus to approach the observer design for
motion systems with nonlinear friction in a simple and systematic way.

The rest of the paper is organized as follows. In the next section, we first
formalize the problem statement by introducing the class of motion systems
with nonlinear friction, for which the asymptotic Luenberger-type state
observer is proposed. In the main section 3, we make assumptions about the
dynamic friction state and introduce the corresponding asymptotic state
observer, also providing the robust design of observer gains. The developed
tribological experimental setup used in this work is described in section 4.
Experimental evaluation of the proposed observer is provided in section 5.
Conclusions are given in section 6.

##  2 Problem formulation

The following observation problem is addressed in this work. Consider the
dynamic motion system

$m\ddot{x}(t)+f\bigl{(}\dot{x}(t)\bigl{)}=u(t),$ (1)

where an inertial mass $m$ is actuated by the input $u$ and counteracted by
the nonlinear kinetic friction $f(\cdot)$. Recall that the latter appears due
to the normal contacts. The available input and output signals are the drive
force $u(t)$ and relative displacement $x(t)$, both in the generalized
coordinates of a 1DOF system, here without distinguishing between the
translational and rotational motion. The inertial mass is assumed to be
accurately known, while for the nonlinear friction the following can be
assumed:

  1. (i)

The total kinetic friction in steady-state is a superposition of the Coulomb
and viscous friction forces

$f(t)=F_{c}+F_{v}.$ (2)

The superposition (2) is consistent with the established approaches to
modeling friction, especially when using the Newton-Euler dynamics equations,
and also considering the Stribeck weakening effects. The superposition
principle can be temporary lost during the dynamic transients, particularly at
the reversals of motion, where the viscous friction effects can subside and
also lose their linearity.

  2. (ii)

The Coulomb friction force in steady-state depends on the motion direction
only, i.e.

$F_{c}=C_{f}\,\mathrm{sign}(\dot{x}),$ (3)

and is parameterized by an uncertain Coulomb coefficient, for which a nominal
value $C_{f}>0$ is known.

  3. (iii)

The viscous friction force depends linearly on the relative displacement rate
only, i.e.

$F_{v}=\sigma\,\dot{x}.$ (4)

The proportional friction law constitutes the standard linear system damping
and is parameterized by the (generally) uncertain viscous friction
coefficient, for which the nominal value $\sigma>0$ is known.

  4. (iv)

The transient behavior of kinetic friction $f(t)$ is to a large extent
unknown, while assuming two of its commonly stated properties. (a) The viscous
friction term is subject to the so-called frictional lag during sufficiently
large velocity rates, i.e. at high accelerations and decelerations. (b) The
Coulomb friction during the so-called _presliding_ 111The term presliding is
associated with a regime of friction where adhesive forces due to asperity
contacts dominate, and thus the friction force is primarily a local function
of relative displacement rather than velocity, see e.g. in Armstrong-Hélouvry
et al. (1994); Al-Bender and Swevers (2008) for further details., i.e. at the
motion beginning, stop, and reversals, is not discontinuous (cf. with eq. (3))
and undergoes some smooth hysteresis-shaped transitions. These transitions are
subject to uncertainties due to the spatially and temporally varying nature of
the friction surfaces.

For the above system (1), with the nonlinear kinetic friction satisfying
(i)–(iv), one wishes to design a robust asymptotic state observer so that the
estimates of dynamic states $\dot{\tilde{x}}(t)\rightarrow\dot{x}(t)$,
$\tilde{f}(t)\rightarrow f(t)$ as $t\rightarrow\infty$. The initial values of
the real system states $(\dot{x},f)(0)$ are unknown. And the only available
system measurements are $-\infty<\bigl{(}x(t),u(t)\bigr{)}<\infty$, both
affected by the sensing noise.

##  3 State observer

###  3.1 Dynamic friction states

At fist, we have to consider both frictional terms in superposition, cf. eq.
(2), as the dynamic friction states. This will enable their observation and
corresponding integration into the observer scheme to be designed.

For the viscous friction term, one can write

$\dot{F}_{v}=\beta^{-1}(\sigma\dot{x}-F_{v}),$ (5)

instead of (4), thus capturing the so-called frictional lag, see e.g. Al-
Bender and Swevers (2008) for details. Recall that the latter appear (but not
necessarily as linear) in the relative $(\dot{x},F_{v})$ coordinates. One can
recognize that if the time constant $\beta$ of the frictional lag is zero,
then the dynamic equation (5) will collapse, and one will obtain the standard
viscous damping relationship which is static, cf. with eq. (4). Here it is
important to emphasize that the introduced time constant $\beta>0$ is
relatively low and mostly uncertain all the time. This makes its proper
identification to a sufficiently challenging task. If no nominal $\beta$ value
is available, assigning an arbitrary $0<\beta\ll m\sigma^{-1}$ may suffice.
This will ensure the time constant of the friction lag is significantly lower
than the time constant of the mechanical drive, cf. with eq. (1).

