# Scalar field models driven by Dirac-Born-Infeld dynamics and their relatives

Elisama E. M. Lima<EMAIL_ADDRESS>Federal Institute of Education,
Science and Technology of Bahia, Barreiras, Bahia, 47808-006, Brazil
Francisco A. Brito<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil

###### Abstract

In this paper, we investigate novel kinklike structures in a scalar field
theory driven by Dirac-Born-Infeld (DBI) dynamics. Analytical features are
reached through a first-order formalism and a deformation procedure. The
analysis ensures the linear stability of the obtained solutions, and the
deformation method permits to detect new topological solutions given some
systems of known solutions. The proposed models vary according to the
parameters of the theory. However, in a certain parameter regime, their defect
profiles are precisely obtained by standard theories. These are the models
relatives. Besides that, we investigate the $\beta-$Starobinsky potential in
the perspective of topological defects; and we have shown that it can support
kinklike solutions, for both canonical and non-canonical kinetics. As a
result, we propose two new kinds of generalizations on the $\beta-$Starobinsky
model, by considering the DBI approach. Finally, we explore the main
characteristics of such structures in these new scenarios.

##  I Introduction

The Dirac-Born-Infeld scalar theory was inspired by the non-linear
electromagnetism Born:1934gh ; Dirac:1962iy , which coincides with the
standard theory for small field excitations. Initially, the theory was
proposed to eliminate divergences in the classical electrodynamics and after,
it also proved useful to investigate topological structures in non-ordinary
field contexts Brown ; Sarangi ; Babichevkvortices ; Andrews ; Garcia ;
Bazeia:2017mnc . For example, it can support instanton solutions Brown ,
global string solutions Sarangi , vortex solutions Babichevkvortices , besides
other kinds of topological defects Andrews ; Garcia ; Bazeia:2017mnc ; tops .

The present work considers novel scalar field models with DBI-like kinetic
term, and we analyze how that generalization changes the topological
solutions. In this situation, the equations of motion include non-linear terms
that can be diminished by a first-order framework FOFGD , and the linear
stability of the solutions is ensured. For some cases, the non-linearities
provoked by the modified dynamics mimic an inner structure, featured by a
plateau on the stability potential brane .

The novel analytical models are discovered through the deformation procedure,
which consists of proposing a deforming function connecting two potentials,
one of the known solutions and another unknown which one desires to find
Bazeia:2002xg ; DMGD . The Dirac-Born-Infeld kink like solutions, in a certain
parameter regime, encodes the same field profile of ordinary scalar theories;
in such a way, the energy density and the stability potential of their
canonical counterparts are precisely recovered Bazeia:2017mnc . That is the
reason why we are calling these models relatives.

The models and their relatives can be useful to provide both the topological
and cosmological inflaton solutions. While in the first case we look for
static solutions with spatial dependence, in the second case we look for
homogeneous solutions, i.e., time dependent only solutions. From the
inflationary cosmology point of view, we need scalar dynamics that can roll
slowly enough to get the precise amount of inflation to solve the horizon,
flatness and monopole problem. One of the main problems in string theory or
supergravity is to find such potentials able to develop this precise behavior.
One has found several potentials in these contexts that are are easily ruled
out by the cosmological observational data. Thus several other approaches
keeping some characteristics around fundamental theories as brane inflation
dvali-tye , flux compactifications KKLT , to quote a few have been addressed
in the literature. On the other hand, the DBI dynamics for being directly
related to brane dynamics seems to be a good place to look for potentials that
can overcome such difficulties. Same can be said to their deformed
counterparts since the deformation can make an initial unsuitable potential to
a final completely acceptable potential to describe inflationary cosmology. In
this sense we consider one of the most celebrated potentials to describe
dynamics of inflation, the Starobinsky potential, to address its deformation
in the DBI dynamics. As we shall show, their relatives may find applications
in both cosmology and (non-)topological solutions.

The paper is organized as follows. In Sec. II we introduce generalities about
the first-order formalism and deformation mechanism. Then, we focus on the
standard dynamics in Sec. III and DBI dynamics in Sec. IV in the context of
deformed theories. The final considerations are presented in Sec. V.

##  II General description

In this section, we shall describe general properties of the formalism used in
this work, to characterize a real scalar field in a non-standard theory, as
shown in Refs. FOFGD . The generalized action in the $(1+1)-$dimensional
space-time is

$S=\int d^{2}x{\cal L}\left(\phi,X\right),$ (1)

where $X=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$. For the specific
case where the Lagrangian density is of the type

${\cal L}\left(\phi,X\right)=F(X)-V(\phi);$ (2)

the equation of motion reads

$\partial_{\mu}(F_{X}\partial^{\mu}\phi)=-V_{\phi};$ (3)

and the energy-momentum tensor is

$T_{\mu\nu}=F_{X}\partial_{\mu}\phi\partial_{\nu}\phi-g_{\mu\nu}{\cal{L}}.$
(4)

Since we are interested in topological solutions, one considers static field
configurations. Then the equation of motion reduces to the form

$(2XF_{XX}+F_{X})\phi^{\prime\prime}=V_{\phi}.$ (5)

The energy-momentum components $T_{00}$ and $T_{ii}$ display the energy
density and stress of the solution, respectively:

$\displaystyle\rho(x)$ $\displaystyle=$ $\displaystyle-{\cal{L}}$ (6)
$\displaystyle\tau(x)$ $\displaystyle=$ $\displaystyle F_{X}\phi^{\prime
2}+{\cal{L}}.$ (7)

Assuming the stressless condition required by stability reasons,
$F_{X}\phi^{\prime 2}+{\cal{L}}=0$. The energy density will be written in the
form

$\rho(x)=-{\cal{L}}=F_{X}\phi^{\prime 2}.$ (8)

It is useful to introduce the first-order framework so that the equation of
motion is satisfied. For that reason, we define a function $W=W(\phi)$ such as

$F_{X}\phi^{\prime}=W_{\phi}.$ (9)

The second-order equation, $(F_{X}\phi^{\prime})^{\prime}=V_{\phi}$, requires
that the function $W$ also obeys

$W_{{\phi}{\phi}}\phi^{\prime}=V_{\phi}.$ (10)

It means that $F_{X}V_{\phi}=W_{\phi}W_{\phi\phi}$, which is verified for the
potential under the stressless condition,
$V(\phi)=F(\phi^{\prime})+W_{\phi}(\phi)\phi^{\prime}$.

The energy density assumes the form

$\rho(x)=W_{\phi}\phi^{\prime}=\frac{dW}{dx},$ (11)

and the total energy can be given by the function $W$ at the asymptotic limits

$\displaystyle E=|\Delta W|$ $\displaystyle=$
$\displaystyle|W(\phi(x\rightarrow\infty))-W(\phi(x\rightarrow-\infty))|.$

The standard situation is recovered assuming $F(X)=X$ bogomol .

###  II.1 Linear stability

The analysis of the linear stability is made assuming small time-dependent
perturbations around the static solution:
$\phi(x,t)=\phi_{s}(x)+\eta_{n}(x)\text{cos}(\omega_{n}t)$. The linear
perturbations lead to

$\displaystyle X$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\phi_{s}^{\prime
2}-\phi_{s}^{\prime}\eta^{\prime}_{n}\text{cos}(\omega_{n}t)$ (13)
$\displaystyle=$ $\displaystyle X_{s}+\overline{X}$ (14)

with
$\overline{X}=-\phi_{s}^{\prime}\eta^{\prime}_{n}\text{cos}(\omega_{n}t)$. The
equation of motion (3) becomes

$-\left[\left(F_{X_{s}}+2X_{s}F_{X_{s}X_{s}}\right)\eta^{\prime}_{n}\right]^{\prime}=\left(F_{X_{s}}\omega_{n}^{2}-V_{\phi_{s}\phi_{s}}\right)\eta_{n},$
(15)

which is a generalized Sturm-Liouville (SL) equation.

The Liouville normal form can be obtained, given the conditions:
$A^{2}F_{X_{s}}>0$ and $F_{X_{s}}>0$, where

$A^{2}=\frac{2X_{s}F_{X_{s}X_{s}}+F_{X_{s}}}{F_{X_{s}}}.$ (16)

Besides, for non-singular SL problems, $A^{2}F_{X_{s}}$ and $F_{X_{s}}$ must
be continuously differentiable, and $V_{\phi_{s}\phi_{s}}$ is required to be
continuous.

Applying the Liouville transformation

$\displaystyle
dz=\frac{1}{A(x)}dx\qquad\mbox{and}\qquad\varepsilon_{n}=\sqrt{F_{X_{s}}A}\,\eta_{n}.$
(17)

The SL equation transforms into an eigenvalue problem similar to the one-
dimensional time-independent Schrödinger equation,

$\left(-\frac{d^{2}}{dz^{2}}+U(z)\right)\varepsilon_{n}(z)=\omega^{2}\varepsilon_{n}(z),$
(18)

where the stability potential is given by

$U(z)=\left.\frac{\sqrt{F_{X_{s}}A}}{F_{X_{s}}}\frac{d}{dx}\left(A\frac{d\left(\sqrt{F_{X_{s}}A}\right)}{dx}\right)+\frac{V_{\phi_{s}\phi_{s}}}{F_{X_{s}}}\,\right|_{x=x(z)}.$
(19)

The zero mode related to $\omega_{0}=0$ gives

$U(z)=\frac{1}{\varepsilon_{0}(z)}\frac{d^{2}}{dz^{2}}\varepsilon_{0}(z).$
(20)

Comparing the SL equation for $\omega_{0}=0$ that is
$-(A^{2}F_{X_{s}}\eta^{\prime}_{0})^{\prime}+V_{\phi_{s}\phi_{s}}\eta_{0}=0$
and the derivative of the static motion equation (5) that is
$-(A^{2}F_{X_{s}}\phi^{\prime\prime})^{\prime}+V_{\phi_{s}\phi_{s}}\phi^{\prime}=0$;
we note that the zero mode is $\eta_{0}=\phi^{\prime}$. Therefore, linear
stability potential can also be written as

$U(z)=\left.\frac{\sqrt{F_{X_{s}}A}}{\phi^{\prime}F_{X_{s}}}\frac{d}{dx}\left(A\frac{d(\phi^{\prime}\sqrt{F_{X_{s}}A})}{dx}\right)\right|_{x=x(z)}.$
(21)

###  II.2 Deformation procedure

The deformation procedure is a practical method used to formulate novel
analytical scalar field models Bazeia:2002xg . That method consists in to
connect an analytical system $V(\phi)$ with another one $V(\chi)$ having
unknown solutions, through a deforming function $f(\chi)$. The deformed system
describing the scalar field $\chi(x,t)$ is given by
$\mathcal{L}=\mathcal{L}(\chi,Y)$ analogous to the non-standard Lagrangian
density (2). In this situation, the deformation procedure follows the steps
proposed in Ref. DMGD .

