A note on integrability loss
in fuzzball geometries
Maxim Emelin, Stefano Massai
Dipartimento di Fisica e Astronomia “Galileo Galilei”
Università di Padova, Via Marzolo 8, 35131 Padova, Italy
&

INFN, Sezione di Padova
Via Marzolo 8, 35131 Padova, Italy

###### Abstract

We study the dynamics of certain string configurations in a class of fivebrane
supertube backgrounds. In the decoupling limit of the fivebranes, these
solutions are known to admit an exact description in worldsheet string theory
and string propagation is integrable. For the asymptotically flat solutions,
we prove, by using analytic tools of classical Hamiltonian systems, the non-
integrability of classical string motion. This suggests that string dynamics
in circular supertube geometries exhibit a regime of chaotic behaviour.

##  1 Introduction

In recent years there has been considerable progress in constructing black
hole microstate geometries (see [1] for a recent review) and in studying their
dynamical properties. In some special cases, it has been shown that the
dynamics of strings on such geometries is described by exact null-gauged WZW
models [2, 3, 4], which allow to study stringy aspects that are invisible in
supergravity. The simplest example is the circular NS5-F1 supertube. The exact
description is found after taking the decoupling limit for the fivebranes. The
resulting geometry interpolates between a linear dilaton region in the UV and
an orbifold of $AdS_{3}$ in the IR. The null-gauging construction that “adds
back the one” in the F1 harmonic function has been generalized to encompass a
larger class of Lunin-Mathur solutions in [5]. It is natural to ask to what
extent one can extend these constructions in order to describe the full
backgrounds by similarly “adding back the one” in the fivebrane harmonic
function, effectively recoupling the flat space asymptotics.

In this note we will show that there is a considerable difference between the
interpolation from the $AdS_{3}$ region to the linear dilaton region, and the
interpolation to the asymptotically flat region. By studying the classical
Hamiltonian system that describes the dynamics of strings on such geometries,
we will show that properties of certain string configurations in the solutions
with linear dilaton asymptotics are consistent with integrability, in
agreement with the fact that they are described by an explicit gauged WZW
model. On the other hand, coupling back the asymptotically flat region has the
consequence of breaking this integrability. We will prove this fact by using
analytic tools that have been developed in the context of Hamiltonian systems
and that have been already applied to study the (non)integrability of strings
in different contexts (see for example [6, 7, 8, 9, 10, 11, 12]).

The notion of integrability of a classical Hamiltonian system, in the sense of
Liouville, relies on the behaviour of trajectories in phase space, as
specified by the Arnold-Liouville theorem. In particular, integrable systems
are non-chaotic. The transition to chaotic behaviour for small perturbations
around an integrable system is described by KAM theory. The work of Ziglin
provides effective criteria for the integrability and non-integrability of
Hamiltonian systems in terms of analytic properties of the corresponding
differential equations, through the analysis of the monodromy group of the
linearized system normal to an integrable plane of solutions. These criteria
have been simplified by the use of differential Galois theory in what is known
as the Ziglin-Morales-Ramis theory. The main result is that a necessary
condition for a system to be integrable is that the Galois group of the normal
variational equation (NVE) has an abelian identity component. In practice,
this means that one can prove non-integrability of a system by perturbing the
equations of motion around a particular solution to linear order (while
respecting the necessary constraints) and showing that the resulting equations
do not admit Liouvillian solutions.

This criterion is particularly useful because the work of Kovacic [13]
provides an explicit algorithm, based on Picard-Vessiot theory, that allows to
check the properties of the Galois group of a given differential equation and
in particular, either find Liouvillian solutions or prove their non-existence.
This algorithm is known as the Kovacic algorithm (see Appendix A for a brief
review). For a general introduction see for example [14]. We will apply these
tools in the string theory context to check whether the dynamics of classical
strings in supertube backgrounds are integrable or not.

The paper is organized as follows. In Section 2 we describe the supergravity
solutions for supertubes and their decoupling limits. In Section 3 we derive
the equations of motion for a particular configuration of strings moving in
such geometries and we derive the corresponding NVEs. We then apply the
Kovacic algorithm to these equations in order to check the necessary condition
for integrability of the system and find that it fails for the asymptotically
flat backgrounds, implying non-integrability. In Section 4 we conclude and
discuss some consequences of our result. We relegate a review of the Kovacic
algorithm and details of its application to our NVEs to Appendix A.

##  2 Supergravity solutions

In this Section we review the solutions corresponding to a circular
distribution of NS5 branes and to the circular fuzzball geometry in the NS5
frame.

###  2.1 Coulomb branch of NS5 branes

The solution sourced by a collection of $n_{5}$ NS5 branes parallel to the
$0-5$ directions and placed at points $x_{n}^{i}$, $i=6,7,8,9$ in the
transverse space is:

$\displaystyle ds^{2}$
$\displaystyle=\eta_{\mu\nu}dx^{\mu}dx^{\nu}+Hdx^{i}dx_{i}\,,$ (2.1)
$\displaystyle e^{2\Phi}$ $\displaystyle=g_{s}^{2}H\,,$ $\displaystyle
H_{ijk}$ $\displaystyle=-\epsilon^{l}_{ijk}\partial_{l}H\,,$

where the harmonic function $H$ is given by111We set $\alpha^{\prime}=1$.

$H=1+\sum_{n=1}^{n_{5}}\frac{1}{|x^{i}-x^{i}_{n}|^{2}}\,.$ (2.2)

We consider the NS5 branes equally distributed on a circle of radius $a$ in
the $(x_{6},x_{7})$ plane. It is useful to use the following coordinates on
the transverse $\mathbb{R}^{4}$:

$\displaystyle x_{6}+x_{7}=re^{i\phi}\,,\quad x_{8}+ix_{9}=Re^{i\psi}\,.$
(2.3)

The fivebrane harmonic function can then be written as [15, 16]

$H=1+\frac{n_{5}\Lambda_{n_{5}}}{2ar\sinh(\chi)}\,,\quad\Lambda_{n_{5}}=\frac{\sinh(n_{5}\chi)}{\cosh(n_{5}\chi)-\cos(n_{5}\phi)}\,,$
(2.4)

with

$2ar\cosh\chi=r^{2}+R^{2}+a^{2}\,.$ (2.5)

We will consider the limit $\Lambda_{n_{5}}\rightarrow 1$, which corresponds
to a smeared distribution of branes along the circle. Setting, in the notation
of [2]

$\displaystyle r=a\cosh\rho\sin\theta\,,\quad R=a\sinh\rho\cos\theta\,,$ (2.6)

one finds

$H=1+\frac{n_{5}}{a^{2}\Sigma}\,,\quad\Sigma=\cosh^{2}\rho-\sin^{2}\theta\,,$
(2.7)

and the transverse metric is:

$\displaystyle ds^{2}_{\perp}$
$\displaystyle=(a^{2}\Sigma+n_{5})\Bigl{[}d\rho^{2}+d\theta^{2}+\frac{1}{\Sigma}\Bigl{(}{\cosh}^{2}\\!\rho\sin^{2}\\!\theta\,d\phi^{2}+{\sinh}^{2}\\!\rho\cos^{2}\\!\theta\,d\psi^{2}\Bigr{)}\Bigr{]}\,.$
(2.8)

In the decoupling limit of the fivebranes the metric and B-field are:

$\displaystyle ds^{2}_{\perp}$
$\displaystyle=n_{5}\Bigl{[}d\rho^{2}+d\theta^{2}+\frac{1}{\Sigma}\Bigl{(}{\cosh}^{2}\\!\rho\sin^{2}\\!\theta\,d\phi^{2}+{\sinh}^{2}\\!\rho\cos^{2}\\!\theta\,d\psi^{2}\Bigr{)}\Bigr{]},$
(2.9) $\displaystyle B$
$\displaystyle=\frac{n_{5}\cos^{2}\theta\cosh^{2}\rho}{\Sigma}\,d\phi\wedge
d\psi\,.\qquad$

String theory in this decoupled background is exactly solvable and it
corresponds to the coset orbifold [15, 17]

$\left(\frac{SL(2)}{U(1)}\times\frac{SU(2)}{U(1)}\right)\big{/}Z_{n_{5}}\,,$
(2.10)

or, equivalently, to the null gauging [18]

$\frac{SL(2)\times SU(2)}{U(1)_{L}\times U(1)_{R}}\,.$ (2.11)

###  2.2 NS5-P supertube

Starting from the fivebrane configuration described in the previous section
one can construct a round supertube in the following way. We compactify the
$1-5$ directions on $T^{4}\times S^{1}$; call the $S^{1}$ direction $y$; we
identify $y\sim y+2\pi R$. We can then tilt the NS5 brane strands along the
$y,\phi$ directions and join them together to make a helical profile. This
system can be stabilised by spinning the fivebranes. The supergravity solution
sourced by such a configuration can be found by following a series of
dualities starting from the F1-P system [19] (see also [20]). The result is

$\displaystyle ds^{2}$
$\displaystyle=-du\,dv+ds_{\scriptscriptstyle\mathbf{T}^{4}}^{2}+(a^{2}\Sigma+{n_{5}})\Bigl{(}d\rho^{2}+d\theta^{2}\Bigr{)}$
$\displaystyle\hskip
17.07182pt+\frac{(a^{2}\Sigma+{n_{5}})}{\Sigma}\Bigl{(}{\cosh}^{2}\\!\rho\sin^{2}\\!\theta\,d\phi^{2}+{\sinh}^{2}\\!\rho\cos^{2}\\!\theta\,d\psi^{2}\Bigr{)}$
$\displaystyle\hskip
17.07182pt+\frac{1}{\Sigma}\Bigl{[}\frac{2k}{R}\sin^{2}\\!\theta\,dv\,d\phi+\frac{k^{2}}{{n_{5}}R^{2}}dv^{2}\Bigr{]},$
(2.12) $\displaystyle B$
$\displaystyle=\frac{n_{5}\cos^{2}\theta\cosh^{2}\rho}{\Sigma}d\phi\wedge
d\psi+\frac{k\cos^{2}\theta}{R\,\Sigma}dv\wedge d\psi\leavevmode\nobreak\
,\qquad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
e^{2\Phi}=1+\frac{n_{5}}{a^{2}\Sigma}\;.$

where $u=t+y$, $v=t-y$ and the supertube radius is fixed in terms of the
fivebrane and momentum charge by

$a=\frac{\sqrt{Q_{p}Q_{5}}R}{k}\,.$ (2.13)

The fivebrane decoupling limit of this solution admits an exact description in
worldsheet string theory as a null gauging that modifies (2.11)[2]