The dynamics of the nonlinear Coulomb friction captures the presliding
transition curves, this way leading to

$\dot{F}_{c}=\left\\{\begin{array}[]{ll}\dot{x}\cdot\partial F_{c}/\partial
x\,,&\hbox{if }|F_{c}|<C_{f},\\\\[2.84526pt] 0,&\hbox{otherwise.}\\\
\end{array}\right.$ (6)

Recall that describing the Coulomb friction force which should include
presliding and, therefore, avoid discontinuity at motion reversals, relies on
a smooth multi-valued mapping $x\mapsto f$ within the presliding range. The
latter means that the motion undergoes an onset or reversal, so that
$|\dot{x}|$ is relatively low and the non-viscous friction mechanisms
predominate. As long as a progressing $F_{c}(x)$-curve is not saturated at
$\pm C_{f}$, the dynamic transitions (6) take place. Afterwards, the constant
Coulomb friction force characterizes the steady-state, cf. with eq. (3).

Since both dynamic friction states (5), (6) act on the motion dynamics (1)
simultaneously, i.e. in superposition (2), their proper decomposition can pose
serious (probably unsolvable) challenge when observing them individually.
Therefore, the overall dynamic friction state $f(\cdot)$ will be used next in
the observer design, while assuming that the nominal value $\sigma$ and
presliding map (6) are available and the parameter $\beta$ is sufficiently
preestimated.

###  3.2 Asymptotic Luenbeger-type observer

For the class of motion systems given by (1), with the dynamic friction (2),
we introduce the state vector
$w\equiv(w_{1},w_{2},w_{3})^{T}:=(x,\dot{x},f)^{T}$ and derive the simplified
dynamic state-space model as

$\dot{w}=\underset{A}{\underbrace{\left(\begin{array}[]{ccc}0&1&0\\\
0&0&-1/m\\\ 0&\bigl{(}\partial F_{c}/\partial
x+\sigma/\beta\bigr{)}&0\end{array}\right)}}w+\underset{B}{\underbrace{\left(\begin{array}[]{c}0\\\
1/m\\\ 0\end{array}\right)}}u.$ (7)

The associated output coupling vector is $C=(1,0,0)$, since the relative
displacement is the single available output value, i.e. $x=Cw$. One should
notice that the system matrix $A$ has one time-dependent term $\partial
F_{c}/\partial x$ so that one deals with $A(t)$ during the presliding. Outside
the presliding range, the $\partial F_{c}/\partial x$ term becomes zero. On
the contrary, $\partial F_{c}/\partial x\rightarrow\kappa$ after each reversal
point $x(t_{r}^{+})\equiv x_{r}$, where $t_{r}$ is the time instant of the
last motion reversal, and $\kappa\gg 0$ is a large positive constant
representing the initial stiffness of the given frictional surface pair. Note
that one can have $\kappa\rightarrow\infty$ at $t_{r}$, that is in line with
temporary stiction during the motion reversals, cf. Ruderman and Rachinskii
(2017). Also recall that at $t_{r}$ the sign of $\dot{x}$ changes, i.e. the
system (1) either starts or stops to move, or it changes the direction of
relative displacement. For both of the above cases, i.e. for $\partial
F_{c}/\partial x=0$ and $\partial F_{c}/\partial x\gg 0$, the $(A,C)$-pair
proves to be observable in the Kalman sense. This allows designing the
standard Luenberger-type observer, cf. Luenberger (1964), so that an
asymptotic convergence of the states estimation, i.e. $\tilde{w}(t)\rightarrow
w(t)$, can be guaranteed.

In order to simplify the observer design and, moreover, to improve the
convergence properties of $\tilde{w}(t)$, we transform the state-space
representation (7) into the regular form

$\displaystyle\left(\begin{array}[]{c}\dot{\bar{w}}\\\ \dot{z}\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{array}\right)\left(\begin{array}[]{c}\bar{w}\\\ z\\\
\end{array}\right)+\left(\begin{array}[]{c}B_{\bar{w}}\\\ B_{z}\\\
\end{array}\right)\,u$ (16) $\displaystyle x$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}C_{\bar{w}}&C_{z}\\\
\end{array}\right)\left(\begin{array}[]{c}\bar{w}\\\ z\\\ \end{array}\right),$
(20)

where $\bar{w}=w_{1}$ and $z=(w_{2},w_{3})^{T}$. Then, we apply a standard
reduced-order asymptotic observer, cf. Luenberger (1971). Recall that the
above transformation of (7) into the regular form, with breakdown of $A$, $B$,
$C$ into the matrices of an appropriate dimension in (16), (20), provides a
separation into the measurable and unmeasurable states $\bar{w}$ and $z$,
respectively. Subsequently, an exclusion of $\bar{w}$ from the estimated state
vector reduces the order of the observer dynamics from three to two and, this
way, the complexity of the associated poles assignment.