Given a known system with analytical solutions, the first-order equation may
be written as a function of the field itself: $\phi^{\prime}=R(\phi)$.
Similarly, the deformed model $V(\chi)$ presents $\chi^{\prime}=S(\chi)$. The
deformation function connects these two models through the transformation
$\phi\rightarrow f(\chi)$, then

$S(\chi)=\frac{R(\phi\rightarrow f(\chi))}{f_{\chi}}.$ (22)

Since $\phi(x)$ is the known solution of the first model ${\cal L}(\phi,X)$,
then the inverse function $\chi(x)=f^{-1}(\phi(x))$ is the static solution of
the second model ${\cal L}(\chi,Y)$.

Figure 1: The general $n-$Starobinsky model (25) and the kink solutions, for
$n=1,2,4$, represented respectively by dashed-dot (blue), dashed (red) and
solid (black) lines.

##  III Standard dynamics

The standard theory is obtained by choosing $F(X)=X$, and the first-order
formalism reduces to Bogomol’nyi-Prasad-Sommerfield (BPS) method bogomol . One
of the most well-known theories is the $\phi^{4}$ scalar field model which is
given by

$V(\phi)=\frac{1}{2}\left(1-\phi^{2}\right)^{2}.$ (23)

The static solutions are $\phi(x)=\pm\text{tanh}(x)$, which have energy
density $\rho(x)=\text{sech}^{4}(x)$ and stability potential of Pöschl-Teller
type, $U(x)=4-6\,\text{sech}^{2}(x)$, with one zero mode and one excited state
teller . Besides that, several other scalar field models of this kind with
polynomial interactions are described in Ref. bazeiaLeon  and references
therein.

###  III.1 The $n-$Starobinsky model

Another model we suggest to analyze in this paper is the $\beta$-Starobinsky,
proposed by the authors in Ref. epjp-2021 , which is

$V(\phi)=\frac{V_{0}}{2}\left(1-\left(1-\sqrt{\frac{2}{3}}\beta\frac{\phi}{M_{Pl}}\right)^{\frac{1}{\beta}}\right)^{2}.$
(24)

The $\beta$-Starobinsky potential can be rewritten in terms of dimensionless
quantities, $V/V_{0}\rightarrow V_{n}$ and
$\frac{\phi}{2\sqrt{6}M_{Pl}}\rightarrow\phi$, then

$V_{n}(\phi)=\frac{1}{2}\left(1-\left(1-\frac{2}{n}\phi\right)^{2n}\right)^{2},$
(25)

where we have replaced the parameter $\beta$ to $\beta=1/2n$, with $n$ being a
positive integer. In this situation, the potential presents a topological
sector for each value of $n$. The minima are localized at $\phi_{min}=0,n$ and
the maxima at $\phi_{max}=n/2$, as can be seen in Figure 1. Employing the BPS-
formalism, we note that

$W(\phi)=\phi+\frac{n}{2(2n+1)}\left(1-\frac{2}{n}\phi\right)^{2n+1}.$ (26)

Then, we can calculate the energy of the topological solutions using the
simplest form

$E_{n}=W(n)-W(0)=\frac{2n^{2}}{2n+1}.$ (27)

This way, the energies of the kink-like defects are known, without calculating
the solutions explicitly.

Furthermore, the expansion for $n>>1$ gives

$\left(1-\frac{2}{n}\phi\right)^{2n}=\text{e}^{-4\phi}-\frac{4}{n}\phi^{2}\text{e}^{-4\phi}+{\cal
O}\left(\frac{1}{n^{2}}\right),$ (28)

so that at the limit $n\rightarrow\infty$, the original Starobinsky model is
recovered, as indicated on the first panel of Figure 1.

Figure 2: Comparison between the $n-$Starobinsky model (with $n=1$) and the
$\phi^{4}$ model, represented by solid (black) and dashed (red) lines,
respectively. The potentials (23) and (29) (first panel), their kink solutions
(second panel), the energy densities (third panel), and stability potentials
(fourth panel).

Considering the particular case $n=1$, the $n$-Starobinsky potential reads

$V_{1}(\phi)=8\phi^{2}\left(1-\phi\right)^{2},$ (29)

which is like a $\phi^{4}$-model with an additional term $\phi^{3}$. The
first-order equation is $\phi^{\prime}=4\phi(1-\phi)$, and the solutions are

$\phi(x)=\frac{1\pm\text{tanh}(2x)}{2}.$ (30)

The energy density is $\rho(x)=\text{sech}^{4}(2x)$, that furnishes the energy
$E=2/3$. The stability potential is

$U(x)=4(4-6\,\text{sech}^{2}(2x)),$ (31)

which has two bound states with eigenvalues $\omega_{0}=0$ and
$\omega_{1}^{2}=12$. Figure 2 shows a parallel between the $\phi^{4}$-model
and the $n-$Starobinsky on this particular case $n=1$.

The $n-$Starobinsky general case can also be obtained through the deformation
procedure. We select a deforming function that leads $V_{1}(\phi)\rightarrow
V_{n}(\chi)$, which is

$f(\chi)=\frac{1}{2}\left[1-\text{tanh}\left(n\left(1-2\chi/n\right){F_{1}}\right)\right].$
(32)

The functions $F_{1}\equiv{F_{1}}(1,1/2n;1+1/2n;\left(1-2\chi/n\right)^{2n})$
are ordinary hypergeometric functions. Then,

$W_{\chi}=\frac{4f(1-f)}{f_{\chi}}$ (33)

becomes

$W_{\chi}=\left(\frac{d}{du}\left\\{u\;{F_{1}}(1,1/2n;1+1/2n;u^{2n})\right\\}\right)^{-1},$
(34)

where we have called $u=\left(1-2\chi/n\right)$. Using the relation

$u\;{F_{1}}(1,1/2n;1+1/2n;u^{2n})=\int\frac{1}{1-u^{2n}}du,$ (35)

the $n-$Starobinsky model is obtained

$W_{\chi}=1-\left(1-2\chi/n\right)^{2n},$ (36)

and the kink solutions are given through the transcendental equation

$\left(1-\frac{2}{n}\chi\right)\;{F_{1}}\left(1,\frac{1}{2n};1+\frac{1}{2n};\left(1-\frac{2}{n}\chi\right)^{2n}\right)=-\frac{2}{n}x,$
(37)

which can also be written in terms of Lerch’s transcendent
$\Phi\left(u^{2n},1,1/2n\right)$, that is

$\left(1-\frac{2}{n}\chi\right)\Phi\left(\left(1-\frac{2}{n}\chi\right)^{2n},1,\frac{1}{2n}\right)=-4x.$
(38)

These solutions are shown on the second panel of Figure 1 for some values of
$n$. The antikinks are obtained changing $x\rightarrow-x$. The energy
densities and stability potentials are represented in Fig. 3.

Figure 3: The energy densities and stability potentials of the $n-$Starobinsky
solutions, considering the same values of $n$ used in Fig. 1.

##  IV Dirac-Born-Infeld dynamics

Here, we propose to alter the standard Lagrangian by selecting a scalar field
theory in which the kinetic term is non-canonical, according to Dirac-Born-
Infeld dynamics

$F(X)=-\alpha^{2}\left(\sqrt{1-2X/\alpha^{2}}-1\right),$ (39)

where $\alpha$ is a non-null real parameter. Taking the expansion for
$|\alpha|>>1$, one gets

$F(X)=X+\frac{1}{2\alpha^{2}}X^{2}+\frac{1}{2\alpha^{4}}X^{3}+{\cal
O}\left(\frac{1}{\alpha^{6}}\right).$ (40)

As can be seen, the parameter $\alpha$ controls high-order powers of the field
derivatives. For $|\alpha|\rightarrow\infty$, the ordinary kinetic term is
recovered.

Considering the first-order description previously discussed, the static field
derivative is written in terms of the function $W_{\phi}$

$\phi^{\prime}=\frac{W_{\phi}}{\sqrt{1-W_{\phi}^{2}/\alpha^{2}}}.$ (41)

It implies that $F_{X}=\sqrt{1-W_{\phi}^{2}/\alpha^{2}}$. Furthermore, another
equation related to (41) is obtained taking $x\rightarrow-x$.

The potential, $V=F+\phi^{\prime 2}F_{X}$, in the context of the DBI kinetics
may also be written as $V=\alpha^{2}(1-F_{X})$. That in terms of $W$ is

$V(\phi)=\alpha^{2}\left(1-\sqrt{1-W_{\phi}^{2}/\alpha^{2}}\right).$ (42)

In this work, we are interested in constructing potentials supporting
topological structures; such that the equations of motion can be analytically
soluble employing first-order equations. The standard Lagrangian modification
leads to a square root on the potential, which introduces some restrictions on
the scalar field. In the limit, $|\alpha|>>1$, the scalar field acquires a
usual dynamics, and the standard Lagrangian is recovered

${\cal
L}(\phi,X)=X-\frac{1}{2}W_{\phi}^{2}+\frac{1}{2\alpha^{2}}\left(X^{2}-\frac{1}{4}W_{\phi}^{4}\right)+O\left(\frac{1}{\alpha^{4}}\right),$
(43)

ensuring an approach compatible with the original BPS description bogomol .

The energy density (8) becomes

$\rho(x)=\frac{\phi^{\prime 2}}{\sqrt{1+\phi^{\prime
2}/\alpha^{2}}}=\frac{W_{\phi}^{2}}{\sqrt{1-W_{\phi}^{2}/\alpha^{2}}}.$ (44)

The linear stability potential, Eq. (21), becomes

$U(z)=\left.\frac{1}{\phi^{\prime}}\frac{d}{dx}\left(F_{X}V_{\phi}\right)\right|_{x=x(z)},$
(45)

since in the DBI case, the static equation of motion is
$(\phi^{\prime}F_{X})^{\prime}=V_{\phi}$ and $F_{X}=A$ .