$\frac{SL(2)\times SU(2)\times S_{y}\times\mathbb{R}_{t}}{U(1)_{L}\times
U(1)_{R}}\,,$ (2.14)

where we gauge two independent left and right null currents in the upstairs
group. After integrating out the gauge fields, one finds a sigma model that
describes the following metric and B-field:

$\displaystyle ds^{2}$
$\displaystyle=\Bigl{(}-du\,dv+ds_{\scriptscriptstyle\mathbf{T}^{4}}^{2}\Bigr{)}+{n_{5}}\Bigl{[}d\rho^{2}+d\theta^{2}+\frac{1}{\Sigma}\Bigl{(}{\cosh}^{2}\\!\rho\sin^{2}\\!\theta\,d\phi^{2}+{\sinh}^{2}\\!\rho\cos^{2}\\!\theta\,d\psi^{2}\Bigr{)}\Bigr{]}$
$\displaystyle\hskip
56.9055pt+\frac{1}{\Sigma}\Bigl{[}\frac{2k}{R}\sin^{2}\\!\theta\,dv\,d\phi+\frac{k^{2}}{{n_{5}}R^{2}}dv^{2}\Bigr{]},$
(2.15) $\displaystyle B$
$\displaystyle=\frac{n_{5}\cos^{2}\theta\cosh^{2}\rho}{\Sigma}d\phi\wedge
d\psi+\frac{k\cos^{2}\theta}{R\,\Sigma}dv\wedge d\psi\leavevmode\nobreak\
.\qquad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $

This indeed coincides with the fivebrane decoupling limit of the solution
(2.2).

###  2.3 NS5-F1 supertube

Starting from the metric of the NS5-P supertube, a T-duality along the $y$
circle brings the metric to the following form:

$\displaystyle ds^{2}$
$\displaystyle=ds_{\scriptscriptstyle\mathbf{T}^{4}}^{2}-\Bigl{(}1-\frac{\alpha^{2}}{{n_{5}}\Sigma_{1}}\Bigr{)}du\>\\!dv+(a^{2}\Sigma+n_{5})\Bigl{(}d\rho^{2}+d\theta^{2}\Bigr{)}$
$\displaystyle\hskip
17.07182pt+\Bigl{(}a^{2}\cosh^{2}\rho+n_{5}+\frac{n_{5}\sin^{2}\theta}{\Sigma_{1}}\Bigr{)}\sin^{2}\theta
d\phi^{2}$ $\displaystyle\hskip
17.07182pt+\Bigl{(}a^{2}\sinh^{2}\rho+n_{5}-\frac{n_{5}\cos^{2}\theta}{\Sigma_{1}}\Bigr{)}\cos^{2}\theta
d\psi^{2}+\frac{2\alpha}{\Sigma_{1}}\Bigl{(}{\sin^{2}\\!\theta\,dt\,d\phi+\cos^{2}\\!\theta\,dy\,d\psi}\Bigr{)}\,,$
$\displaystyle B$
$\displaystyle=n_{5}\cos^{2}\theta\Bigl{(}1+\frac{\sin^{2}\theta}{\Sigma_{1}}\Bigr{)}d\phi\wedge
d\psi-\frac{\alpha^{2}}{n_{5}\Sigma_{1}}\,dt\wedge dy$ $\displaystyle\hskip
17.07182pt{}+\frac{\alpha\cos^{2}\\!\theta}{\Sigma_{1}}dt\wedge
d\psi+\frac{\alpha\sin^{2}\\!\theta}{\Sigma_{1}}dy\wedge
d\phi\leavevmode\nobreak\ ,$ $\displaystyle\hskip
17.07182pte^{2\Phi}=\frac{a^{2}\Sigma+n_{5}}{a^{2}\Sigma_{1}}\leavevmode\nobreak\
,\quad\Sigma_{1}=\frac{\alpha^{2}}{{n_{5}}}+\Sigma\,,$ (2.16)

where $\alpha=k\tilde{R}$, $a=\sqrt{Q_{1}Q_{5}}/(k\tilde{R})$, with
$\tilde{R}$ the T-dual radius. The fivebrane decoupled limit of this solution
is again described by a null gauging of the form (2.14). In fact T-duality
just amounts to flipping a sign in the left current being gauged. This
decoupled geometry provides an interpolation between a linear dilaton region
in the UV to an orbifold of $AdS_{3}$ in the IR. The fact that it arises as a
gauged WZW model implies, in particular, that string dynamics on this geometry
is integrable.

##  3 String motion in NS5 backgrounds

We now turn to the classical dynamics of strings in the configuration of NS5
branes described in the previous section. In order to illustrate the procedure
for a simple case, we first consider the NS5 decoupling limit of the circular
distribution of fivebranes, before proceeding to the asymptotically flat cases
of the circular NS5 distribution and the supertube background.

###  3.1 Decoupled limit

We are interested in studying the motion of classical strings in the geometry
(2.9), which is obtained by taking a decoupling limit of the asymptotically
flat solution describing a circular distribution of NS5 branes. The string
action in conformal gauge is

$\mathcal{L}=-\frac{1}{4\pi}\int d\tau
d\sigma\left[\partial_{a}X^{\mu}\partial^{a}X^{\nu}G_{\mu\nu}+\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}\right]\,.$
(3.1)

The Virasoro constraints are

$\displaystyle
G_{\mu\nu}\partial_{\tau}X^{\mu}\partial_{\sigma}X^{\nu}=0\,,\qquad
G_{\mu\nu}\left(\partial_{\tau}X^{\mu}\partial_{\tau}X^{\nu}+\partial_{\sigma}X^{\mu}\partial_{\sigma}X^{\nu}\right)=0\,.$
(3.2)

We consider a string configuration defined by the following embedding:

$t=t(\tau)\,,\quad\rho=\rho(\tau)\,,\quad\theta=\theta(\tau)\,\quad\psi=\psi(\sigma)\,,\quad\phi=\phi(\tau)\,\,.$
(3.3)

The string equations of motion for $t,\psi,\phi$ are solved by

$t(\tau)=\tau\,,\quad\psi(\sigma)=c_{\psi}\sigma\,,\quad\dot{\phi}(\tau)=-c_{\psi}\tanh^{2}\rho\,,$
(3.4)

and the remaining system of equations for $\theta,\rho$ are

$\ddot{\theta}=0\,,\quad\cosh^{2}\rho\ddot{\rho}+c_{\psi}^{2}\tanh\rho=0\,.$
(3.5)

This system admits the particular solution

$\theta=0\,,\quad n_{5}\dot{\rho}^{2}=1-c_{\psi}^{2}n_{5}\tanh^{2}\rho\,,$
(3.6)

where the last relation is found by solving the Virasoro constraints. The NVE
is derived by considering fluctuations around the particular solution
$\theta=0$. Setting $\theta=0+\delta\theta$ one trivially obtains the equation
$\ddot{\delta\theta}=0$. Clearly this equation admits the Liouvillian solution
$\delta\theta=a\theta+b$, with $a,b$ arbitrary constants, which is consistent
with the fact that the decoupled geometry is described by a gauged WZW model
and is therefore integrable. One could make different choices of the string
embedding, and still obtain an integrable NVE equation.

###  3.2 Asymptotically flat circular NS5

We now study string propagation in the full solution (2.8). In what follows we
set $a=1$ for convenience. We consider the same embedding (3.3), for which the
equations of motion become:

$\displaystyle 0$
$\displaystyle=2(n_{5}+\Sigma)\cos^{2}\theta\sinh^{2}\rho\psi^{\prime\prime}\,,$
(3.7) $\displaystyle 0$
$\displaystyle=\psi^{\prime}\Big{(}n_{5}\dot{r}\sin^{2}2\theta\sinh
2\rho+n_{5}\dot{\theta}\sinh^{2}2\rho\sin
2\theta\Big{)}+4\dot{\phi}\,\dot{r}\sin^{2}\theta\sinh
2\rho\Big{[}\Sigma(n_{5}+\Sigma)$ $\displaystyle-
n_{5}\cosh^{2}\rho\Big{]}+4\dot{\phi}\,\dot{\theta}\sin
2\theta\cosh^{2}\rho\Big{[}\Sigma(n_{5}+\Sigma)+n_{5}\sin^{2}\theta\Big{]}+4\ddot{\phi}\,\sin^{2}\theta\cosh^{2}\rho\Sigma(n_{5}+\Sigma)$
$\displaystyle 0$ $\displaystyle=\psi^{\prime 2}\cos^{2}\theta\Big{[}\sinh
2\rho\Sigma(n_{5}+\Sigma)-2n_{5}\cosh\rho\sinh^{3}\rho\
\Big{]}+\dot{\rho}^{2}\sinh 2\rho\Sigma^{2}$ $\displaystyle-2\sin
2\theta\Sigma^{2}\dot{\rho}\dot{\theta}-\sinh
2\rho\Sigma^{2}\dot{\theta}^{2}-2n_{5}\cos^{2}\theta\sinh
2\rho\Big{[}\cosh^{2}\rho-\Sigma\Big{]}\psi^{\prime}\dot{\phi}$
$\displaystyle+\sin^{2}\theta\sinh
2\rho\Big{[}n_{5}\cosh^{2}\rho-\Sigma(n_{5}+\Sigma)\Big{]}\dot{\phi}^{2}+2\Sigma^{2}(n_{5}+\Sigma)\ddot{\rho}$
$\displaystyle 0$ $\displaystyle=\psi^{\prime 2}\sin
2\theta\sinh^{2}\rho\Big{[}n_{5}\cos^{2}\theta-\Sigma(n_{5}+\Sigma)\Big{]}+\dot{\rho}^{2}\sin
2\theta\Sigma^{2}$ $\displaystyle+2\sinh
2\rho\Sigma^{2}\dot{\rho}\dot{\theta}-\sin
2\theta\Sigma^{2}\dot{\theta}^{2}+2n_{5}\sin
2\theta\cosh^{2}\rho\Big{[}\cos^{2}\theta-\Sigma\Big{]}\psi^{\prime}\dot{\phi}$
$\displaystyle-\cosh^{2}\rho\Big{[}2n_{5}\cos\theta\sin^{3}\theta+\sin
2\theta\Sigma(n_{5}+\Sigma)\Big{]}\dot{\phi}^{2}+2\Sigma^{2}(n_{5}+\Sigma)\ddot{\theta}\,.$

The Virasoro constraints give the condition

$\displaystyle 0$ $\displaystyle=\psi^{\prime
2}(n_{5}+\Sigma)\cos^{2}\theta\sinh^{2}\rho+\dot{\rho}^{2}\Sigma(n_{5}+\Sigma)$
(3.8)
$\displaystyle-\dot{t}^{2}\Sigma+\dot{\theta}^{2}\Sigma(n_{5}+\Sigma)+\dot{\phi}^{2}(n_{5}+\Sigma)\cosh^{2}\rho\sin^{2}\theta\,.$

This system has the following particular solution

$\displaystyle t$ $\displaystyle=\tau\,,\quad\psi=c_{\psi}\sigma\,,$
$\displaystyle\theta$ $\displaystyle=0\,,\quad\phi=0\,,$ (3.9)
$\displaystyle\dot{\rho}^{2}$
$\displaystyle=\frac{1}{n_{5}+\cosh^{2}\rho}-c_{\psi}^{2}\tanh^{2}\rho\,.$

As before, we perturb the equations of motion around $\theta=0$ in order to
derive the NVE. Setting $\theta(\tau)=0+\eta(\tau)$ we find the
equation222Strictly speaking, the NVE is derived by expanding in all
directions normal to the invariant submanifold of phase space to which our
solution belongs. For our chosen solution, this requires also perturbing the
$\rho$ and $\phi$ solutions while enforcing the Virasoro constraint. However,
these perturbations can be shown to only contribute at quadratic order and
thus do not enter the NVE.