The reduced-order Luenberger observer is given by

$\displaystyle\dot{\tilde{z}}$ $\displaystyle=$
$\displaystyle(A_{22}-LA_{12})\tilde{z}+$ $\displaystyle+$
$\displaystyle(A_{21}-LA_{11}+A_{22}L-LA_{12}L)\bar{w}+(B_{z}-LB_{\bar{w}})u,$

where $L\equiv(L_{1},L_{2})^{T}$ is the vector of observer gains, which are
the design parameters. Here $\tilde{z}$ is the estimate vector of the
unmeasurable system states $z$. It is worth recalling that the state observer
(3.2) provides an asymptotic convergence $\tilde{z}(t)\rightarrow z(t)$ under
one and the single condition – the observer system matrix $(A_{22}-LA_{12})$
has to be Hurwitz. In order to transform back the dynamic state
$\tilde{z}(t)$, which was input-output-scaled according to the right-hand-side
of (3.2), the back-transformation

$(\tilde{w}_{2},\tilde{w}_{3})^{T}=\tilde{z}+L\bar{w},$ (22)

is subsequently required, cf. Luenberger (1971).

###  3.3 Presliding transitions

In order to capture the $\partial F_{c}/\partial x$ term, which is entering
the system matrix $A$ in (7), one needs to map the continuous presliding
friction transitions in the sense of $\bigl{(}x(t)-x_{r}\bigr{)}\mapsto
F_{c}$. Note that a particular modeling approach can be of secondary
importance when capturing the presliding friction force. Most important is
rather to ensure that such mapping is (i) piecewise continuous on the interval
between the reversal state $f_{r}\equiv F_{c}(t_{r})$ at $t_{r}$ and the
saturated friction state $|F_{c}|=C_{f}$; (ii) the force-displacement curves
are at least $\mathcal{C}^{1}$ smooth, (iii) the force-displacement curves
follow the closed hysteresis loops between two reversal points and, then, they
saturate at the $\pm C_{f}$ Coulomb friction level, cf. Fig. 1.

Figure 1: Presliding friction force transitions at reversals. Each change in
the sign of $\dot{x}$ gives rise to a new hysteresis branch and sets the
frictional system into presliding. Once the friction force saturates at $\pm
C_{f}$, the hysteresis memory is erased, and the frictional system is in gross
sliding.

While different fairly suitable modeling approaches (most importantly, those
that have a small number of free parameters) exist, for instance the Dahl
model (see Dahl (1976)) or the modified Maxwell-slip model (see Ruderman and
Bertram (2011)), in this work we use an approach (see Ruderman (2017) for
details) which is based on the tribological study Koizumi and Shibazaki (1984)
of presliding hysteresis loops. Using the scaling factor $s>0$, which relates
an after-reversal motion to the presliding distance as

$z=s\int\limits^{t}_{t_{r}}\dot{x}\,dt,$ (23)

and is defined on the interval $z\in[-1,\,0)\cup(0,\,1]$, one can describe the
branching of friction force in presliding by

$f_{0}(z)=z\bigl{(}1-\ln|z|\bigr{)}.$ (24)

Recall that each motion reversal at $t_{r}$ gives rise to a new presliding
transition, so that the total (normalized) presliding friction map is, cf.
Ruderman (2017),

$f_{p}(t)=\bigl{|}\mathrm{sign}(\dot{x})-f_{r}\bigr{|}\,z\bigl{(}1-\ln|z|\bigr{)}+f_{r}.$
(25)

Note that the presliding friction dynamics memories the state of the last
reversal transition, i.e. $f_{r}:=f_{p}(t_{r})$, see Fig. 1. Since the
presliding mapping (25) is defined for $|z|\leq 1$ only, the entire continuous
Coulomb friction law becomes, cf. with eq. (3),