Taking into account the first-order formalism, we have
$F_{X}V_{\phi}=W_{\phi}W_{\phi\phi}$; consequently the stability potential
reads

$U(z)=\left.W_{\phi\phi}^{2}+W_{\phi}W_{\phi\phi\phi}\,\right|_{x=x(z)}.$ (46)

The Schr$\ddot{\text{o}}$dinger-like equation (18) can be factorizable as
$Q^{\dagger}Q\varepsilon_{n}=\omega_{n}^{2}\varepsilon_{n}$, with

$Q=-\frac{d}{dz}+W_{\phi\phi}\>\>\>\>\>\>\mbox{and}\>\>\>\>\>\>Q^{\dagger}=\frac{d}{dz}+W_{\phi\phi},$
(47)

since $\phi_{z}=W_{\phi}$. Therefore, the eigenvalues $\omega_{n}^{2}$ are
positive, and the static solutions are linearly stable. Besides, there is at
least a zero mode given by

$\varepsilon_{0}(z)={\cal N}\phi_{z}={\cal N}W_{\phi},$ (48)

where ${\cal N}$ is a normalization constant.

The deformation procedure described in Sec. II gives

$V(\chi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+S(\chi)^{2}/\alpha^{2}}}\right).$
(49)

The stability potential becomes

$U(z)=\left.\frac{1}{S(\chi)}\frac{d}{dx}\left(\frac{S(\chi)}{\left(1+S(\chi)^{2}/\alpha^{2}\right)^{2}}\frac{dS}{d\chi}\right)\right|_{x(z)}.$
(50)

###  IV.1 Models relatives of polynomial potentials

Figure 4: The potential (51) (first panel), its kink solution (second panel),
the energy density (52) (third panel), and the stability potential $U(z)$
calculated through (45) (fourth panel). We have chosen ${\alpha}=0.5,1,10$,
represented respectively by solid (black), dashed (red), and dashed-dot (blue)
lines.

In order to find novel analytical models relatives of polynomial potentials,
we select a known model which is going to be deformed

$V(\phi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+\dfrac{1}{\alpha^{2}}(1-\phi^{2})^{2}}}\right).$
(51)

This potential is proposed in Ref. Bazeia:2017mnc , which shall be a deforming
model. That is a non-negative potential with minima at $\bar{\phi}=\pm 1$ and
maximum at the origin, as can be seen at the leftmost panel of Fig. 4. The
kink solution is known and given by $\phi(x)=\text{tanh}(x)$, that is the same
solution found for the standard $\phi^{4}$ model. However, the DBI kinetic
modifies the energy density and linear stability, in a way that

$\rho(x)=\frac{\text{sech}^{4}(x)}{\sqrt{1+\frac{1}{\alpha^{2}}\text{sech}^{4}(x)}},$
(52)

and

$F_{X}V_{\phi}=-\frac{2\,\text{sech}^{2}(x)\text{tanh}(x)}{\left(1+\frac{1}{\alpha^{2}}\text{sech}^{4}(x)\right)^{2}},$
(53)

as represented on the rightmost panels of Fig. 4 for some values of $\alpha$.
To obtain $U(z)$ is necessary to perform a numerical integration, since the
change of variables $z(x)$ carries a non-analytical integral equation.

In addition, the first-order equation is $\phi^{\prime}=1-\phi^{2}\equiv
R(\phi)$. Through the deformation procedure, the function $S(\chi)$ becomes

$S(\chi)=\frac{1-f^{2}}{f_{\chi}}.$ (54)

The solutions of novel field models shall be found by
$\chi(x)=f^{-1}(\text{tanh}(x))$. The same expression for $S(\chi)$ is
provided by the deformation function $g(\chi)=1/f(\chi)$. Such distinct
deforming functions lead exactly to the same deformed potential and furnish
solutions to distinct topological sectors of $V(\chi)$ familySine2009 .

####  IV.1.1 The modified polynomial models

The novel deformed models shall be obtained using the deformation procedure
presented in the last sections, where we will introduce deforming functions
based on the ones presented in Ref. Bazeia:2017mnc  and references therein.
However, we are going to suggest the change of
$\chi\rightarrow\chi(1+\chi/4\alpha^{2})$.

Figure 5: The potential (56), the kink solutions (57), the energy density
$\rho_{+}(x)$ given by Eq. (58), and the stability potential $U_{+}(z)$. For
${\alpha}=1,10$, represented respectively by solid (black) and dashed (red)
lines.

The first deformation function is
$f_{1}(\chi)=\chi\left(1+\chi/4\alpha^{2}\right)$, then

$S_{1}(\chi)=\frac{1-\chi^{2}\left(1+{\chi}/{4\alpha^{2}}\right)^{2}}{1+{\chi}/{2\alpha^{2}}}.$
(55)

The potential given by Eq. (49) becomes

$V=\alpha^{2}\left(1-\frac{|1+{\chi}/{2\alpha^{2}}|}{\sqrt{(1+\frac{\chi}{2\alpha^{2}})^{2}+\frac{1}{\alpha^{2}}\left(1-\chi^{2}\left(1+\frac{\chi}{4\alpha^{2}}\right)^{2}\right)^{2}}}\right)$
(56)

which has minima at $\chi_{min}=2\alpha^{2}(-1\pm\sqrt{1\pm 1/\alpha^{2}})$.
For the particular case $\alpha=1$, there are three minima at the points
$-2(1\pm\sqrt{2})$ and $-2$, and two maxima at $-2(1\pm\sqrt{2/3})$,
constituting two topological sectors as presented in the first graph of Fig.
5. In the situation $\alpha=1$, the potential behaves like a $\phi^{6}$ model;
but when $|\alpha|>>1$ the standard $\phi^{4}$ model is recovered.

The solutions are found by $\chi(x)=f_{1}^{-1}(\text{tanh}(x))$, which are

$\chi_{\pm\pm}(x)=2\alpha^{2}\left(-1\pm\sqrt{1\pm\frac{1}{\alpha^{2}}\text{tanh}(x)}\right),$
(57)

where $\chi_{++}$ is the kink solution corresponding to the right topological
sector, and $\chi_{--}$ to the left one. Real value solutions require
$|\alpha|\geq 1$. In the limit $|\alpha|>>1$, these solutions become
$\chi_{+\pm}(x)=\pm\text{tanh}(x)$ and
$\chi_{-\pm}(x)=-4\alpha^{2}\pm\text{tanh}(x)$, in such way that
$\chi_{-\pm}(x)$ is displaced to minus infinity, and the standard framework is
recovered.

The energy density is

$\rho_{\pm}(x)=\frac{\text{sech}^{4}(x)}{\left(1\pm\frac{1}{\alpha^{2}}\text{tanh}(x)\right)\sqrt{1+\dfrac{\text{sech}^{4}(x)}{\alpha^{2}\pm\text{tanh}(x)}}},$
(58)

$\rho_{+}(x)$ is related to the kink $\chi_{++}$ (anti-kink $\chi_{-+}$) of
the right sector (left sector), and $\rho_{-}(x)$ is equal to $\rho_{+}(x)$
mirrored. These quantities and the stability potential $U(z)$ are represented
at the rightmost panels in Fig. 5.

Figure 6: The potential (60), the kink solutions (61), the energy density
$\rho$, and the stability potential $U(z)$. For the same set of parameters
chosen in Fig. 5.

Figure 7: The novel DBI models (56) and (60), pictured in the left and right
panels respectively, using ${\alpha}=1.2$.

The second deformation function is $f_{2}=2f_{1}^{2}-1$, that is
$f_{2}(\chi)=2\chi^{2}\left(1+\chi/4\alpha^{2}\right)^{2}-1$. The function
$S_{2}(\chi)$ is

$S_{2}(\chi)=\chi\left(1+\frac{\chi}{4\alpha^{2}}\right)\left[\frac{1-\chi^{2}\left(1+{\chi}/{4\alpha^{2}}\right)^{2}}{1+{\chi}/{2\alpha^{2}}}\right].$
(59)

It can also be written in terms of the first deforming function, that is
$S_{2}(\chi)=f_{1}(\chi)S_{1}(\chi)$. And the deformed potential will be

$V(\chi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+f_{1}(\chi)^{2}S_{1}(\chi)^{2}/\alpha^{2}}}\right).$
(60)

The minima are obtained through $f_{1}(\chi_{min})S_{1}(\chi_{min})=0$, which
are $\chi_{min}=0,-4\alpha^{2},2\alpha^{2}(-1\pm\sqrt{1\pm 1/\alpha^{2}})$. In
the case $\alpha=1$, there are five minima localized at the points $0,-2,-4,$
and $-2(1\pm\sqrt{2})$; and so four topological sectors. Then the potential
behaves like a standard $\phi^{10}$ model, as shown in the first graph of Fig.
6.

As the parameter $\alpha$ increases, one part of the potential tends to
approximate to the $\phi^{6}$ model, in such a way that for $|\alpha|>>1$, the
standard $\phi^{6}$ model is restored. At the same time, another part of the
potential is displaced along the negative $\chi$-axis. Such behaviour is
presented in Fig. 7, including the model previous seen.

The solutions are found by $\chi(x)=f_{2}^{-1}(\text{tanh}(x))$, which are

$\chi(x)=2\alpha^{2}\left(-1\pm\sqrt{1\pm\frac{1}{\alpha^{2}}\sqrt{\frac{1\pm\text{tanh}(x)}{2}}}\right).$
(61)

Real valued solutions require $|\alpha|\geq 1$. In the limit $|\alpha|>>1$,
the $\phi^{6}$ solutions are recovered which are
$\chi_{6}(x)=\pm\sqrt{\frac{1\pm\text{tanh}(x)}{2}}$.

The energy densities are calculated through (44), where

$\chi^{\prime}(x)=\frac{\text{sech}^{2}(x)}{4\chi_{6}(x)\sqrt{1+\chi_{6}(x)/\alpha^{2}}},$
(62)

and $\chi_{6}(x)$ is the static solution of the standard $\chi^{6}-$model.

Figure 6 shows the model (60) and its kink solutions (leftmost panels) for
some values of $\alpha$. Also, it shows the energy densities and stability
potentials (rightmost panels) for the larger and smaller sectors of the
potential. In this situation, the DBI modifications do not induce a plateau on
the stability potential, as happened on the previous models. The effect here
is a flattening on the stability potential of the larger sectors.

Figure 8: Family of DBI potentials (66) fixing $\alpha=1$. In the left panel
$n=1,2$; and in the right panel $n=3,4$.