$\mathcal{A}\partial_{\tau}^{2}\eta+\mathcal{B}\partial_{\tau}\eta+\mathcal{C}\eta=0\,,$
(3.10)

where

$\displaystyle\mathcal{A}$ $\displaystyle=n_{5}+\cosh^{2}\rho$ (3.11)
$\displaystyle\mathcal{B}$
$\displaystyle=2\cosh\rho\sinh\rho\left(\frac{1-c_{\psi}^{2}\sinh^{2}\rho-
n_{5}c_{\psi}^{2}\tanh^{2}\rho}{n_{5}+\cosh^{2}\rho}\right)^{1/2}$ (3.12)
$\displaystyle\mathcal{C}$ $\displaystyle=\frac{1-c_{\psi}^{2}\sinh^{2}\rho-
n_{5}c_{\psi}^{2}\tanh^{2}\rho}{n_{5}+\cosh^{2}\rho}-c_{\psi}^{2}\sinh^{2}\rho-
n_{5}c_{\psi}^{2}\tanh^{4}\rho\,,$ (3.13)

where $\Sigma=n_{5}+\Sigma$. By changing variable $\tau\rightarrow
x=\cosh\rho(\tau)$ we bring the equation in the form

$\mathcal{A}\dot{x}^{2}\partial_{x}^{2}\eta+(\mathcal{A}\ddot{x}+\mathcal{B}\dot{x})\partial_{x}\eta+\mathcal{C}\eta=0\,.$
(3.14)

Finally, we can bring the NVE in normal form by defining $\xi=g\eta$, with

$bg-2g^{\prime}=0\,.$ (3.15)

We then get

$\xi^{\prime\prime}-U\xi=0\,,$ (3.16)

with $U=F/G$,

$\displaystyle F$
$\displaystyle=-c_{\psi}^{4}x^{16}+\left(-4n_{5}c_{\psi}^{4}-4c_{\psi}^{4}-2c_{\psi}^{2}\right)x^{14}$
(3.17)
$\displaystyle+\left(-10n_{5}^{2}c_{\psi}^{4}-16n_{5}c_{\psi}^{4}-2n_{5}c_{\psi}^{2}+14c_{\psi}^{4}+14c_{\psi}^{2}\right)x^{12}$
$\displaystyle+\left(-12n_{5}^{3}c_{\psi}^{4}-8n_{5}^{2}c_{\psi}^{4}+56n_{5}c_{\psi}^{4}+6n_{5}^{2}c_{\psi}^{2}+34n_{5}c_{\psi}^{2}-12c_{\psi}^{4}-18c_{\psi}^{2}+2n_{5}-6\right)x^{10}$
$\displaystyle+\big{(}-5n_{5}^{4}c_{\psi}^{4}+16n_{5}^{3}c_{\psi}^{4}+60n_{5}^{2}c_{\psi}^{4}-48n_{5}c_{\psi}^{4}+6n_{5}^{3}c_{\psi}^{2}$
$\displaystyle\qquad+14n_{5}^{2}c_{\psi}^{2}-50n_{5}c_{\psi}^{2}+3c_{\psi}^{4}+6c_{\psi}^{2}-n_{5}^{2}-14n_{5}+3\big{)}x^{8}$
$\displaystyle+\left(12n_{5}^{4}c_{\psi}^{4}+8n_{5}^{3}c_{\psi}^{4}-56n_{5}^{2}c_{\psi}^{4}+12n_{5}c_{\psi}^{4}-6n_{5}^{3}c_{\psi}^{2}-34n_{5}^{2}c_{\psi}^{2}+18n_{5}c_{\psi}^{2}-2n_{5}^{2}+6n_{5}\right)x^{6}$
$\displaystyle+\left(-10n_{5}^{4}c_{\psi}^{4}-16n_{5}^{3}c_{\psi}^{4}+14n_{5}^{2}c_{\psi}^{4}-2n_{5}^{3}c_{\psi}^{2}+14n_{5}^{2}c_{\psi}^{2}\right)x^{4}$
$\displaystyle+\left(4n_{5}^{4}c_{\psi}^{4}+4n_{5}^{3}c_{\psi}^{4}+2n_{5}^{3}c_{\psi}^{2}\right)x^{2}-n_{5}^{4}c_{\psi}^{4}$
(3.18) $\displaystyle G$
$\displaystyle=4x^{2}\left(x^{2}-1\right)^{2}\left(n_{5}+x^{2}\right){}^{2}\left(n_{5}c_{\psi}^{2}\left(x^{2}-1\right)+x^{2}\left(c_{\psi}^{2}\left(x^{2}-1\right)-1\right)\right){}^{2}$

In this form one can implement Kovacic’s algorithm (see Appendix A) and see
that the algorithm fails, and thus the NVE does not admit, for generic values
of the parameters, a Liouvillan solution. This in turn implies that the
Hamiltonian system describing the dynamics of strings in the background
sourced by a circular array of NS5 branes is not integrable.

###  3.3 Supertube background

We can repeat the process above for the NS5-P supertube solution. The
equations of motion are more complicated due to the additional metric and
B-field components. For the decoupled case, one can find a solution analogous
to what we had in the decoupled circular NS5 case. We have checked that the
corresponding NVE, despite being significantly more complicated, does in fact
produce Liouvillian solutions, as expected. For the asymptotically flat case,
starting with the same initial string embedding ansatz (3.3) we find a
particular solution to the string equations of motion of the form

$\displaystyle t=$
$\displaystyle\tau\,,\quad\psi=c_{\psi}\sigma=\frac{k}{n_{5}R}\sigma,$ (3.19)
$\displaystyle\quad y$ $\displaystyle=0\,,\quad\theta=0\,,\quad\phi=0\,,$
$\displaystyle\dot{\rho}^{2}(\tau)$
$\displaystyle=\frac{n_{5}\left(n_{5}R^{2}-k^{2}\right)-k^{2}\sinh^{2}\rho}{n_{5}^{2}R^{2}\left(n_{5}+\cosh^{2}\rho\right)}\,.$

Note that the previously unconstrained $c_{\psi}$ now acquires a fixed value
that depends on the supertube parameter $k/R$. Expanding around this solution
as $\theta(\tau)=0+\eta(\tau)$ we find, as before, the NVE of the form

$\mathcal{A}\partial_{\tau}^{2}\eta+\mathcal{B}\partial_{\tau}\eta+\mathcal{C}\eta=0\,,$
(3.20)

where now

$\displaystyle\mathcal{A}$ $\displaystyle=n_{5}+a^{2}\cosh^{2}\rho$ (3.21)
$\displaystyle\mathcal{B}$
$\displaystyle=a^{2}\sinh(2\rho)\left(\frac{n_{5}(n_{5}R^{2}-k^{2})-a^{2}k^{2}\sinh^{2}\rho}{n_{5}^{2}R^{2}(n_{5}+a^{2}\cosh^{2}\rho)}\right)^{1/2}$
$\displaystyle\mathcal{C}$
$\displaystyle=-\frac{a^{4}k^{2}\sinh^{2}\rho(\cosh(2\rho)+3)+4a^{2}k^{2}n_{5}\cosh^{2}\rho+2n_{5}^{2}\left(k^{2}-a^{2}R^{2}\right)}{2n_{5}^{2}R^{2}\left(a^{2}\cosh^{2}\rho+n_{5}\right)}\,,$

and $\rho$ is a function of $\tau$ obtained by solving the last equation in
(3.19). Through the same variable transformations as in the circularn NS5
case, we reduce the NVE to an equivalent equation in normal form:

$\xi^{\prime\prime}(x)-U(x)\xi(x)=0$ (3.22)

with $U=F/G$ and 333For the sake of notational simplicity we display $U$ by
setting $a=1$.

$\displaystyle F$
$\displaystyle=-k^{4}x^{10}+\left(-4k^{4}n_{5}-4k^{4}-2k^{2}n_{5}^{2}R^{2}\right)x^{8}$
(3.23)
$\displaystyle+\left(-10k^{4}n_{5}^{2}-14k^{4}n_{5}+14k^{4}-2k^{2}n_{5}^{3}R^{2}+14k^{2}n_{5}^{2}R^{2}\right)x^{6}$
$\displaystyle+\big{(}-12k^{4}n_{5}^{3}-10k^{4}n_{5}^{2}+42k^{4}n_{5}-12k^{4}+6k^{2}n_{5}^{4}R^{2}$
$\displaystyle\qquad+40k^{2}n_{5}^{3}R^{2}-18k^{2}n_{5}^{2}R^{2}+2n_{5}^{5}R^{4}-6n_{5}^{4}R^{4}\big{)}x^{4}$
$\displaystyle+\big{(}-5k^{4}n_{5}^{4}+2k^{4}n_{5}^{3}+29k^{4}n_{5}^{2}-30k^{4}n_{5}+3k^{4}+6k^{2}n_{5}^{5}R^{2}+20k^{2}n_{5}^{4}R^{2}$
$\displaystyle\qquad-44k^{2}n_{5}^{3}R^{2}+6k^{2}n_{5}^{2}R^{2}-n_{5}^{6}R^{4}-14n_{5}^{5}R^{4}+3n_{5}^{4}R^{4}\big{)}x^{2}$
$\displaystyle-2n_{5}^{6}R^{4}+6n_{5}^{5}R^{4}+2k^{4}n_{5}^{4}+4k^{4}n_{5}^{3}-12k^{4}n_{5}^{2}+6k^{4}n_{5}-14k^{2}n_{5}^{4}R^{2}+12k^{2}n_{5}^{3}R^{2}$
$\displaystyle G$
$\displaystyle=4\left(x^{2}-1\right)^{2}\left(n_{5}+x^{2}\right){}^{2}\left(k^{2}\left(n_{5}+x^{2}-1\right)-n_{5}^{2}R^{2}\right){}^{2}\,.$

Once again, the Kovacic algorithm fails to produce a solution. This proves the
absence of Liouvillian solutions to the NVE and thus the non-integrability of
string dynamics in the full NS5-P supertube background. Since integrability is
preserved by T-duality, this result also implies the non-integrability of the
full NS5-F1 supertube solution (2.3).