$F_{c}=\left\\{\begin{array}[]{ll}C_{f}\,f_{p}(t),&\hbox{if }|z|\leq
1,\\\\[2.84526pt]
C_{f}\,\mathrm{sign}\bigl{(}\dot{x}(t)\bigr{)},&\hbox{otherwise.}\\\
\end{array}\right.$ (26)

For the presliding distance $z$, which is linked to the output displacement by
the scaling parameter $s$ and integral (23), which is reset at each $t_{r}$,
one can obtain $\partial F_{c}/\partial x$ out from

$\frac{\partial F_{c}}{\partial
z}=-C_{f}\bigl{|}\mathrm{sign}(\dot{x})-f_{r}\bigr{|}\,\ln|z|.$ (27)

###  3.4 Robust observer design

Recalling the time-variance of the system matrix $A$ and $A_{22}$,
respectively, one can show the characteristic polynomial of the matrix
$(A_{22}-LA_{12})$ to be

$\Bigl{(}\lambda+L_{1}\Bigr{)}\lambda+\frac{1}{m}\Bigl{(}\sigma/\beta+\partial
F_{c}/\partial x-L_{2}\Bigr{)}=0.$ (28)

Obviously $\lambda$ is the complex Laplace variable so that both eigenvalues
of (28) can be computed explicitly as

$\lambda_{1,2}=\frac{1}{2}\Bigl{(}-L_{1}\pm\sqrt{L_{1}^{2}+\frac{4}{m}\bigl{(}L_{2}-\partial
F_{c}/\partial x-\sigma/\beta\bigr{)}}\,\Bigr{)}.$ (29)

Based on that, imposing the conditions

$(\mathrm{a})\quad L_{1}>0\quad\text{ and }\quad(\mathrm{b})\quad
L_{1}>2\sqrt{\frac{\kappa+\sigma/\beta-L_{2}}{m}}$ (30)

one can ensure the observer dynamics is (i) asymptotically stable and (ii) has
no complex poles, which otherwise would lead to transient oscillations.
Furthermore, one can recognize that a sufficiently large $L_{1}$ will
determine the most left-hand-side pole and, thus, a faster convergence of the
$\tilde{w}_{2}$ estimate, which is the relative velocity. At the same time,
$L_{2}$ is expected to be negative in the most cases, i.e. for
$0<\sigma\ll\kappa$, so as to guarantee that (30) holds also for $\partial
F_{c}/\partial x=0$. Recall that the latter characterizes the sliding phases
where $F_{c}$ is already saturated. A selected ratio between the $L_{1},L_{2}$
gains, while satisfying (30), is controlling the distance between both real
poles (29). This distance becomes maximal for $\partial F_{c}/\partial x=0$,
and it is decreasing during the presliding as the $\partial F_{c}/\partial x$
value grows. This way, a robust observer (3.2), (22) can be realized for all
phases of the nonlinear friction, also for uncertain parameters and during
transient behavior of the dynamic friction, cf. sections 1 and 3.1.

##  4 Tribological setup

The specially designed tribological setup (see Fig. 2) consists of a linear
moving platform which is placed under the mechanical frame and guides the
lumped disks (from various materials). This way, an exposed disk and the
linear platform have a homogenous horizontal frictional contact. The used
steel disk (lacquered for a better laser reflection) with the total mass
$m=52$ gram can slide over the moving platform which has an equally polished
surface out of steel. The Coulomb friction coefficient is calculated as
$C_{f}=\mu mg=0.2143$ N, where $g$ is the gravitational acceleration constant.
The frictional coefficient $\mu$ is taken from the tribological literature for
a kinetic (sliding) behavior of a clean and dry steel on steel pair. The
moving platform can be actuated by a servo-drive with the stiff high-precision
ball-screw, so that its linear displacement is known based on the motor
encoder (not used in this work). The absolute position of the sliding disk is
measured by a high-resolution laser sensor, which beam is touching the disk in
its center (see Fig. 2). The diameter of the disk is only half millimeter
lower than the width of the guiding slot of the frame. This way, the motion of
the disk is accurately constrained from both sides, and the only one
horizontal degree of freedom (i.e. denoted with the coordinate $x$) can be
assumed. Lateral contacts between the disc and the walls of the frame-slot are
minimized by the specially machined side-edging on the disc. Therefore, the
by-effect due to the side-contact with the walls can be neglected, in
comparison to the main frictional contact area of the disc which is placed on
the moving platform.

Figure 2: Tribological experimental setup (laboratory view).

In this work, the decelerating trajectories of the sliding disk were used
without actuating the motion platform. More specifically, a series of
mechanical impulses was manually provided to the disk, while the motion
platform remained non-driven. This way, the measured with the laser sensor
signal $x(t)$ represents an absolute displacement of the disk, while an
initial relative velocity after each new impulse is unknown. Note that the
unfiltered real-time data of the disk displacement are recorded in the control
board, with the sampling rate set to 2 kHz.