These results can be extended for a family of potentials, where the relatives
of standard models are approached by a DBI-like perspective. Let us select the
general deforming function $f_{n}=\text{cos}(n\arccos(f_{1})-m\pi)$, with $n$
non-null positive integer, $m$ integer, and
$f_{1}(\chi)=\chi\left(1+\chi/4\alpha^{2}\right)$ bazeiaLeon . The modified
$S_{n}(\chi)$ functions are such that

$S_{modified}(\chi)=\frac{S_{standard}(\chi\rightarrow
f_{1}(\chi))}{f_{1,\chi}},$ (63)

where $S_{standard}(\chi)$ are the $\alpha-$independent functions obtained at
the scenarios bazeiaLeon ; Bazeia:2017mnc , and $f_{1,\chi}$ is the derivative
of $f_{1}(\chi)$ with respect to its argument. Therefore

$S_{n}(f_{n})=\frac{(1-f_{1}^{2})}{nf_{1,\chi}}U_{n-1}(f_{1}),$ (64)

that is

$S_{n}(\chi)=\frac{1}{n}S_{1}(\chi)U_{n-1}\left(\chi\left(1+\chi/4\alpha^{2}\right)\right),$
(65)

where $S_{1}(\chi)$ is given by expression (55), and $U_{n-1}$ are the
Chebyshev polynomials of second kind. Particularly, $n=1,2$ constitute the
last modified polynomial models presented in this section.

The deformed potentials are given by

$V_{n}(\chi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+\frac{1}{\alpha^{2}n^{2}}S_{1}^{2}(\chi)U_{n-1}^{2}\left(\chi\left(1+\chi/4\alpha^{2}\right)\right)}}\right),$
(66)

which is represented in Fig. 8 for some values of $n$, and $\alpha=1$. The
minima are

$\chi_{n,l}^{min}=-2\alpha^{2}\left(1\pm\sqrt{1+\frac{1}{\alpha^{2}}\text{cos}\left(\frac{l\pi}{n}\right)}\right)$
(67)

with $l=0,1,..,n$. The solutions are

$\chi_{n,m}(x)=2\alpha^{2}\left(-1\pm\sqrt{1+\frac{1}{\alpha^{2}}\text{cos}\left(\frac{\theta(x)+m\pi}{n}\right)}\right),$
(68)

where $\theta(x)=\arccos(\text{tanh}(x))$ and $m=0,..,2n-1$. The energy
density is obtained by

$\chi^{\prime}_{n,m}(x)=\frac{\sin\left(\frac{\theta+m\pi}{n}\right)\text{sech}(x)}{n\sqrt{1+\frac{1}{\alpha^{2}}\text{cos}\left(\frac{\theta+m\pi}{n}\right)}},$
(69)

and the stability potential reads

$\displaystyle
U(x)=\frac{1}{S_{1}U_{n-1}}\frac{d}{dx}\left(\frac{S_{1}U_{n-1}\left\\{\left(S_{1,\chi}+(1-n)f_{1}\right)U_{n-1}+nU_{n-2}\right\\}}{n\left(1+\frac{1}{\alpha^{2}n^{2}}S_{1}U_{n-1}^{2}\right)^{2}}\right).$

Figure 9: The potential (71) (left panel), the energy densities (centre panel)
and stability potentials (right panel) for the same values of $\alpha$ used in
Fig. 4.

###  IV.2 Models relatives of the $n-$Starobinsky

####  IV.2.1 The first generalized $n-$Starobinky

In this subsection, we are going to propose a first modification on the
standard $n-$Starobinsky potential. Initially, we suggest

$V(\phi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+\dfrac{16}{\alpha^{2}}\phi^{2}(1-\phi)^{2}}}\right),$
(71)

which will be our deforming model used to get a more general $n-$Starobinsky.
In this case, $\phi^{\prime}=4\phi(1-\phi)$ and the solutions are the same
obtained by the standard scenario, Eq. (30). The energy density is

$\rho(x)=\frac{\text{sech}^{4}(2x)}{\sqrt{1+\frac{1}{\alpha^{2}}\text{sech}^{4}(2x)}},$
(72)

and the stability potential is obtained by

$F_{X}V_{\phi}=-\frac{4\,\text{sech}^{2}(2x)\text{tanh}(2x)}{\left(1+\frac{1}{\alpha^{2}}\text{sech}^{2}(2x)\right)^{2}}.$
(73)

Fig. 9 shows how this model behaves as $\alpha$ varies. In Fig. 10, the same
quantities are compared with the ones found by the $\phi^{4}$ relative (51)
for a small $\alpha$. The kink solutions are already compared on the second
panel in Fig. 2. If $\alpha$ is large, the results become the same seen in
Fig. 2 for the standard situation.

Figure 10: A comparative between the modified models (51) and (71) fixing
$\alpha=0.5$, represented respectively by dashed (red) and solid (black)
lines.

Figure 11: The deformed $n-$Starobinsky potential (76), varying $\alpha$ and
fixing $n=2$, for the same $\alpha$ values used in Fig. 4.

From that, it is possible to propose a set of modified $n-$Starobinsky models,
by deforming the previous model

$S(\chi)=\frac{4f(1-f)}{f_{\chi}}.$ (74)

Through the same function adopted in the standard case (32), we have

$S(\chi)=1-\left(1-2\chi/n\right)^{2n},$ (75)

and the deformed potential (49) becomes

$V_{n}(\chi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+\frac{1}{\alpha^{2}}\left(1-\left(1-\frac{2}{n}\chi\right)^{2n}\right)^{2}}}\right),$
(76)

which has the same minima and maxima found by the ordinary $n-$Starobinsky.
The topological solutions also are the same found in the standard scenario,
Eqs. (37) and (38). The energy density and stability potential are modified
according to the DBI dynamics. The case $n=1$ is shown in Fig. 9, and the case
$n=2$ is shown in Fig. 11, for some $\alpha$ values.

The deformed potential (76) constitutes a generalization of the
$n-$Starobinsky model, as can be seen through the expansion $\alpha>>1$

$\displaystyle
V_{n}(\chi)=\frac{1}{2}\left(1-\left(1-\frac{2}{n}\chi\right)^{2n}\right)^{2}-\frac{3}{8\alpha^{2}}\left(1-\left(1-\frac{2}{n}\chi\right)^{2n}\right)^{4}+{\cal
O}\left(\frac{1}{\alpha^{4}}\right).$

At the limit $n\rightarrow\infty$, the expression (76) provides the following
modification on the original Starobinsky model

$V_{\alpha}(\chi)=\alpha^{2}\left(1-\frac{1}{\sqrt{1+\frac{1}{\alpha^{2}}\left(1-\text{e}^{-4\chi}\right)^{2}}}\right);$
(78)

for $\alpha>>1$, the original Starobinsky model is recovered at leading order

$V(\chi)=\frac{1}{2}\left(1-\text{e}^{-4\chi}\right)^{2}-\frac{3}{8\alpha^{2}}\left(1-\text{e}^{-4\chi}\right)^{4}+{\cal
O}\left(\frac{1}{\alpha^{4}}\right),$ (79)

as suggested on Fig. 12.

Figure 12: The first $\alpha-$Starobinsky potential $V_{\alpha}(\chi)$, Eq.
(78), varying the parameter $\alpha$.

####  IV.2.2 The second generalized $n-$Starobinsky

Figure 13: The second generalized $n-$Starobinsky model for $n=1$, and the
same $\alpha$ values used in Fig. 4. The potentials $V_{n,\alpha}(\chi)$, Eq.
(IV.2.2) (first panel), their kink solutions (IV.2.2) (second panel), the
energy densities (third panel), and the stability potentials (fourth panel).

Figure 14: The second generalized $n-$Starobinsky model for $n=2$, and the
same quantities adopted in Fig. 13.

Another possible generalization of the $n-$Starobinsky model is made applying
a deformation $\chi\rightarrow f(\phi)$ on the previous general model (76),
then

$S_{n}(\phi)=\frac{1-\left(1-\frac{2}{n}f\right)^{2n}}{f_{\phi}}.$ (80)

Selecting $f(\phi)=\phi(1+\phi^{2}/4\alpha^{2})$, we have

$S_{n}(\phi)=\frac{1-\left(1-\frac{2}{n}\phi\left(1+\frac{\phi^{2}}{4\alpha^{2}}\right)\right)^{2n}}{1+3\phi^{2}/4\alpha^{2}};$
(81)

and the novel modified $n-$Starobinsky potential is

$\displaystyle
V_{n,\alpha}(\phi)=\alpha^{2}\left(1-\frac{1+{3\phi^{2}}/{4\alpha^{2}}}{\sqrt{\left(1+\frac{3\phi^{2}}{4\alpha^{2}}\right)^{2}+\frac{1}{\alpha^{2}}\left(1-\left(1-\frac{2}{n}\phi\left(1+\frac{\phi^{2}}{4\alpha^{2}}\right)\right)^{2n}\right)^{2}}}\right).$

It forms another generalization of the standard $n-$Starobinsky that is
restored when $\alpha\rightarrow\infty$.

The kink solutions can be obtained by the substitution of $\chi\rightarrow
f(\phi)$ in the transcendental solution (38), that is

$\displaystyle\left(1-\frac{2}{n}\phi\left(1+\frac{\phi^{2}}{4\alpha^{2}}\right)\right)\Phi\left(\left(1-\frac{2}{n}\phi\left(1+\frac{\phi^{2}}{4\alpha^{2}}\right)\right)^{2n},1,\frac{1}{2n}\right)=-4x.$

Figure 15: The second $\alpha-$Starobinsky model, Eq. (IV.2.2), for the same
$\alpha$ values adopted in Fig. 13.

Figures 13 and 14 show the behavior of the potentials $V_{n,\alpha}(\chi)$,
the kink solutions, their energy densities and stability potentials for some
values of $\alpha$, and $n=1,2$. As the parameter $\alpha$ increases, the
results approach to the canonical $n-$Starobinsky; and for small $\alpha$, an
asymmetrical behavior is observed.

On the other hand, the limit $n\rightarrow\infty$ furnishes the following
modification on the original Starobinsky model

$\displaystyle
V_{\alpha}(\phi)=\alpha^{2}\left(1-\frac{1+{3\phi^{2}}/{4\alpha^{2}}}{\sqrt{\left(1+\frac{3\phi^{2}}{4\alpha^{2}}\right)^{2}+\frac{1}{\alpha^{2}}\left(1-\text{e}^{-4\phi\left(1+\frac{\phi^{2}}{4\alpha^{2}}\right)}\right)^{2}}}\right),$

which we are calling of second $\alpha-$Starobinsky model. As suggested in
Fig. 15, the ordinary Starobinsky is precisely recovered for $\alpha>>1$.