##  4 Conclusions

In this note we proved that the Hamiltonian system that describes the dynamics
of strings on the asymptotically flat circular NS5-F1 fuzzball geometry is not
integrable, in contrast with the case of the fivebrane decoupling limit, where
an explicit description in terms of a gauged WZW model is available.

This result implies that, although a gauged sigma model description along the
lines of [5] that describes an asymptotically flat fuzzball might be found,
the dynamics of strings in this case is expected to be considerably more
intricate. In particular, we expect that at least in some regimes it will
exhibit a chaotic behaviour. It would be interesting to show explicitly that
the dynamics of strings on the circular two-charge geometry is chaotic by
computing the typical signatures of chaos on the phase space of the system by
a numerical analysis of the equations of motion and the computation of the
Lyapunov exponents (see for example [21, 22, 23, 9] for a similar analysis in
different contexts). In particular, one could explore the idea of using the
Lyapunov exponent to distinguish fuzzballs from black hole solutions. Some
aspects of chaos on circular fuzzball geometries (in the S-dual D1-D5 frame)
have been reported in [24], although their analysis was limited to the point
particle limit, i.e. geodesic motion.

It could also be interesting to set up a perturbative expansion of the
worldsheet models for the asymptotically linear region solutions that
describes the coupling to flat space, and to see if one can access some
signatures of chaotic behaviour, and to explore the consequences for flat
space holography.

## Acknowledgements

The work of SM is supported by the MIUR program for young researchers “Rita
Levi Montalcini”.

##  Appendix A Kovacic algorithm

Consider a second order linear ODE in normal form

$\xi^{\prime\prime}(x)-U(x)\xi(x)=0\,,$ (A.1)

where $U(x)$ is a rational function. The Kovacic algorithm provides an
explicit way to produce Liouvillian solutions if they exists, or prove that
they do not exist, that can easily be implemented in a programming language.
Here we review the algorithm itself. Proofs of all the steps can be found in
the original paper [13].

In the following, the “order at infinity” of $U(x)$ will refer to its order as
a zero, i.e. the difference between the highest power of $x$ in the
denominator and the highest power of $x$ in the numerator, while the order of
the pole $c$ is the highest power of $(x-c)^{-1}$ in the Laurent expansion
around $c$.

There are three cases to consider, corresponding to three types of possible
solutions to the ODE. The necessary conditions for these three cases are as
follows:

  * •

Case 1: Every pole of $U$ has even order, or else has order 1. At infinity,
the order of $U$ is even, or else is greater than 2.

  * •

Case 2: $U$ has at least one pole of odd order greater than 2, or else has
order 2.

  * •

Case 3: The order of poles of $U$ does not exceed 2, and the order of $U$ at
infinity is at least 2. In this case the partial fraction decomposition of $U$
can be written as

$U(x)=\sum_{i}\frac{\alpha_{i}}{(x-c_{i})^{2}}+\sum_{j}\frac{\beta_{j}}{x-d_{j}}$
(A.2)

and we further require that $\sum_{j}\beta_{j}=0$,
$\sqrt{1+4\alpha_{i}}\in\mathbb{Q}$ $i$ and furthermore that

$\sqrt{1+4\left(\sum_{i}\alpha_{i}+\sum_{j}\beta_{j}d_{j}\right)}\in\mathbb{Q}$
(A.3)

If none of the above are satisfied by $U$, then the equation does not admit
Liouvillan solutions, and the system whose NVE is given by the above equation
is not integrable. If the necessary conditions for any of the cases are
satisfied, we must check the cases sequentially. In each case we construct
several candidate algebraic equations based on the behaviour of $U$ at its
poles and at infinity. The exact procedure varies by case, which is reviewed
below. The solutions to these algebraic equations, if they exist, are then
used to construct the solution to the original ODE. If the solution doesn’t
exist, we move to the next available case. If all cases fail, there are no
Liouvillian solutions to the ODE.

The general equation (3.17) has order at infinity of $2$ and several poles
each of order $2$. This means all three cases have their necessary conditions
satisfied and must be checked sequentially.

For demonstration purposes, we choose $c_{t}=1,c_{\psi}=1,n_{5}=1$ in (3.17)
so that

$U=-\frac{x^{16}+10x^{14}-42x^{10}+10x^{8}+42x^{6}-10x^{2}+1}{4\left(x^{9}-x^{7}-2x^{5}+x^{3}+x\right)^{2}}$
(A.4)

but the final non-integrability result holds for general values. The nine
order 2 poles are located at $0,\pm 1,\pm i$ and $\pm x_{\pm}$ where

$x_{\pm}=\sqrt{\frac{1}{2}(\sqrt{5}\pm 1)}.$ (A.5)

For other values of $n_{5}$ the poles at $\pm i$ and $\pm x_{\pm}$ will move
along the imaginary axis, but never become degenerate, so the orders of the
poles do not change.

### Case 1

For each pole, denoted $c$, we compute three pieces of data
$\\{[\sqrt{U}]_{c},\alpha_{c}^{+},\alpha_{c}^{-}\\}$ in the following way

  * •

If the order of the pole is 1, then

$\\{[\sqrt{U}]_{c},\alpha_{c}^{+},\alpha_{c}^{-}\\}=\\{0,1,1\\}$ (A.6)

  * •

If the order of the pole is 2, and $b$ is the coefficient of $(x-c)^{-2}$ in
the expansion around $c$, then

$\\{[\sqrt{U}]_{c},\alpha_{c}^{+},\alpha_{c}^{-}\\}=\\{0\ ,\
\frac{1}{2}+\frac{1}{2}\sqrt{1+4b}\ ,\ \frac{1}{2}-\frac{1}{2}\sqrt{1+4b}\\}$
(A.7)

  * •

If the order of the pole is $2\nu\geq 4$ then $[\sqrt{U}]_{c}$ is the polar
part of the Laurent expansion of $\sqrt{U}$ around $c$. Let $a$ be the
coefficient of $(x-c)^{-\nu}$ in $[\sqrt{U}]_{c}$ and $b$ be the difference of
coefficients of $(x-c)^{-\nu-1}$ in the expansions of $U$ and
$([\sqrt{U}]_{c})^{2}$ respectively. Then

$\alpha_{c}^{\pm}=\frac{1}{2}\left(\nu\pm\frac{b}{a}\right)$ (A.8)

We also compute
$\\{[\sqrt{U}]_{\infty},\alpha_{\infty}^{+},\alpha_{\infty}^{-}\\}$ in the
following way

  * •

If the order at infinity of $U$ is greater than 2, then

$\\{[\sqrt{U}]_{\infty},\alpha_{\infty}^{+},\alpha_{\infty}^{-}\\}=\\{0,0,1\\}$
(A.9)

  * •

If the order at infinity of $U$ is equal to 2, and $b$ is the coefficient of
$x^{-2}$ in the expansion around infinity, then

$\\{[\sqrt{U}]_{\infty},\alpha_{\infty}^{+},\alpha_{\infty}^{-}\\}=\\{0\ ,\
\frac{1}{2}+\frac{1}{2}\sqrt{1+4b}\ ,\ \frac{1}{2}-\frac{1}{2}\sqrt{1+4b}\\}$
(A.10)

  * •

If the order at infinity of $U$ is $-2\nu\leq 0$ then $[\sqrt{U}]_{\infty}$ is
the sum of non-positive powers of $x$ in the expansion around infinity. Let
$a$ be the coefficient of $x^{\nu}$ in $[\sqrt{U}]_{\infty}$ and $b$ be the
difference of coefficients of $x^{\nu-1}$ in the expansions of $U$ and
$([\sqrt{U}]_{\infty})^{2}$ respectively. Then

$\alpha_{\infty}^{\pm}=\frac{1}{2}\left(-\nu\pm\frac{b}{a}\right)\ .$ (A.11)

Having constructed the above data, we compute

$d=\alpha_{\infty}^{\pm}-\sum_{\\{c\\}}\alpha_{c}^{\pm}$ (A.12)

for all choices of $\pm$ for each $c$ independently. If $d$ is a non-negative
integer, then for the _same choice of signs_ we compute

$\omega=\pm[\sqrt{U}]_{\infty}\sum_{\\{c\\}}\pm[\sqrt{U}]_{c}+\frac{\alpha_{c}^{\pm}}{x-c}\
.$ (A.13)

For each such $\omega$ we search for a monic polynomial $P$ of degree $d$ that
satisfies

$P^{\prime\prime}+2\omega P^{\prime}+(\omega^{\prime}+\omega^{2}-U)P=0$ (A.14)

If such a polynomial exists, then $Pe^{\int\omega}$ is a solution to the
original ODE. If it doesn’t exist for any of the $\omega$’s, then case 1
fails.

For our example we obtain for the poles

$\displaystyle\\{[\sqrt{U}]_{0},\alpha_{0}^{+},\alpha_{0}^{-}\\}$
$\displaystyle=\\{0,\ \frac{1}{2},\ \frac{1}{2}\\}$ (A.15)
$\displaystyle\\{[\sqrt{U}]_{c},\alpha_{c}^{+},\alpha_{c}^{-}\\}$
$\displaystyle=\\{0,\ \frac{3}{4},\ \frac{1}{4}\\}\qquad c=\pm 1,\pm i,\pm
x_{\pm}$

and at infinity we have

$\\{[\sqrt{U}]_{\infty},\alpha_{\infty}^{+},\alpha_{\infty}^{-}\\}=\\{0,\frac{1}{2},\frac{1}{2}\\}$
(A.16)

It’s easy to see that no choice of signs produces a non-negative $d$ as
defined in (A.12), since the $\alpha_{0}^{\pm}$ already cancel positive
contribution from the point at infinity. This rules out case 1.