##  5 Experimental evaluation

While the overall moving mass $m$ and the Coulomb friction coefficient $C_{f}$
are assumed to be known from the available system data, cf. section 4, the
residual frictional parameters $\sigma$, $\beta$, $s$ are uncertain and can
only be estimated roughly, based on the (even though accurate) experimental
measurements. The least-squares best fit of the modeled dynamics (1) when
tuning the unknown frictional parameters in (4), (5), (23), is shown in Fig.
3.

Figure 3: Measured impulse response versus the motion dynamics (1) with least-
squares best fit of friction (2).

Note that the manually injected input impulse cannot be ideally executed, i.e.
with a time-of-impact $\rightarrow 0$. Therefore, the measured impulse
response is inherently affected by some by-effecting transient input dynamics,
cf. Fig. 3, even though the impulse is also fitted in the model together with
the identified parameters. Despite some visible transient discrepancy, the
principal shape of the fictionally damped convergence is well captured by the
modeled $f(\cdot)$, cf. later with Fig. 5 (c). This allows implementing the
state-space model (7) and the asymptotic observer (3.2), (22).

The measured motion profile, shown in Fig. 4 for a sequence of manually
injected mechanical impulses, is used for evaluating the designed observer.

Figure 4: Relative motion of the sliding disk: measured displacement in
response to the series of impulses.

Note that the applied impulses provide a series of unknown disturbing initial
conditions, for which the robust observer has to converge fast enough during
the relatively short periods of non-zero velocity, cf. Fig. 4. It is also
important to notice that the excited motion steps are distributed over the
entire length of the sliding surface of the supporting platform. This results
in the varying, correspondingly uncertain, parameters of the frictional
behavior, depending on spatial properties of the contact surface.

The assigned poles of the observer are $\lambda_{1,2}=(-350,-10)$. These
result in the feedback gain values $L^{T}=(360,-182)$. The evaluated observer
performance is demonstrated in Fig. 5. The $\tilde{w}_{3}$-estimate, which is
the observed dynamic friction state, is compared with the model-predicted
Coulomb friction force in the diagram (a). Note that the high-frequency
pattern below the $C_{f}$-level is not discontinuous, and it constitutes a
smooth pres-sliding friction behavior driven by the noisy $\bar{w}(t)$ and
$\tilde{w}_{2}(t)$ values. The corresponding $\tilde{w}_{2}$-estimate of the
relative velocity is shown in the diagram (b), cf. with the measured motion
profile from Fig. 4. The estimated velocity peaks are clearly visible despite
the noise of displacement measurement is propagated into the
$\tilde{w}_{2}(t)$ signal. For a further assessment of the observer
performance, the position error is compared between the nominal model, i.e.

$e_{model}=\bar{w}-x,$

where $x$ is predicted based on (1), and the observer-based

$e_{obs}=\bar{w}-\int\tilde{w}_{2}\,dt,$

where $\tilde{w}_{2}$ is the velocity estimate. Despite the integrative
behavior, the $e_{obs}(t)$ is clearly superior comparing to $e_{model}(t)$,
especially during the transients of non-zero velocities, as it is visible in
the diagram (c). This speaks further in favor of the
$(\tilde{w}_{2},\tilde{w}_{3})(t)$ estimate.

Figure 5: Observer evaluation: model-predicted $F_{c}$ versus observed
$\tilde{w}_{3}$ (a), observed $\tilde{w}_{2}$ (b), model-predicted position
error $e_{model}$ versus observer error $e_{obs}$ (c).

##  6 Conclusions

A simple and robust asymptotic observer of the nonlinear friction state is
provided. Using the derived time-varying state-space notation of the motion
dynamics and reduced-order Luenberger observer, it is shown that both dynamic
states of the relative velocity and overall nonlinear friction force can be
estimated sufficiently fast and accurately. In particular, the estimation is
feasible for both, the transient presliding and sliding phases of an excited
and damped relative motion. The observer provided can be parameterized by a
simple and straightforward pole placement manner, which makes it well
accessible for standard control engineering applications. An experimental
evaluation was demonstrated on a specially designed tribological setup of
moving bodies on a sliding surface. The relatively short motion instances,
excited by a series of the external force-impulses, where used as a
challenging scenario of relative motion in the transient phase.

## Acknowledgment

The author is grateful to the laboratory engineering support by Roy Werner
Folgero and Harald Sauvik for assembling the tribological setup.

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