##  V Comments and conclusions

In this paper we have addressed the existence of scalar field solutions in DBI
dynamics in the presence of several potentials and their relatives, i.e.,
their deformed counterparts. The mechanism may find several applications from
topological solutions that can be considered as kinks in two-dimensional
spacetimes to domain walls or braneworlds in higher dimensions, such as four
or five-dimensional spacetimes brane . Furthermore, up to properly changing in
the coordinates, the static one-dimensional spatial solutions can be
analytical continued to provide cosmological solutions — see e.g., cvetic-
soleng  in four and bbc  in five-dimensions. Thus the scalar potentials of the
original theory have inflaton potentials counterparts in cosmology and their
form may satisfy the requirements to describe observational cosmology, mainly
in the inflationary phase. Interestingly enough we have found some relatives
of the Starobinsky model that approaches the shape found in the KKLT potential
KKLT  which has been found through flux compactification in string theory.
Several interesting investigations such as tunneling effects among the vacua
can be considered in the present scenario to address the issue of vacuum
stability. This can be crucial for both cosmology and bubble nucleation driven
by potentials whose relatives have a barrier shape in the presence of a softly
breaking term in the Lagrangian to render non-topological solutions kolb .
These are interesting points that merit further consideration in upcoming
studies.

## Acknowledgments

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant no.
165/2018), for partial financial support. FAB acknowledges support from CNPq
(Grant no. 312104/2018-9). The author E.E.M Lima also thank to PRPGI/IFBA and
PROEX/IFBA for their support.

## References

  * (1) M. Born and L. Infeld, Proc. Roy. Soc. Lond. A 144, n. 852, 425 (1934).
  * (2) P. A. M. Dirac, Proc. Roy. Soc. Lond. A 268, 57 (1962) .
  * (3) A. R. Brown, S. Sarangi, B. Shlaer, and A. Weltman, Phys. Rev. Lett. 99, 161601 (2007).
  * (4) S. Sarangi, JHEP 0807, 018 (2008).
  * (5) E. Babichev, Phys. Rev. D 77, 065021 (2008).
  * (6) M. Andrews, M. Lewandowski, M. Trodden and D. Wesley, Phys. Rev. D 82, 105006 (2010).
  * (7) D. Rubiera-Garcia and C. dos Santos, J. Phys. A: Math. Theor. 47, 105402 (2014).
  * (8) D. Bazeia, E. E. M. Lima and L. Losano, Annals Phys. 388, 408 (2018).
  * (9) R. Rajaraman, Solitons and Instantons (North-Holland, 1982); A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Others Topological Defects (Cambridge, 1994); N. Manton and P. Sutcliffe, Topological Solitons (Cambridge, 2004); P. Dorey, K. Mersh, T. Romanczukiewicz, and Ya. Shnir, Phys. Rev. Lett. 107, 091602 (2011); M. Kawasaki, K. Saikawa, and T. Sekiguchi, Phys. Rev. D 91, 065014 (2015); R. P. L. Azevedo and C. J. A. P. Martins, Phys. Rev. D 95, 043537 (2017).P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, England, 1995); D. Walgraef, Spatio- Temporal Pattern Formation (Springer-Berlag, New York, 1997); G. Bertotti, Hysteresis in Magnetism (Academic, San Diego, CA, 1998). A. Vanhaverbeke, A. Bischof, and R. Allenspach, Phys. Rev. Lett. 101, 107202 (2008); J. C. Y. Teo and C. L. Kane, Phys. Rev. B 82, 115120 (2010); D. Claudio-Gonzalez, A. Thiaville, and J. Miltat, Phys. Rev. Lett. 108, 227208 (2012); F.J. Buijnsters, A. Fasolino, and M.I. Katsnelson, Phys. Rev. Lett. 113, 217202 (2014); A. Bauer at al. Phys. Rev. B 95, 024429 (2017); A. O. Sorokin, Phys. Rev. B 95, 094408 (2017).
  * (10) D. Bazeia, L. Losano, R. Menezes and J. C. R. E. Oliveira, Eur. Phys. J. C 51, 953 (2007); D. Bazeia, L. Losano and R. Menezes, Phys. Lett. B 668, 246 (2008); Bazeia, D., L. Losano, and R. Menezes. Physics Letters B 731, 293 (2014).
  * (11) D. Bazeia, C. Furtado, A. R. Gomes, Journal of Cosmology and Astroparticle Physics 2004, 002 (2004); D. Bazeia, A. R. Gomes, Journal of High Energy Physics 2004, 012 (2004); F.A. Brito, L. Losano, J.R.L. Santos, Int. J. Mod. Phys. A (2021) doi:10.1142/S0217751X21500755] [arXiv:1911.00191[hep-th]].
  * (12) D. Bazeia, L. Losano and J. M. C. Malbouisson, Phys. Rev. D 66, 101701 (2002).
  * (13) D. Bazeia, L. Losano and R. Menezes, Phys. Lett. B 731, 293 (2014).
  * (14) G.R. Dvali, S.H.H. Tye, Phys. Lett. B 450, 72 (1999).
  * (15) S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, Phys. Rev. D 68, 046005 (2003).
  * (16) E. B. Bogomol’nyi, Sov. J. Nucl. Phys 24, 449 (1976); M. K. Prasad, C. M. Sommerfield, Phys. Rev. Lett. 35, 760 (1975).
  * (17) N. Rosen and P. M. Morse, Phys. Rev. 42, 210 (1932); G. Pöschl and E. Teller, Z. Phys. 83, 143 (1933).
  * (18) D. Bazeia, M. G. Leon, L. Losano, and J. M. Guilarte, Physical Review D 73, 105008 (2006).
  * (19) S. Santos da Costa, M. Benetti, R.M.P. Neves, F.A. Brito, R. Silva, J.S. Alcaniz, Eur. Phys. J. Plus 84, 136 (2021).
  * (20) D. Bazeia, L. Losano, R. Menezes, M.A.M. Souza, Europhys. Lett. 87 (2009) 21001.
  * (21) M. Cvetic and H. H. Soleng, Phys. Rev. D 51, 5768 (1995); M. Cvetic and H. H. Soleng, Phys. Rep. 282, 159 (1997); K. Skenderis and P. K. Townsend, Phys. Rev. Lett. 96, 191301 (2006).
  * (22) D. Bazeia, F.A. Brito, F.G. Costa, Phys. Lett. B 661, 179 (2008).
  * (23) E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley Publishing Company, New York, 1990).


# Learning-based Perception Contracts and Applications

Dawei Sun, Benjamin C. Yang and Sayan Mitra Coordinated Science Laboratory,
University of Illinois at Urbana-Champaign {daweis2, bcyang2,
<EMAIL_ADDRESS>
###### Abstract

Perception modules are integral in many modern autonomous systems, but their
accuracy can be subject to the vagaries of the environment. In this paper, we
propose a learning-based approach that can automatically characterize the
error of a perception module from data and use this for safe control. The
proposed approach constructs a perception contract (PC) which generates a set
that contains the ground-truth value that is being estimated by the perception
module, with high probability. We apply the proposed approach to study a
vision pipeline deployed on a quadcopter. With the proposed approach, we
successfully constructed a PC for the vision pipeline. We then designed a
control algorithm that utilizes the learned PC, with the goal of landing the
quadcopter safely on a landing pad. Experiments show that with the learned PC,
the control algorithm safely landed the quadcopter despite the error from the
perception module, while the baseline algorithm without using the learned PC
failed to do so.

##  I Introduction

Perception plays a crucial role in making modern autonomous systems react and
adapt in unstructured environments. In the past decade, we have seen
perception algorithms invented to perform a variety of tasks such as detection
and classification of obstacles [1], pose estimation [2], and localization
[3].

However, any interpretation of signals from the real world has to deal with
noise, which is sometimes poorly understood and characterized. As the output
of the perception module drives downstream decisions or control modules in an
autonomous system, the errors in perception may result in unwanted and even
catastrophic actions. Therefore, the problem of designing reliable perception-
based control algorithms for safety-critical systems has become an active
research topic, and attempts have been made in recent works, for example [4,
5].

One way to design reliable perception-based controllers is to first
characterize the perception error and then design controllers that are robust
to the perception error. In a series of recent works [6, 7], the notion of
perception contracts (PC) has been proposed to address this problem. A PC
characterizes the error of perception modules in a way that can be useful for
proving system-level invariants. The authors have shown that such perception
contracts can be automatically constructed from data and they used PCs to
verify several vision-based lane-keeping systems [6, 7]. In this paper, we aim
to solve the controller synthesis problem inspired by the idea of PCs.
Specifically, the perception contract used in this paper maps a perceived
value to a perception error, which characterizes the uncertainty of the
perception result111In the earlier works, the PC mapped the ground truth
values to perception error.. Such a perception contract can then be used by
the downstream controller and decision-making modules to compensate for the
uncertainty of the perception, which contributes to enhancing the safety and
robustness of the autonomous system.

In this paper, we consider perception modules that aim to estimate the value
of a specific quantity, for example, the position of an obstacle. Let $y$ be
the ground-truth value of this quantity. Most perception algorithms only
provide a single-point estimate $\hat{y}$, which can be seen as a noisy
version of $y$. However, in order to design robust controllers, the
uncertainty of the estimate $\hat{y}$ is also needed. To this end, we
construct a perception contract (PC) for the perception module, which is
simply a mapping from the perceived value $\hat{y}$ to a set that contains the
ground-truth value $y$ with a high probability. This containment should hold
even under environmental variations. We model the PC using a neural network
and carefully design loss functions that balance the error and
conservativeness of the PC. The only requirement for applying the proposed
approach is a data set that consists of pairs of estimate $\hat{y}$ and ground
truth $y$. Once trained on the data set, the PC can be executed online and
quickly compute the uncertainty of any estimate $\hat{y}$. The computed
uncertainty can then be used by downstream modules to compute safe control
signals.

We evaluated the proposed approach on a quadcopter equipped with a camera. The
perception module of interest is a program that aims to estimate the pose of a
landing pad from camera images. By applying the proposed approach, we
successfully constructed an accurate PC for the perception module.
Experimental results show that the error of the learned PC was below $0.2\%$
on the testing set. In order to study the conservativeness of the learned PC,
we tuned some parameters of the perception module to change the
characteristics of the perception error. We measured and reported the error of
the PC under these unseen perception parameters. Results show that the output
of the learned PC is tight and adheres to the perception parameter under which
it was trained. Then, in order to show the PC’s capability of being used by
downstream modules, we design a robust control algorithm that utilizes the
learned PC and aims to safely land the quadcopter on the landing pad.
Experiments show that with the learned PC, the quadcopter can safely complete
the landing task despite the significant perception error.