### Case 2

For case 2, we define a set of integers $E_{c}$ associated to each pole of
$U$, denoted $c$, as follows:

  * •

If the order of the pole is 1, then $E_{c}={4}$

  * •

If the order of the pole is 2 and $b$ is the coefficient of $(x-c)^{-2}$ in
the partial fraction decomposition of $U$, then

$E_{c}=\\{2+k\sqrt{1+4b}\ |\ k\in\\{-2,0,2\\}\\}\cap\mathbb{Z}$ (A.17)

  * •

If the order of the pole is $\nu>2$ then $E_{c}=\\{\nu\\}$.

We also define $E_{\infty}$ as follows

  * •

If the order of $U$ at infinity is greater than 2, $E_{\infty}=\\{0,2,4\\}$

  * •

If the order of $U$ at infinity is equal to 2 and $b$ is the coefficient of
$x^{-2}$ in the expansion around infinity, then

$E_{\infty}=\\{2+k\sqrt{1+4b}\ |\ k\in\\{-2,0,2\\}\\}\cap\mathbb{Z}$ (A.18)

  * •

If the order at infinity is $\nu<2$ then $E_{\infty}=\\{\nu\\}$

We now consider all families $\\{e_{c}\\}\cup\\{e_{\infty}\\}$ where $e_{c}\in
E_{c}$ and $e_{\infty}\in E_{\infty}$. For each family where at least one of
the integers is odd we compute

$d=\frac{1}{2}\left(e_{\infty}-\sum_{\\{c\\}}e_{c}\right)$ (A.19)

For each family that gives a non-negative integer $d$ we define

$\theta=\frac{1}{2}\sum_{\\{c\\}}\frac{e_{c}}{x-c}$ (A.20)

and search for a monic polynomial $P$ of degree $d$ that satisfies

$\displaystyle P^{\prime\prime\prime}$ $\displaystyle+3\theta
P^{\prime\prime}+(3\theta^{2}+3\theta^{\prime}-4U)P^{\prime}$ (A.21)
$\displaystyle+(\theta^{\prime\prime}+3\theta\theta^{\prime}+\theta^{3}-4U\theta-2U^{\prime})P=0\
.$ (A.22)

If such a solution is found, define $\phi=\theta+P^{\prime}/P$ and let
$\omega$ be a solution to

$\omega^{2}+\phi\omega+(\frac{1}{2}\phi^{\prime}+\frac{1}{2}\phi^{2}-U)=0$
(A.23)

Then $e^{\int\omega}$ is a solution to the original ODE. If no polynomials $P$
can be found, then case 2 fails.

For our example, we have

$\displaystyle E_{0}$ $\displaystyle=\\{2\\}$ (A.24) $\displaystyle E_{c}$
$\displaystyle=\\{2,3,1\\}\qquad c=\pm 1,\pm i,\pm x_{\pm}$ $\displaystyle
E_{\infty}$ $\displaystyle=\\{2\\}$

Again, no combination results in a positive $d$ as defined in (A.19). This
rules out case 2.

### Case 3

Choose $n=4,6$ or $12$. In principle one can immediately start with $n=12$,
but it is easier to consider the cases sequentially if one expects to find
solutions. To rule out solutions we must verify the $n=12$ case.

Similar to case 2, define a set of integers $E_{c}$ for each pole $c$ as
follows:

  * •

If the order of the pole is 1, then $E_{c}=\\{12\\}$

  * •

If the order of the pole is 2, and $b$ is the coefficient of $(x-c)^{-2}$ in
the partial fraction decomposition of $U$, then

$E_{c}=\\{6+\frac{12k}{n}\sqrt{1+4b}|k\in{-n/2,...-1,0,1,...,n/2}\\}\cap\mathbb{Z}$
(A.25)

Let $\gamma$ be the coefficient of $x^{-2}$ in the expansion of $U$ around
infinity. Then define

$E_{\infty}=\\{6+\frac{12k}{n}\sqrt{1+4\gamma}|k\in{-n/2,...-1,0,1,...,n/2}\\}\cap\mathbb{Z}$
(A.26)

We now consider all families $\\{e_{c}\\}\cup\\{e_{\infty}\\}$ where $e_{c}\in
E_{c}$ and $e_{\infty}\in E_{\infty}$. For each family we compute

$d=\frac{n}{12}\left(e_{\infty}-\sum_{\\{c\\}}e_{c}\right)\ .$ (A.27)

For each family that gives a non-negative integer $d$ we define

$\theta=\frac{n}{12}\sum_{\\{c\\}}\frac{e_{c}}{x-c}$ (A.28)

and we also define $S=\prod_{\\{c\\}}(x-c)$. Next, consider the recursive
relation

$P_{i-1}=-SP_{i}^{\prime}+((n-i)S^{\prime}-S\theta)P_{i}-(n-i)(i+1)S^{2}UP_{i+1}\
.$ (A.29)

We must now look for a monic polynomial $P$ such that for $P_{n}=-P$, the
$P_{-1}$ generated by the above relation is identically zero. If such a
polynomial is found, then let $\omega$ be a solution to

$\sum_{i=0}^{n}\frac{S^{i}P_{i}}{(n-i)!}\omega^{i}=0\ .$ (A.30)

Then $e^{\int\omega}$ is a solution to the original ODE. If no such $P_{n}$
can be found (for all choices of $n$), case 3 fails.

In our example, taking directly $n=12$, we have

$\displaystyle E_{0}$ $\displaystyle=\\{6\\}$ (A.31) $\displaystyle E_{c}$
$\displaystyle=\\{3,4,5,6,7,8,9\\}\qquad c=\pm 1,\pm i,\pm x_{\pm}$
$\displaystyle E_{\infty}$ $\displaystyle=\\{6\\}$

And once again, no combination can give a non-negative $d$ as defined in
(A.27). This rules out case 3, and therefore the original ODE has no
Liouvillian solutions.

### NS5-P supertube NVE

For the NS5-P supertube NVE, given by (3.23), the poles are order 2 and the
order at infinity is 2, so once again, all three cases of the Kovacic
algorithm must be checked. The poles are located at $\pm 1,\pm i\sqrt{n_{5}}$
and $\pm\sqrt{1-n_{5}+n_{5}^{2}R^{2}/k^{2}}$ Note that the pole at $x=0$ is
absent in the supertube NVE.

For case 1, we obtain

$\displaystyle\\{[\sqrt{U}]_{c},\alpha_{c}^{+},\alpha_{c}^{-}\\}$
$\displaystyle=\\{0,\ \frac{3}{4},\ \frac{1}{4}\\}$ (A.32)
$\displaystyle\\{[\sqrt{U}]_{\infty},\alpha_{\infty}^{+},\alpha_{\infty}^{-}\\}$
$\displaystyle=\\{0,\frac{1}{2},\frac{1}{2}\\}\ ,$

for case 2, we obtain

$\displaystyle E_{c}$ $\displaystyle=\\{2,3,1\\}$ (A.33) $\displaystyle
E_{\infty}$ $\displaystyle=\\{2\\}\ ,$

and for case 3, we obtain

$\displaystyle E_{c}$ $\displaystyle=\\{3,4,5,6,7,8,9\\}$ (A.34)
$\displaystyle E_{\infty}$ $\displaystyle=\\{6\\}\ .$

As in the case of the circular NS5 distribution, all three cases fail to
produce a non-negative integer $d$ as defined in either (A.12),(A.19) or
(A.27). Thus the NVE has no Liouvillian solutions and string motion in the
NS5-P supertube background is non-integrable.

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# Geometric Tracking on $\mathcal{S}^{3}$ Based on Sliding Mode Control

Eduardo Espindola<EMAIL_ADDRESS>Yu Tang∗<EMAIL_ADDRESS>Ningbo
Institute of Technology, Zhejiang University, Ningbo, CHINA.

###### Abstract

Attitude tracking on the unit sphere of dimension $3$ based on sliding mode is
considered in this paper. The tangent bundle of Lagrangian dynamics that
describes the rotational motion of a rigid body is first shown to be a Lie
group, and then a sliding surface that emerged on it is defined. Next, a
sliding-mode controller is designed for attitude tracking that relies on an
intrinsic error defined on the Lie group. Almost global asymptotic stability
of the closed loop is demonstrated using the Lyapunov analysis. Numerical
simulations are included to compare the performance of the sliding mode
controller designed on the Lie group with that designed in the embedding
Euclidean space.

###### keywords:

Attitude control, Geometric tracking, Lagrangian dynamics, Lie Groups, Sliding
subgroups.

††thanks:  ∗Corresponding author Yu Tang, on leave from the National
Autonomous University of Mexico, Mexico City, MEXICO.

,

##  1 Introduction

There has been a growing interest in designing controllers (Bullo & Lewis,
2019; Bullo & Murray, 1999; Spong & Bullo, 2005) and observers (Aghannan &
Rouchon, 2003; Bonnabel et al., 2008; Lageman et al., 2009; Zlotnik & Forbes,
2018; Wang et al., 2021) for nonlinear systems on Lie groups due to the
stronger stability and robustness that can be achieved by exploiting geometric
structures of the configuration manifold such as symmetries and invariance
(Marsden & Ratiu, 1999). Attitude tracking for rigid bodies using a geometric
approach has been reported in Markdahl et al. (2017); Lee (2011); Lee et al.
(2011); Saccon et al. (2013); Vang & Tron (2019) and references cited therein.
In Refs. Maithripala et al. (2006); Maithripala & Berg (2015) an intrinsic
configuration error is defined on a Lie group, and PD and PID controllers are
developed with local exponential stability. These intrinsic controllers do not
depend on a particular coordinate chosen, and therefore can be applied to any
attitude representation in attitude tracking, e.g. rotation matrices or
quaternions, with the established local exponential stability.

Sliding mode control is a widely known control methodology recognized for its
ability to deal with uncertainties and external disturbances, and has been
applied to design robust controller in Euclidean space (Utkin, 1978; Fridman
et al., 2011). Its extension to nonlinear manifolds to address attitude
control problems with a geometric approach has been studied in Gómez et al.
(2019); Ghasemi et al. (2020); Meng et al. (2023). Under a sliding mode
controller, the error dynamics first reaches a sliding surface, designed to be
invariant and to describe the desired behavior. Upon arriving the sliding
surface, the error dynamics slides to the origin according to the desired
behavior assigned to the sliding surface. Several sliding mode controllers
applied to the attitude problem of a rigid body have been proposed in
Crassidis & Markley (1996); Lo & Chen (1995); Lee & Utkin (2007); Zhu et al.
(2011). One drawback of these schemes compared to the sliding mode control
design with a geometric approach is that the non-Euclidean geometry of the
manifold in which the system evolves is not considered in the control design,
which could cause undesired behavior of the closed-loop system, such as the
presence of undesired equilibria and discontinuity of the closed-loop dynamics
(Gómez et al., 2019).