In summary, our contribution is threefold.

  * •

We proposed an approach that can automatically learn a perception contract
from data;

  * •

We demonstrated that the learned perception contract can be used by downstream
modules to compensate for the error of the upstream perception module;

  * •

We evaluated the proposed approach in a real-world application.

##  II Related work

#### Perception-based control

As new types of sensors emerge, the problem of integrating these sensors and
the corresponding perception modules into the control pipeline has attracted
interest. In recent works [8, 9, 10], perception-based controllers have been
studied to enable aggressive control for quadcopters. Further, data-driven
approaches have been developed by the machine learning community. For example,
imitation learning [11] and reinforcement learning [12] have been used to
learn vision-based control policies. However, those approaches cannot provide
any guarantees for the synthesized controller, which limits their application
in safety-critical applications.

As autonomous system enters more and more safety-critical domains, designing
perception-based control with formal safety guarantees has become a key
problem in robotics research. In a series of recent works by Dean et al., the
authors studied the robustness guarantees of perception-based control
algorithms under simple perception noises: In [13], the authors proposed a
perception-based controller synthesis approach for linear systems and provided
a theoretical analysis. In [4], the authors proposed robust barrier functions
that provide an approach for synthesizing safety-critical controllers under
uncertainty of the state. More complex perception modules have also been
studied. In [5], the authors studied the perception-based control synthesis
problem for a vehicle with a Lidar. The authors use neural networks to learn a
control Lyapunov function (CLF) and a control barrier function (CBF) in the
observation space, which enable safe navigation in an environment with unknown
obstacles. In [14], a perception module is learned with neural networks from
data. Instead of a single state, the perception module outputs a set of
states. Then, the authors apply contraction theory and robust motion planning
algorithms to synthesize control that is robust to the perception error. Our
approach is similar to the one proposed in [14] in the sense that they both
introduce a high-confidence set for the perception results. In contrast, our
approach can be applied to construct such high-confidence sets for an existing
perception module, while in [14], the authors have to construct these sets
while designing the perception module.

#### Analysis of systems with black-box modules

The proposed approach is also related to works concerning the analysis of
systems with black-box modules. As the complexity of autonomous systems
increases, it becomes impossible to analyze them exactly. Many modules in
modern autonomous systems are black boxes, for example, modules based on
neural networks. Data-driven approaches have been applied to analyze those
systems instead. The approach of constructing a perception contract from data
is mainly adapted from the reachability analysis approach proposed in [15],
where machine learning is used to analyze the reachability of black-box
systems. VerifAI [16] provides a complete framework for analyzing autonomous
systems with ML modules in the loop. In [6], the authors study safe
abstractions of ML-based perception modules, which inspires the approach
proposed in this paper. [6] is focused on verifying properties of an existing
system and thus concerned with predicting the estimate $\hat{y}$ given the
ground truth $y$. In contrast, this paper is focused on the synthesis problem
and thus aims to predict the ground truth $y$ given the estimate $\hat{y}$.

##  III Learning perception contracts

Figure 1: An autonomous system that entails a perception module and the
proposed perception contract.

In this paper, we consider general control systems that entail perception
modules as shown in Figure 1. The state of the system is denoted by
$x\in\mathcal{X}$, where $\mathcal{X}$ is the state space of the system. For
example, if the system is a quadcopter, the state vector $x$ may include the
position, velocity, and attitude of the quadcopter.

For a control system that interacts with the real world, access to only the
state $x$ of the system itself is not enough. It is very common that the value
of an external quantity $y\in\mathcal{Y}\subseteq\mathbb{R}^{n}$ is also
needed in order to complete a task. For example, for a quadcopter that is
asked to land, $y$ can be the position of the landing pad. However, access to
the ground-truth value of $y$ is often unavailable. To this end, a perception
module is used to estimate the value of $y$.

As shown in Figure 1, we assume that a perception module is given a priori.
The perception module aims at estimating the value of $y$. We denote the
output of the perception module by $\hat{y}$ and call it the perceived value.
Due to the complexity and uncertainty of the real world, the perceived value
does not always coincide with the ground-truth value, and there is a non-zero
perception error $e=\hat{y}-y$.

###  III-A Perception contracts

In order to design a robust controller that utilizes the perception results,
one has to analyze the perception module and characterize the perception
error. That is, given the state $x$ and the perceived value $\hat{y}$, one has
to be able to know some characteristics of the perception error $e$, for
example, the maximum of $\|e\|$. However, it is a challenging task. Although
there is a relationship between the state $x$, the ground-truth value $y$, and
the perceived value $\hat{y}$, it is usually not possible to find a closed-
form expression of this relationship. Inspired by [17], we propose to use
perception contracts as a unified way of characterizing perception errors.
Perception contracts then serve as an interface between the perception module
and the controller synthesis algorithm. Perception modules and controller
synthesis algorithms that follow the same interface can be freely paired.

In general, the relationship between the state $x$, the ground-truth value
$y$, and the perceived value $\hat{y}$ is not deterministic. Thus, we assume
that $x$, $y$ and $\hat{y}$ conform to a joint distribution $\mathcal{D}$,
i.e., $x,y,\hat{y}\sim\mathcal{D}$. We aim at finding a way to recover $y$
from $x$ and $\hat{y}$. To this end, we define a perception contract as a
mapping $\mathcal{A}:\mathcal{X}\times\mathcal{Y}\to 2^{\mathcal{Y}}$,
$x,\hat{y}\mapsto\mathcal{A}(x,\hat{y})$, where $2^{\mathcal{Y}}$ is the power
set of $\mathcal{Y}$. Ideally, given the state $x$ and the perceived value
$\hat{y}$, the output of the perception contract $\mathcal{A}(x,\hat{y})$
should contain the ground-truth value $y$, i.e., $y\in\mathcal{A}(x,\hat{y})$.
However, such a requirement for the perception contract is too strong, given
the fact that the relationship between $x$, $y$, and $\hat{y}$ is stochastic.
Instead, we allow approximately correct perception contracts, and the error of
a perception contract $\mathcal{A}$ is defined as follows.

######  Definition 1 (Error of a perception contract).

The error of a perception contract
$\mathcal{A}:\mathcal{X}\times\mathcal{Y}\to 2^{\mathcal{Y}}$ is

$\Pr_{x,y,\hat{y}\sim\mathcal{D}}\left(y\notin\mathcal{A}(x,\hat{y})\right).$
(1)

In this paper, we study the problem of constructing a perception contract of a
given perception module. Next, we will propose an approach to this problem,
where we model the perception contract using a neural network.

###  III-B Learning perception contracts from data

In this section, we propose an algorithm for learning a perception contract
from data. The perception contract is modeled with a neural network. By
definition, the output of the perception contract is a set. In order to use a
neural network to model a perception contract, we need to assume that the
output of the perception contract is from a finite-dimensional domain, i.e.,
the elements of this domain can be represented with finite-dimensional
vectors. In this paper, we adopt ellipsoids as the domain. That is, the output
of the perception contract is always an ellipsoid.

In theory, any finite-dimensional domain such as hyper-rectangles or zonotopes
could be used as the output of the perception contract. We adopt ellipsoids
due to their smoothness and simple representations. An ellipsoid in
$\mathbb{R}^{n}$ is defined by a center $c\in\mathbb{R}^{n}$ and a non-
singular matrix $C\in\mathbb{R}^{n\times n}$ and is denoted by
$\mathcal{E}\left(c,C\right):=\\{x\in\mathbb{R}^{n}:\|C(x-c)\|_{2}\leq 1\\}$.
Thus, in order to model a perception contract whose output is an ellipsoid, we
only need two parametric functions
$c_{\theta}:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}^{n}$ and
$C_{\theta}:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}^{n\times n}$ with
parameters $\theta\in\mathcal{W}$. Given a state $x$ and a perceived value
$\hat{y}$, $c_{\theta}$ and $C_{\theta}$ defines an ellipsoid with center
$c_{\theta}(x,\hat{y})$ and shape $C_{\theta}(x,\hat{y})$. Therefore, the
parametric perception contract is as follows

$\mathcal{A}_{\theta}(x,\hat{y}):=\mathcal{E}\left(c_{\theta}(x,\hat{y}),C_{\theta}(x,\hat{y})\right).$
(2)

As will be shown in Section IV-B, $c_{\theta}$ and $C_{\theta}$ are actually
modeled as two heads of the same neural network.

For the sake of simplicity, we denote $X=(x,y,\hat{y})$ and define a helper
function

$g_{\theta}(X):=\left\|C_{\theta}(x,\hat{y})\left(y-c_{\theta}(x,\hat{y})\right)\right\|_{2}.$
(3)

Clearly, $g_{\theta}(X)\leq 1$ if and only if
$y\in\mathcal{A}_{\theta}(x,\hat{y})$.

We aim at designing a learning algorithm to find a $\theta$ that minimizes the
error of the parametric perception contract $\mathcal{A}_{\theta}$. In other
words, we want to minimize the following loss function.

$L(\theta):=\mathop{\mathbb{E}}\limits_{X\sim\mathcal{D}}\left[\mathbb{I}_{\mathbb{R}^{\geq
0}}\left(g_{\theta}(X)-1\right)\right],$ (4)

where $\mathbb{I}_{\mathbb{R}^{\geq 0}}\left(\cdot\right)$ is the indicator
function, i.e., $\mathbb{I}_{\mathbb{R}^{\geq 0}}\left(x\right)=1$ if $x>0$,
and $\mathbb{I}_{\mathbb{R}^{\geq 0}}\left(x\right)=0$ otherwise.

In practice, the closed-form expression of the underlying distribution
$\mathcal{D}$ is not available, and one can only sample from $\mathcal{D}$.
Thus, the above loss function cannot be minimized directly since the
expectation over $\mathcal{D}$ cannot be computed exactly. Therefore, we
resort to empirical risk minimization as follows. First, a training set
$S=\\{X_{i}\\}_{i=1}^{N}$ is constructed, where the samples
$X_{i}=(x^{(i)},y^{(i)},\hat{y}^{(i)})$ are independently drawn from the data
distribution $\mathcal{D}$. Then, the empirical loss on $S$ is defined as
follows.

$L_{ERM}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\ell\left(g_{\theta}(X_{i})-1\right),$
(5)

where $\ell(x):=\max\\{0,\frac{x}{\alpha}+1\\}$ is the hinge loss function
with the hyper-parameter $\alpha>0$, which can be seen as a soft proxy for the
indicator function.