To address these issues, in this paper we propose a sliding mode control
design on the Lie group $\mathcal{S}^{3}$ using unit quaternions. The proposed
sliding subgroup enables us to define a new sliding variable with geometric
interpretations and allows us to design tracking controllers with stronger
robustness properties. The almost global asymptotic stability of the desired
equilibrium in the closed-loop system is shown by using the Lyapunov analysis.
Numerical simulations are also included to compare this geometric controller
with a non-geometric controller designed based on a conventional sliding
variable in Euclidean space under the uncertainty in the inertia matrix and
noisy measurements. Compared to the group $SO(3)$, the unit quaternions have
the advantage of easier manipulation for the implementation of control laws.
The main contributions of this paper are the design of sliding mode
controllers on the unit quaternion group to track a desired attitude and
rigorous proof of the stability of the proposed scheme. Instead of
$\mathcal{S}^{3}\times\mathbb{R}^{3}$ (Espindola & Tang, 2023b), where
$\mathbb{R}^{3}$ is the tangent space at the group identity, the sliding mode
control design is developed directly on the state space defined by the tangent
bundle $\mathcal{S}^{3}\times T_{q}\mathcal{S}^{3}$. This allows us to explore
the Lagrangian dynamics for unit quaternions in the controller design. Towards
this end, the state manifold of the Lagrangian dynamics based on quaternions
that describe the rotational motion of a rigid body is first shown to be a Lie
group under a properly defined group operation. Then a Lie subgroup embedded
in the Lie group is designed with the desired property to asymptotically
stabilize the identity $\bar{1}\in\mathcal{S}^{3}$ of the quaternions group.
Finally, an attitude-tracking control law is devised based on the proposed
sliding subgroup and the intrinsic error defined on the Lie groups.

Section 2 describes the rotational motion of a rigid body based on a 4-DOF
(Degree of Freedom) Lagrangian dynamics using unit quaternions proposed in
Espíndola & Tang (2023a). Some results and properties of the Lagrangian
dynamics useful for the subsequent sliding mode control design are summarized.
In Section 3, the definition of the group operation is first defined, then the
state space of the rotational Lagrangian dynamics is shown to be a Lie group.
At last, a sliding surface is defined, which is shown to be embedded on the
state space with desired sliding properties. In Section 4, the evolution of
the attitude error is described by a Lagrangian dynamical equation, and a
tracking control law is designed. Section 5 provides the simulation results to
illustrate the theoretical results and compare the proposed geometric
controller with a non-geometric controller under parameter uncertainty and
noise in the measurements. Section 6 draws the conclusions.

###  1.1 Notation

The $l_{2}$-norm is denoted by $\|x\|=\left(x^{T}x\right)^{1/2}$ for a vector
$x\in\mathbb{R}^{n}$, whereas, for a matrix $A\in\mathbb{R}^{n\times m}$ its
induced norm is given by
$\|A\|=\left(\lambda_{\max}\left(A^{T}A\right)\right)^{1/2}$, which describes
the spectral norm, where $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$
respectively denote the maximum and minimum eigenvalues of $A$. The notation
$\mathrm{det}(\cdot)$ and $\mathrm{tr}(\cdot)$ define the determinant and
trace operation, respectively. The unit sphere and the unit ball embedded in
$\mathbb{R}^{n}$ are, respectively, defined as
$\mathcal{S}^{n-1}=\\{x\in\mathbb{R}^{n}\;|\;\|x\|^{2}=1\\}$ and
$\mathcal{B}^{n-1}=\\{x\in\mathbb{R}^{n}\;|\;\|x\|^{2}\leq 1\\}$. The matrix
$I_{n}$ describes the identity matrix of size $n\times n$. Similarly, a
$n\times m$ matrix with zero entries is denoted by $0_{n\times m}$. The
special orthogonal group $n$ is defined as $SO(n)=\\{R\in\mathbb{R}^{n\times
n}\;|\;R^{T}R=RR^{T}=I_{n},\;\mathrm{det}(R)=1\\}$. The Lie algebra of $SO(n)$
is $\mathfrak{so}(n)=\\{A\in\mathbb{R}^{n\times n}\;|\;A=-A^{T}\\}$. Finally,
the map $\cdot^{\wedge}\vcentcolon\mathbb{R}^{3}\to\mathfrak{so}(3)$
represents the cross-product operator $u^{\wedge}v=u\times v$, for all
$u,v\in\mathbb{R}^{3}$.

##  2 Rotational motion of the rigid body

Given an inertial reference frame $\mathbf{I}$ and a body frame $\mathbf{B}$
fixed to the center of mass of the body, a rotation matrix $R\in SO(3)$
determines the attitude of the rigid body relative to the inertial frame. The
rotational motion of the rigid body can be globally described by the following
kinematic and dynamic equations

$\displaystyle\dot{q}$ $\displaystyle=\frac{1}{2}J(q)\omega,$ (1)
$\displaystyle M\dot{\omega}$ $\displaystyle=(M\omega)^{\wedge}\omega+\tau,$
(2)

where $q=\left[q_{0},\vec{q}^{T}\right]^{T}\in\mathcal{S}^{3}$ is the unitary
quaternion, which is related to a rotation matrix through Rodriguez formula
$R(q)=I_{3}+2q_{0}\vec{q}^{\wedge}+2\left(\vec{q}^{\wedge}\right)^{2}$, with
$q_{0}\in[-1,1]$ and $\vec{q}\in\mathcal{B}^{2}$ denoting the scalar and
vector part of the unit quaternion, respectively. $\omega\in\mathbb{R}^{3}$ is
the angular velocity, $\tau\in\mathbb{R}^{3}$ is the torque control vector,
and $M=M^{T}\in\mathbb{R}^{3\times 3}$ is a constant and positive definite
inertia tensor of the rigid body, all expressed in the body frame
$\mathbf{B}$. Map $J\vcentcolon\mathbb{R}^{4}\to\mathbb{R}^{4\times 3}$ is
defined as

$\displaystyle J(x)$ $\displaystyle=\left[\begin{array}[]{c}-\vec{x}^{T}\\\
x_{0}I_{3}+\vec{x}^{\wedge}\end{array}\right],\;\forall
x=\left[\begin{array}[]{c}x_{0}\\\
\vec{x}\end{array}\right],\;x_{0}\in\mathbb{R},\;\vec{x}\in\mathbb{R}^{3}.$
(7)

Similarly, maps $L\vcentcolon\mathbb{R}^{4}\to\mathbb{R}^{4\times 3}$, and
$Q,W\vcentcolon\mathbb{R}^{4}\to\mathbb{R}^{4\times 4}$, which will be used in
the Lagrangian dynamics, are defined as follows

$\displaystyle L(x)=\left[\begin{array}[]{c}-\vec{x}^{T}\\\
x_{0}I_{3}-\vec{x}^{\wedge}\end{array}\right],\;\forall
x=\left[\begin{array}[]{c}x_{0}\\\
\vec{x}\end{array}\right],\;x_{0}\in\mathbb{R},\;\vec{x}\in\mathbb{R}^{3}.$
(12) $\displaystyle Q(x)$
$\displaystyle=\left[x\;J(x)\right]=\left[\begin{array}[]{cc}x_{0}&-\vec{x}^{T}\\\
\vec{x}&x_{0}I_{3}+\vec{x}^{\wedge}\end{array}\right],$ (15) $\displaystyle
W(x)$
$\displaystyle=\left[x\;L(x)\right]=\left[\begin{array}[]{cc}x_{0}&-\vec{x}^{T}\\\
\vec{x}&x_{0}I_{3}-\vec{x}^{\wedge}\end{array}\right].$ (18)

Some useful properties of these maps are summarized in Appendix A.

The motion equations (1)-(2) can be described by a $4$-DOF Lagrangian dynamics
proposed in Espíndola & Tang (2023a), given by

$D(q)\ddot{q}+C(q,\dot{q})\dot{q}=\bar{\tau},$ (19)

where

$\displaystyle D(q)$ $\displaystyle=J(q)MJ^{T}(q)+m_{0}qq^{T},$ (20)
$\displaystyle C(q,\dot{q})$
$\displaystyle=-J(q)(M\omega)^{\wedge}J^{T}(q)-D(q)Q(\dot{q})Q^{T}(q),$ (21)
$\displaystyle\bar{\tau}$ $\displaystyle=\frac{1}{2}J(q)\tau,$ (22)

and $\lambda_{\min}(M)\leq m_{0}\leq\lambda_{\max}(M)$ a virtual inertia
parameter. Note that the configuration manifold of Lagrangian dynamics (19) is
$\mathcal{S}^{3}$. Moreover, the system (19) verifies the basic properties of
Lagrangian dynamics, related to the energy conservation property, summarized
in the following lemma.

######  Lemma 1 (Properties of the Lagrangian system).

Let $D(q)$ and $C(q,\dot{q})$ defined in (20) and (21), respectively, then the
following statements hold (Espíndola & Tang, 2023a):

  1. 1.

Matrix $D(q)$ is symmetric and positive definite:

$\lambda_{min}(M)I_{4}\leq D(q)\leq\lambda_{max}(M)I_{4},\ \ \forall
q\in\mathcal{S}^{3}.$ (23)

  2. 2.

Matrix $\dot{D}(q)-2C(q,\dot{q})$ is skew-symmetric:

$x^{T}\left(\dot{D}(q)-2C(q,\dot{q})\right)x=0,\ \ \forall
q\in\mathcal{S}^{3},\;\forall x\in{R}^{4}.$ (24)

This Lagrangian dynamics allows one to address the full attitude control
problem, where the attitude may evolve on the whole group $SO(3)$, and to
design control laws $\bar{\tau}$ based on the energy shaping methodology in
Euclidean space $\mathbb{R}^{4}$ (Espíndola & Tang, 2023a). The torque applied
to the rigid body is obtained by $\tau=2J^{T}(q)\bar{\tau}$ using Property 3
of matrix $J(x)$. In this paper, the geometric structure of the configuration
manifold is explored, allowing one to define a sliding subgroup that inherits
the geometric structure and acquires the salient features of the sliding mode
through the proper design of the controller.