In addition to minimizing the error of the perception contract, we also have
to penalize its conservativeness. Otherwise, we might get trivial perception
contracts with extremely low error. For example, the error of the perception
contract $\mathcal{A}(x,\hat{y})\equiv\mathcal{Y}$ is $0$, but such a
perception contract is useless in the design of robust controllers. Thus, the
volume of the ellipsoid must be penalized. Inspired by [18], we use
$-\log(|C^{\intercal}C|)$ as a proxy of the volume of an ellipsoid
$\mathcal{E}\left(c,C\right)$, and the following regularization term is added
to penalize conservative outputs.

$L_{REG}(\theta)=-\frac{1}{N}\sum_{i=1}^{N}\log\Big{(}\left|C_{\theta}(x^{(i)},\hat{y}^{(i)})^{\intercal}C_{\theta}(x^{(i)},\hat{y}^{(i)})\right|\Big{)}.$
(6)

In practice, we solve the following optimization problem with stochastic
gradient descent.

$\hat{\theta}=\arg\min_{\theta}L_{ERM}(\theta)+\lambda L_{REG}(\theta),$ (7)

where $\lambda$ is a hyper-parameter balancing the error and conservativeness
of the perception contract.

###  III-C Probabilistic correctness of the perception contract

With standard techniques from statistical learning theory, we can derive the
following theorem which gives an upper bound on the error of the learned
perception contract.

######  Theorem 1.

For any $\epsilon>0$, and a random training set $S$ with $N$ i.i.d. samples,
with probability at least $1-2\exp(-2N\epsilon^{2})$, the following inequality
holds,

$\mathop{\mathbb{E}}\limits_{X\sim\mathcal{D}}\left[\mathbb{I}_{\mathbb{R}^{\geq
0}}\left(g_{\hat{\theta}}(X)-1\right)\right]\\\
\leq\frac{1}{N}\sum_{i=1}^{N}\tilde{\ell}(g_{\hat{\theta}}(X_{i})-1)+\frac{12}{\alpha}L_{g}\sqrt{\frac{p}{N}}+\epsilon,$
(8)

where $p$ is the number of scalar parameters of the neural network, and
$\tilde{\ell}(\cdot)=\min\\{1,\ell(\cdot)\\}$ is the truncated hinge loss, and
$L_{g}$ is the Lipschitz constant of $g_{\theta}$ w.r.t. $\theta$.

######  Remark.

Theorem 1 shows that by controlling $\epsilon$ and $N$, the actual error
$\mathop{\mathbb{E}}_{X\sim\mathcal{D}}\left[\mathbb{I}_{\mathbb{R}^{\geq
0}}\left(g_{\hat{\theta}}(X)-1\right)\right]$ can be made arbitrarily close to
the empirical loss
$\frac{1}{N}\sum_{i=1}^{N}\tilde{\ell}(g_{\hat{\theta}}(X_{i})-1)$, with
arbitrarily high probability. Furthermore, the empirical loss can be made very
small in practice due to the high capacity of the neural network. Of course,
there is no free lunch in general. In order to drive the empirical loss to
$0$, we might have to increase the number of parameters, which in turn
increases the term $\frac{12}{\alpha}L_{g}\sqrt{\frac{p}{N}}$.

##  IV Application study: safe quadcopter landing

In this section, we evaluate the approach proposed in the last section in a
real-world application, namely, safe quadcopter landing.

(a)

(b)

Figure 2: The quadcopter and the landing pad.

###  IV-A The safe landing problem

#### The quadcopter

In this section, we consider the problem of designing an algorithm for safely
landing the quadcopter as shown in Figure 2(a). The quadcopter is built based
on a DJI® F450 frame. The size of the quadcopter is $36$cm $\times$ $36$cm
$\times$ $10$cm measured without propellers. A Raspberry Pi 3® computer and a
Navio2® board222https://navio2.hipi.io/ are mounted on the quadcopter. The
Navio2® board provides essential sensors for quadcopter control such as
inertial measurement units (IMU) and a barometric pressure sensor. A camera is
connected to the Raspberry Pi computer. The camera can produce $320\times 240$
images at a rate of $60$ frames per second. The state of the quadcopter is

$x:=[p_{x},p_{y},p_{z},v_{x},v_{y},v_{z},\phi_{x},\phi_{y},\phi_{z},\omega_{x},\omega_{y},\omega_{z}],$
(9)

where $p_{x}$, $p_{y}$, and $p_{z}$ are the 3D position of the quadcopter and
similarly, $v$, $\phi$ and $\omega$ are velocity, attitude and angular
velocity respectively.

#### The workspace

The quadcopter is operating in a $6\text{m}\times 6\text{m}\times 3\text{m}$
workspace. Furthermore, a motion capture system (Vicon®) is used for low-
latency and high-accuracy localization of the quadcopter. The state $x$
estimated by the Vicon® system is viewed as the ground-truth state of the
quadcopter. As shown in Figure 2(b), a landing pad is placed on the ground of
the workspace. The goal of the quadcopter is to land on the landing pad.
Although the quadcopter has access to its state $x$ estimated by the Vicon®
system, the position of the landing pad is unknown to the quadcopter. We
denote the ground-truth position of the landing pad by $y\in\mathbb{R}^{3}$.

#### Coordinate frames

There are several coordinate frames that will be used in the experiments. They
are the world frame $W$, the quadcopter frame $Q$, and the camera frame $C$.
For the same quantity, we use subscripts to discriminate among its values
under different coordinate frames. For example, $y_{W}$ is the coordinate of
the landing pad in the world frame, while $y_{C}$ is that in the camera frame.
When the subscript is omitted, it denotes the value in the world frame.

#### The vision-based perception module

In order to land on the landing pad, the quadcopter uses the camera to
estimate the position $y_{W}$. To this end, we attach an ArUco marker [19] to
the landing pad. An ArUco marker is basically a QR-code-like image, which
enables robust and fast pose estimation from images. By calling functions
implemented in the OpenCV library [20], we can estimate the position of the
landing pad in the camera frame, and such an estimate is denoted by
$\hat{y}_{C}$. Then, we transform $\hat{y}_{C}$ into the world frame
$\hat{y}_{W}$ using the extrinsic parameter between the camera and the
quadcopter, and the quadcopter’s pose $x_{W}$ in the world frame. However,
$\hat{y}_{C}$ and $x_{W}$ are measured by two different sensors (camera and
Vicon) without synchronization, and this asynchronization contributes as the
main source of the perception error.

#### The task specification

The goal of the quadcopter is to navigate to a box around the ground-truth
position $y$ of the landing pad and then turn off the motors. The box is
defined as $\\{y\\}\oplus G$ where $G$ is a $0.1m\times 0.1m\times 0.05m$ box
defined as $G:=[-0.05,0.05]\times[-0.05,0.05]\times[0,0.05]$, and $\oplus$ is
the Minkowski addition. The quadcopter is said to violate the safety
requirement if it stops the motors outside $\\{y\\}\oplus G$.

###  IV-B Learning a perception contract

Next, we apply the proposed approach to learn an abstraction of the perception
module. The most straightforward idea is to directly apply the proposed
approach to learn a perception contract as a function of $x$ and $\hat{y}$ in
the world frame. However, by doing this, we bind the learned perception
contract to a specific transformation between the quadcopter and the camera.
Such a perception contract will not be usable after the transformation between
the quadcopter and camera is changed, for example, when the camera direction
is switched from forward to downward. To tackle this issue, we instead learn a
perception contract in the camera frame, i.e., $x_{C}$, $y_{C}$, and
$\hat{y}_{C}$ will be used in learning. As the first step, we construct a data
set.

#### Construction of the data set

During the process of collecting data, we fix the position of the landing pad
and measure its ground-truth position $y$ beforehand. Then, we program the
quadcopter to follow some predefined paths and record the state $x$ and the
perceived value $\hat{y}$ on the fly. The pre-defined paths are designed such
that by following it the quadcopter explores diverse positions with diverse
velocities in the workspace. We then transform all the quantities into the
camera frame and obtained the data set
$\\{(x_{C}^{(i)},y_{C}^{(i)},\hat{y}_{C}^{(i)})\\}_{i=1}^{N}$ with $N=5000$.

#### Machine learning setup

We model the perception contract using a three-layer neural network of which
the hidden layers contain $64$ and $128$ neurons. The input of the neural
network is the concatenation of the state $x_{C}$ and the perceived value
$\hat{y}_{C}$. The output size of the neural network is $12$. The first three
elements of the neural network’s output are used as the center of the
ellipsoid and the remaining nine elements are reshaped into a $3\times 3$
matrix that describes the shape of the ellipsoid. We train the neural network
for $20$ epochs using the Adam [21] optimizer. The learning rate is set to
$0.001$.

#### Test of the learned perception contract

After training, we evaluate the learned perception contract using real data
and estimate its error. To do that, we manually control the quadcopter to fly
in the workspace until the quadcopter finishes $\sim 300$ measurements of the
landing pad. For each measurement, we check whether
$y\in\mathcal{A}(x,\hat{y})$ and compute an empirical error rate. As mentioned
before, the perception error mainly comes from the asynchronization between
the camera and Vicon. By tuning the camera buffer size in the OpenCV library,
we can change the extent to which the camera image lags from the Vicon
measurements. For each buffer size configuration, we conducted the same test
and reported the error of the learned perception contract in Table I. The
training data was collected at buffer size $=1$. As can be seen from Table I,
the error of the learned perception contract is as low as $0.19\%$ at buffer
size $=1$. On the other hand, the learned perception contract adheres to the
buffer size configuration under which it was trained. As buffer size
increases, the error of the learned perception contract drastically increases,
which suggests that the output of the perception contract is not conservative.
Moreover, it takes only $\sim 3$ milliseconds on the Raspberry Pi computer to
compute $\mathcal{A}(x,\hat{y})$ for each query $x$ and $\hat{y}$, which
enables it to be used in online control pipelines.

Table I: Error rate of the learned perception contract under different camera buffer size configurations. Buffer size | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Error | $0.19\%$ | $0.93\%$ | $2.05\%$ | $4.07\%$ | $11.05\%$

###  IV-C Utilizing the learned perception contract

Figure 3: The state machine of the safe landing algorithm.

(a)

(b)

(c)

Figure 4: Illustration of the safe landing algorithm. The green ellipsoids are
the output of the PC. (a) The uncertainty of the measurement is above the
threshold. In this case, the quadcopter will follow the green line and
navigate to a new waypoint (blue). (b) The uncertainty of the measurement is
below the threshold. In this case, the quadcopter first moves horizontally and
then descends to complete the landing. (c) The quadcopter safely lands on the
landing pad.

In order to demonstrate the PC’s capability of being used by downstream
modules, we design a simple algorithm that utilizes the output of the
perception contract to solve the safe landing problem. The proposed algorithm
is a state machine as shown in Figure 3. An overview of the state machine is
as follows. In the beginning, the quadcopter estimates the position of the
landing pad using its perception module. Then, it runs the perception contract
to compute the ellipsoid that contains the ground-truth position. If the size
of the ellipsoid is above a threshold, the quadcopter navigates to a new
waypoint determined by the ellipsoid and returns to the initial state to do
another measurement. The quadcopter keeps doing this until the size of the
ellipsoid is below the threshold. In that case, the quadcopter will navigate
to the center of the ellipsoid and then turn off the motor. We illustrated the
algorithm in Figure 4. Next, we elaborate on each state in the state machine.

#### Measuring

In this state, the quadcopter obtains an image from the camera and feeds it to
the perception module. The perception module estimates the position of the
landing pad as $\hat{y}$. Then, the quadcopter calls the perception contract
with its current state $x$ and the perceived value $\hat{y}$. The perception
contract outputs an ellipsoid $\mathcal{A}(x,\hat{y})$. Intuitively, the size
of the ellipsoid reflects the uncertainty in the measurement. If the ellipsoid
is small, then the quadcopter can trust the measurements. Otherwise, it has to
take another measurement. This is formalized as follows. Let $p_{xmin}$,
$p_{xmax}$, $p_{ymin}$, $p_{ymax}$, $p_{zmin}$, and $p_{zmax}$ be the minimum
or maximum along each dimension of the ellipsoid $\mathcal{A}(x,\hat{y})$. If
$p_{xmax}-p_{xmin}\leq 0.1$, $p_{ymax}-p_{ymin}\leq 0.1$, and
$p_{zmax}-p_{zmin}\leq 0.05$, i.e., the ellipsoid can be completely contained
in the box $G$, then quadcopter trusts the measurement and switches to the
Landing state. Otherwise, it enters the New Waypoint state and navigates to
somewhere else in order to take another measurement.

#### New Waypoint

The quadcopter enters this state because the size of the ellipsoid
$\mathcal{A}(x,\hat{y})$ is beyond the threshold. In this state, the
quadcopter will navigate to another waypoint to take a new measurement. The
new waypoint is selected as follows. Since the ground-truth position of the
landing pad can be at any point in the ellipsoid, to avoid hitting the landing
pad, the quadcopter sets the next waypoint as the intersection point between
the surface of the ellipsoid and the line segment connecting its current
position $[p_{x},p_{y},p_{z}]$ and the center of $\mathcal{A}(x,\hat{y})$.

#### Landing

In this state, the quadcopter will navigate to the point
$P_{\mathit{turnoff}}:=\left[\frac{p_{xmin}+p_{xmax}}{2},\frac{p_{ymin}+p_{ymax}}{2},p_{zmax}\right]$
and shut off the motors. Recall that the quadcopter enters this state because
the ellipsoid $\mathcal{A}(x,\hat{y})$ can be completely contained in $G$.
Therefore, we immediately have the following proposition.

######  Proposition 1.

If the ground truth $y$ is in the ellipsoid $\mathcal{A}(x,\hat{y})$, then
$P_{\mathit{turnoff}}\in\\{y\\}\oplus G$.

Combining Theorem 1 and Proposition 1, it follows that with a high
probability, the quadcopter will shutoff its motors only inside the target
region defined by $\\{y\\}\oplus G$.

In order to evaluate the above algorithm, we put the landing pad at $10$
randomly selected locations and ran the above algorithm to land the
quadcopter. Furthermore, in order to show the difficulty of the landing
problem and emphasize the advantage of the perception contract, we also
compare it with a simple baseline approach, where we adopt exactly the same
state machine as in Figure 3 but replace the learned perception contract with
a trivial one, namely,
$\mathcal{A}(x,\hat{y})=\mathcal{E}\left(\hat{y},10000I_{3}\right)$, i.e., it
always outputs a very small ellipsoid around $\hat{y}$. The results show that,
with the learned PC, the quadcopter safely landed on the landing pad in all of
the $10$ runs, while the baseline approach using the trivial PC failed in $9$
of them. In Figure 5, we visualized the $10$ runs. For each run, we translated
the ground-truth position of the landing pad to the origin. In order to make a
clear visualization without occlusion, we also rotated the data of some runs.
As can be seen from the figure, due to the existence of the perception error,
the baseline approach using the trivial PC landed the quadcopter at locations
far off the origin, while with the learned PC, the control algorithm refused
to trust the first measurement and conduct another more accurate measurement,
which leads the quadcopter to the precise location of the landing pad. A video
of the runs is available at https://youtu.be/Jep_6u_AxD4.

Figure 5: 2D visualization of 10 runs. Each run is visualized in a unique
color. The crosses mark the positions where the baseline approach turns off
the motors. The stars mark the locations where the approach using the learned
PC turns off the motors. Circles correspond to the measurements made by the
proposed approach. The center of the circle is the location where measurement
is made, and size of the circle corresponds to the size of the ellipsoid
computed by PC.

##  V Conclusion

We demonstrated that learned perception contracts can be easily used to
achieve reliable and safe control. However, limitations and research
opportunities still remain. In this paper, the control algorithm is
straightforward and mainly used to demonstrate the perception contract. It
does not have any convergence guarantee, i.e., new measurements are not
necessarily more accurate. In future work, we will explore more sophisticated
algorithms to jointly synthesize the perception contract and downstream
controller.

## References

  * [1] R. Girshick, “Fast r-cnn,” in _Proceedings of the IEEE international conference on computer vision_ , 2015, pp. 1440–1448.
  * [2] Y. Xiang, T. Schmidt, V. Narayanan, and D. Fox, “Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes,” _arXiv preprint arXiv:1711.00199_ , 2017.
  * [3] R. Mur-Artal, J. M. M. Montiel, and J. D. Tardos, “ORB-SLAM: a versatile and accurate monocular slam system,” _IEEE transactions on robotics_ , vol. 31, no. 5, pp. 1147–1163, 2015.
  * [4] S. Dean, A. Taylor, R. Cosner, B. Recht, and A. Ames, “Guaranteeing safety of learned perception modules via measurement-robust control barrier functions,” in _Conference on Robot Learning_. PMLR, 2021, pp. 654–670.
  * [5] C. Dawson, B. Lowenkamp, D. Goff, and C. Fan, “Learning safe, generalizable perception-based hybrid control with certificates,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 1904–1911, 2022.
  * [6] C. Hsieh, Y. Li, D. Sun, K. Joshi, S. Misailovic, and S. Mitra, “Verifying controllers with vision-based perception using safe approximate abstractions,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems_ , vol. 41, no. 11, pp. 4205–4216, 2022.
  * [7] A. Astorga, C. Hsieh, P. Madhusudan, and S. Mitra, “Perception contracts for safety of ml-enabled systems,” _Proceedings of the ACM on Programming Languages_ , vol. 1, no. OOPSLA, 2023.
  * [8] Y. Lin, F. Gao, T. Qin, W. Gao, T. Liu, W. Wu, Z. Yang, and S. Shen, “Autonomous aerial navigation using monocular visual-inertial fusion,” _Journal of Field Robotics_ , vol. 35, no. 1, pp. 23–51, 2018.
  * [9] G. Loianno, C. Brunner, G. McGrath, and V. Kumar, “Estimation, control, and planning for aggressive flight with a small quadrotor with a single camera and imu,” _IEEE Robotics and Automation Letters_ , vol. 2, no. 2, pp. 404–411, 2016.
  * [10] S. Tang, V. Wüest, and V. Kumar, “Aggressive flight with suspended payloads using vision-based control,” _IEEE Robotics and Automation Letters_ , vol. 3, no. 2, pp. 1152–1159, 2018.
  * [11] F. Codevilla, M. Müller, A. López, V. Koltun, and A. Dosovitskiy, “End-to-end driving via conditional imitation learning,” in _2018 IEEE international conference on robotics and automation (ICRA)_. IEEE, 2018, pp. 4693–4700.
  * [12] F. Sadeghi and S. Levine, “Cad2rl: Real single-image flight without a single real image,” in _Proceedings of Robotics: Science and Systems_ , Cambridge, Massachusetts, July 2017.
  * [13] S. Dean, N. Matni, B. Recht, and V. Ye, “Robust guarantees for perception-based control,” in _Learning for Dynamics and Control_. PMLR, 2020, pp. 350–360.
  * [14] G. Chou, N. Ozay, and D. Berenson, “Safe output feedback motion planning from images via learned perception modules and contraction theory,” in _International Workshop on the Algorithmic Foundations of Robotics_. Springer, 2022, pp. 349–367.
  * [15] D. Sun and S. Mitra, “Neureach: Learning reachability functions from simulations,” in _International Conference on Tools and Algorithms for the Construction and Analysis of Systems_. Springer, 2022, pp. 322–337.
  * [16] T. Dreossi, D. J. Fremont, S. Ghosh, E. Kim, H. Ravanbakhsh, M. Vazquez-Chanlatte, and S. A. Seshia, “Verifai: A toolkit for the design and analysis of artificial intelligence-based systems,” _arXiv preprint arXiv:1902.04245_ , 2019.
  * [17] C. Hsieh, K. Joshi, S. Misailovic, and S. Mitra, “Verifying controllers with convolutional neural network-based perception: a case for intelligible, safe, and precise abstractions,” _arXiv preprint arXiv:2111.05534_ , 2021.
  * [18] A. Devonport and M. Arcak, “Estimating reachable sets with scenario optimization,” in _Learning for dynamics and control_. PMLR, 2020, pp. 75–84.
  * [19] S. Garrido-Jurado, R. Muñoz-Salinas, F. J. Madrid-Cuevas, and M. J. Marín-Jiménez, “Automatic generation and detection of highly reliable fiducial markers under occlusion,” _Pattern Recognition_ , vol. 47, no. 6, pp. 2280–2292, 2014.
  * [20] G. Bradski, “The OpenCV Library,” _Dr. Dobb’s Journal of Software Tools_ , 2000.
  * [21] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _arXiv preprint arXiv:1412.6980_ , 2014.