##  3 Lie Group and Sliding Subgroup

###  3.1 The State Manifold as a Lie Group

Consider the state manifold of the Lagrangian system (19), given by the
tangent bundle $T\mathcal{S}^{3}\vcentcolon=\mathcal{S}^{3}\times
T_{q}\mathcal{S}^{3}$, $T_{q}\mathcal{S}^{3}$ being the tangent space at
$q\in\mathcal{S}^{3}$. The following lemma establishes that $T\mathcal{S}^{3}$
is a Lie group endowed with the binary operation

$\displaystyle g_{1}\cdot g_{2}$
$\displaystyle=\Big{(}Q(q_{1})q_{2},\;Q(q_{1})p_{2}+W(q_{2})p_{1}$
$\displaystyle\quad+\lambda Q(q_{1})\big{(}q_{0,2}q_{2}-\bar{1}\big{)}+\lambda
W(q_{2})\big{(}q_{0,1}q_{1}-\bar{1}\big{)}$
$\displaystyle\quad-\lambda\big{(}q_{0,12}Q(q_{1})q_{2}-\bar{1}\big{)}\Big{)}\in
T\mathcal{S}^{3}$ (25)

$\forall g_{1}=(q_{1},p_{1})$ and $g_{2}=(q_{2},p_{2})\in T\mathcal{S}^{3}$,
where $\lambda>0$ is a scalar, $\bar{1}\vcentcolon=[1,0,0,0]^{T}$ is the
identity element on $\mathcal{S}^{3}$, and
$Q(q_{i})q_{j}=\left[q_{0,ij},\vec{q}^{T}_{ij}\right]^{T}\in\mathcal{S}^{3}$,
for $i,j=1,2,3$. Note that $Q(q_{i})q_{j}=q_{i}\otimes
q_{j}\in\mathcal{S}^{3}$ corresponds to the binary operation for the Lie group
$\mathcal{S}^{3}$. Moreover, with a little abuse of notation, it is denoted
the triple product $q_{i}\otimes q_{j}\otimes
q_{k}=\left[q_{0,ijk},\vec{q}^{T}_{ijk}\right]^{T}$, for all $i,j,k=1,2,3$.

######  Lemma 2 (Lie group structure).

The tangent bundle $T\mathcal{S}^{3}$ is a Lie group under the binary
operation (3.1), with

  * •

Group identity: $e=\left(\bar{1},0_{4\times 1}\right)\in T\mathcal{S}^{3}$,

  * •

Inverse: $g^{-1}=\left(q^{-1},p^{-1}\right)\in T\mathcal{S}^{3}$, $\forall
g=(q,p)\in T\mathcal{S}^{3}$, where
$q^{-1}=\left[q_{0},-\vec{q}^{T}\right]^{T}\in\mathcal{S}^{3}$, and
$p^{-1}=\left[p_{0},-\vec{p}^{T}\right]^{T}\in T_{q}S^{3}$.

It is clear that $T\mathcal{S}^{3}$ is a smooth manifold and that the group
operation (3.1) is smooth. The group axioms are verified as follows:

  1. 1.

$\forall g=(q,p)\in T\mathcal{S}^{3}$, it has

$\displaystyle g\cdot e$ $\displaystyle=\Big{(}Q(q)\bar{1},\;Q(q)0_{4\times
1}+W(\bar{1})p+\lambda Q(q)(\bar{1}-\bar{1})$ $\displaystyle\quad+\lambda
W(\bar{1})(q_{0}q-\bar{1})-\lambda(q_{0}Q(q)\bar{1}-\bar{1})\Big{)}$
$\displaystyle=\Big{(}Q(\bar{1})q,\;Q(\bar{1})p+W(q)0_{4\times 1}+\lambda
Q(\bar{1})(q_{0}\vec{q}-\bar{1})$ $\displaystyle\quad+\lambda
W(q)(\bar{1}-\bar{1})-\lambda(q_{0}Q(\bar{1})q-\bar{1})\Big{)}$
$\displaystyle=e\cdot g=\left(q,p\right)=g.$

  2. 2.

Let $g^{-1}=\left(q^{-1},p^{-1}\right)\in T\mathcal{S}^{3}$ be the inverse of
$g=(q,p)\in T\mathcal{S}^{3}$, then

$\displaystyle g\cdot g^{-1}=$
$\displaystyle\Big{(}Q(q)q^{-1},\;Q(q)p^{-1}+W\left(q^{-1}\right)p+\lambda
Q(q)(q_{0}q^{-1}-\bar{1})$ $\displaystyle\quad\quad+\lambda
W\left(q^{-1}\right)(q_{0}q-\bar{1})-\lambda(Q(q)q^{-1}-\bar{1})\Big{)}$
$\displaystyle=\Big{(}Q\left(q^{-1}\right)q,\;Q\left(q^{-1}\right)p+W(q)p^{-1}+\lambda
Q\left(q^{-1}\right)(q_{0}q-\bar{1})$ $\displaystyle\quad\quad+\lambda
W(q)(q_{0}q^{-1}-\bar{1})-\lambda(Q\left(q^{-1}\right)q-\bar{1})\Big{)}$
$\displaystyle=g^{-1}\cdot g=\left(\bar{1},0_{4\times 1}\right)=e,$

where the fact that $q^{T}p=0$ is used, given that $p\in
T_{q}\mathcal{S}^{3}$.

  3. 3.

Associativity: $\forall g_{1}=(q_{1},p_{1})$, $g_{2}=(q_{2},p_{2})$ and
$g_{3}=(q_{3},p_{3})\in T\mathcal{S}^{3}$,

$\displaystyle g_{1}$ $\displaystyle\cdot\left(g_{2}\cdot
g_{3}\right)=g_{1}\cdot\Big{(}Q(q_{2})q_{3},\;Q(q_{2})p_{3}+W(q_{3})p_{2}$
$\displaystyle+\lambda Q(q_{2})(q_{0,3}q_{3}-\bar{1})+\lambda
W(q_{3})(q_{0,2}q_{2}-\bar{1})$
$\displaystyle-\lambda(q_{0,23}Q(q_{2})q_{3}-\bar{1})\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}Q(q_{1})Q(q_{2})q_{3},\;Q(q_{1})\Big{(}Q(q_{2})p_{3}+W(q_{3})p_{2}$
$\displaystyle+\lambda Q(q_{2})(q_{0,3}q_{3}-\bar{1})+\lambda
W(q_{3})(q_{0,2}q_{2}-\bar{1})$
$\displaystyle-\lambda(q_{0,23}Q(q_{2})q_{3}-\bar{1})\Big{)}+W\left(Q(q_{2})q_{3}\right)p_{1}$
$\displaystyle+\lambda Q(q_{1})(q_{0,23}Q(q_{2})q_{3}-\bar{1})$
$\displaystyle+\lambda W\left(Q(q_{2})q_{3}\right)(q_{0,1}q_{1}-\bar{1})$
$\displaystyle-\lambda(q_{0,123}Q(q_{1})Q(q_{2})q_{3}-\bar{1})\Big{)}$
$\displaystyle=$
$\displaystyle\Big{(}Q\left(Q(q_{1})q_{2}\right)q_{3},\;Q\left(Q(q_{1})q_{2}\right)p_{3}+W(q_{3})\Big{(}Q(q_{1})p_{2}$
$\displaystyle+W(q_{2})p_{1}+\lambda Q(q_{1})(q_{0,2}q_{2}-\bar{1})$
$\displaystyle+\lambda
W(q_{2})(q_{0,1}q_{1}-\bar{1})-\lambda(q_{0,12}Q(q_{1})q_{2}-\bar{1})\Big{)}$
$\displaystyle+\lambda Q\left(Q(q_{1})q_{2}\right)(q_{0,3}q_{3}-\bar{1})$
$\displaystyle+\lambda W(q_{3})(q_{0,12}Q(q_{1})q_{2}-\bar{1})$
$\displaystyle-\lambda(q_{0,123}Q\left(Q(q_{1})q_{2}\right)q_{3}-\bar{1})\Big{)}$
$\displaystyle=$
$\displaystyle\Big{(}Q(q_{1})q_{2},\;Q(q_{1})p_{2}+W(q_{2})p_{1}$
$\displaystyle+\lambda Q(q_{1})(q_{0,2}q_{2}-\bar{1})+\lambda
W(q_{2})(q_{0,1}q_{1}-\bar{1})$
$\displaystyle-\lambda(q_{0,12}Q(q_{1})q_{2}-\bar{1})\Big{)}\cdot g_{3}$
$\displaystyle=$ $\displaystyle\left(g_{1}\cdot g_{2}\right)\cdot g_{3},$

where Properties of matrices $Q(\cdot)$ and $W(\cdot)$ in Lemma 8 are used.

###  3.2 Sliding Subgroup

The sliding subgroup, defined below, inherits the structure of the Lie group
$T\mathcal{S}^{3}$, allowing us to design sliding mode controllers on the Lie
group.

######  Lemma 3 (Sliding subgroup).

For $g=(q,p)\in T\mathcal{S}^{3}$, let the sliding variable $s(g)$ be defined
as

$s(g)=p+\lambda\left(q_{0}q-\bar{1}\right)$ (26)

where $\lambda>0$ is a scalar.

Then the set

$H=\left\\{g\in T\mathcal{S}^{3}\;|\;s(g)=0_{4\times 1}\right\\}$ (27)

is a Lie subgroup.

Its clear that (26) defines a smooth map. The associativity of $H$ is
inherited from $T\mathcal{S}^{3}$. Therefore, it remains to verify the
following:

  * •

Identity $e\in H$: $s(e)=s(\bar{1},0_{4\times 1})=0$.

  * •

Inverse: Let $g=(q,p)\in H$. Then $s(g)=0\implies
p=-\lambda\left(q_{0}q-\bar{1}\right)$. Then

$\displaystyle s\left(g^{-1}\right)$
$\displaystyle=p^{-1}+\lambda\left(q_{0}q^{-1}-\bar{1}\right)$
$\displaystyle=-\lambda\left(q_{0}q^{-1}-\bar{1}\right)+\lambda\left(q_{0}q^{-1}-\bar{1}\right)=0_{4\times
1}.$

This shows that $g^{-1}\in H$.

  * •

Closure of the group operation: $\forall g_{1}=(q_{1},p_{1})$ and
$g_{2}=(q_{2},p_{2})\in H$, it has that
$p_{1}=-\lambda\left(q_{0,1}q_{1}-\bar{1}\right)$ and
$p_{2}=-\lambda\left(q_{0,2}q_{2}-\bar{1}\right)$. Thus,

$\displaystyle s\left(g_{1}\cdot g_{2}\right)$
$\displaystyle=s\Big{(}Q(q_{1})q_{2},\;Q(q_{1})p_{2}+W(q_{2})p_{1}$
$\displaystyle\quad+\lambda Q(q_{1})(q_{0,2}q_{2}-\bar{1})+\lambda
W(q_{2})(q_{0,1}q_{1}-\bar{1})$
$\displaystyle\quad-\lambda(q_{0,12}Q(q_{1})q_{2}-\bar{1})\Big{)}$
$\displaystyle=Q(q_{1})p_{2}+W(q_{2})p_{1}+\lambda
Q(q_{1})(q_{0,2}q_{2}-\bar{1})$ $\displaystyle\quad+\lambda
W(q_{2})(q_{0,1}q_{1}-\bar{1})-\lambda(q_{0,12}Q(q_{1})q_{2}-\bar{1})$
$\displaystyle\quad+\lambda(q_{0,12}Q(q_{1})q_{2}-\bar{1})$
$\displaystyle=Q(q_{1})(-\lambda(q_{0,2}q_{2}-\bar{1}))$
$\displaystyle\quad+W(q_{2})(-\lambda(q_{0,1}q_{1}-\bar{1}))$
$\displaystyle\quad+\lambda Q(q_{1})(q_{0,2}q_{2}-\bar{1})+\lambda
W(q_{2})(q_{0,1}q_{1}-\bar{1})$ $\displaystyle=0_{4\times 1}.$

Therefore, $g_{1}\cdot g_{2}\in H$.

###  3.3 Sliding Property of the Sliding Subgroup

One of the salient features of sliding mode control is that the behavior of a
system on the sliding subgroup is independent of the system dynamics. Define
$\mathcal{M}\vcentcolon=\left\\{g=(q,p)\in
T\mathcal{S}^{3}\;|\;p=\dot{q}\right\\}$, then $\mathcal{M}\subset
T\mathcal{S}^{3}$. The following lemma establishes the sliding property of the
subgroup $\bar{H}\vcentcolon=H\cap\mathcal{M}$, where $H$ is defined in (27),
that is, $\bar{g}\to e$ asymptotically provided that $\bar{g}\in\bar{H}$.

######  Lemma 4 (Sliding property).

Let $\bar{g}=(q,\dot{q})\in\bar{H}$, then $\bar{g}\to e=(\bar{1},0_{4\times
1})$ asymptotically.

Since $\left(q,\dot{q}\right)\in H\;\implies\;s(\bar{g})=0_{4\times 1}$, it
follows that

$\frac{d}{dt}(q-\bar{1})=\dot{q}=-\lambda(q_{0}q-\bar{1}).$

Consider the candidate Lyapunov function
$V=\frac{1}{2}\|q-\bar{1}\|^{2}=1-q_{0}$. Its time derivative on the sliding
subgroup is

$\displaystyle\dot{V}$
$\displaystyle=(q-\bar{1})^{T}\dot{q}=(q-\bar{1})^{T}\left(-\lambda(q_{0}q-\bar{1})\right)$
$\displaystyle=\lambda\bar{1}^{T}(q_{0}q-\bar{1})=-\lambda(1-q_{0}^{2})=-\lambda(1+q_{0})V.$

For all $q\in\mathcal{S}^{3}\backslash\\{-\bar{1}\\}$, it has $1+q_{0}<0$.
Therefore, $\dot{V}<0$. This shows the almost global asymptotic stability of
$q=\bar{1}$.

###  3.4 Stability of the Sliding Subgroup

Consider the Lagrangian dynamics (19) with the configuration manifold
$\mathcal{M}$. Let $\bar{g}=\left(q,\dot{q}\right)\in\mathcal{M}$, then the
following control law

$\bar{\tau}=-\lambda\left(D(q)(q_{0}\dot{q}+\dot{q_{0}}q)+C(q,\dot{q})(q_{0}q-\bar{1})\right)-K_{r}s(\bar{g})$
(28)

renders the sliding subgroup $\bar{H}$ attractive, where
$K_{r}=K^{T}_{r}\in\mathbb{R}^{4\times 4}$ is a positive definite gain matrix.

######  Lemma 5 (Stability of the Sliding Subgroup).

The control law (28) in closed loop with the system (19) renders the
equilibrium point $s(\bar{g})=0_{4\times 1}$ globally exponentially stable.

By the definition of the sliding variable (26) for $s(\bar{g})$ and the
Lagrangian dynamics (19), it has

$\displaystyle D(q)\dot{s}$ $\displaystyle=D(q)\ddot{q}+\lambda
D(q)(q_{0}\dot{q}+\dot{q_{0}}q),$
$\displaystyle=-C(q,\dot{q})\dot{q}+\bar{\tau}+\lambda
D(q)(q_{0}\dot{q}+\dot{q_{0}}q).$

In closed loop with the control law (28) it yields

$D(q)\dot{s}=-C(q,\dot{q})s-K_{r}s.$ (29)

Consider the following Lyapunov function candidate

$V_{r}(s)=\frac{1}{2}s^{T}D(q)s,$ (30)

which is positive definite and radially unbounded according to Property 23 of
Lemma 1.

The time derivative of (30) along the trajectories of system (29) is

$\displaystyle\dot{V}_{r}$
$\displaystyle=s^{T}D(q)\dot{s}+\frac{1}{2}s^{T}\dot{D}(q)s$
$\displaystyle=s^{T}\left(-C(q,\dot{q})s-K_{r}s\right)+\frac{1}{2}s^{T}\dot{D}(q)s$
$\displaystyle=-s^{T}K_{r}s$
$\displaystyle\leq-\lambda_{\min}\left(K_{r}\right)\|s\|^{2}$
$\displaystyle\leq-2\frac{\lambda_{\min}\left(K_{r}\right)}{\lambda_{\max}(M)}V_{r},$
(31)

where the skew-symmetry property (24) of Lemma 1 is used. Therefore,

$\|s(t)\|\leq\left(\frac{\lambda_{\max}(M)}{\lambda_{\min}(M)}\right)^{1/2}\|s(t_{0})\|\mathrm{e}^{-\frac{\lambda_{\min}(K_{r})}{\lambda_{\max}(M)}(t-t_{0})},$

giving that $s\to 0_{4\times 1}$ exponentially (Theorem 4.10, Hassan et al.
(2002)).

######  Remark 6.

Due to the double coverage of $SO(3)$ by $\mathcal{S}^{3}$, both $\pm q$ will
give the same attitude. Thus, the sliding variable (26) can be changed to
$s(\bar{g})=\dot{q}+\lambda\left(q_{0}q\pm\bar{1}\right)$ according to the
sign of the scalar part $q_{0}$ of the unit quaternion. Furthermore, to ensure
robustness to arbitrarily small noise in measurements, a switching strategy
may be used in the control law (Mayhew et al., 2011). For the same reason, the
global exponential stability established on the group $T\mathcal{S}^{3}$ must
be interpreted with caution, since there are no continuous control laws that
achieve global stability on any compact group due to topological constraints
(Bhat & Bernstein, 2000).

##  4 Attitude tracking control

###  4.1 Control Objectives

Given a desired trajectory $\dot{q}_{d}=\frac{1}{2}J(q_{d})\omega_{d}$, where
$\omega_{d}\in\mathbb{R}^{3}$ is a desired angular velocity that is twice
differentiable and $q_{d}\in\mathcal{S}^{3}$ is the desired attitude, the goal
of tracking the attitude is to render
$\left(q,\dot{q}\right)\to\left(q_{d},\dot{q}_{d}\right)$ asymptotically.

Let $g_{d}:=\left(q_{d},\dot{q}_{d}\right)$, $g:=(q,\dot{q})\in\mathcal{M}$.
Then the intrinsic error defined on the Lie group $T\mathcal{S}^{3}$ is
defined as

$\displaystyle g_{e}$ $\displaystyle=g^{-1}_{d}\cdot
g=\left(q_{e},p_{e}\right)\in T\mathcal{S}^{3},$ (32) $\displaystyle q_{e}$
$\displaystyle=Q(q^{-1}_{d})q=Q^{T}(q_{d})q,$ (33) $\displaystyle p_{e}$
$\displaystyle=Q(q^{-1}_{d})\dot{q}+W(q)\dot{q}^{-1}_{d}+\lambda
Q(q^{-1}_{d})(q_{0}q-\bar{1})$ $\displaystyle\quad+\lambda
W(q)(q_{0,d}q^{-1}_{d}-\bar{1})-\lambda\left((q^{T}_{d}q)Q(q^{-1}_{d})q-\bar{1}\right).$
(34)

Note that by Eq. (1),

$\dot{q}_{e}=Q(q^{-1}_{d})\dot{q}+W(q)\dot{q}^{-1}_{d}=\frac{1}{2}J(q_{e})\omega_{e},$
(35)

where $\omega_{e}=\omega-R^{T}(q_{e})\omega_{d}$ is the angular velocity
error.

Therefore, $\bar{g}_{e}=(q_{e},\dot{q}_{e})\in\mathcal{M}$ describes the
intrinsic error on the tangent bundle of the Lagrangian system

$D(q_{e})\ddot{q}_{e}+C(q_{e},\dot{q}_{e})\dot{q}_{e}=\bar{\tau}_{c},$ (36)

with

$\displaystyle D(q_{e})$
$\displaystyle=J(q_{e})MJ^{T}(q_{e})+m_{0}q_{e}q_{e}^{T},$ (37) $\displaystyle
C(q_{e},\dot{q}_{e})$
$\displaystyle=-J(q_{e})w^{\wedge}_{r}J^{T}(q_{e})-D(q_{e})Q(\dot{q}_{e})Q^{T}(q_{e}),$
(38) $\displaystyle\bar{\tau}_{c}$
$\displaystyle=\frac{1}{2}J(q_{e})\tau_{c},$ (39)

where

$\displaystyle w_{r}$
$\displaystyle=M\omega_{e}-\left(\mathrm{tr}(M)I_{3}-2M\right)R^{T}(q_{e})\omega_{d},$
(40) $\displaystyle\tau_{c}$
$\displaystyle=S\left(MR^{T}(q_{e})\omega_{d}\right)R^{T}(q_{e})\omega_{d}-MR^{T}(q_{e})\dot{\omega}_{d}+\tau.$
(41)

Therefore, it follows from the results of the previous section that the
control objectives are achieved by designing a control law for
$\bar{\tau}_{c}$ that makes the sliding subgroup $\bar{H}$ attractive.
Subsequently, $\bar{g}_{e}\to e$ as a consequence of the sliding property.

###  4.2 Controller Design

The intrinsic error $g_{e}\in T\mathcal{S}^{3}$ can be expressed in the state
manifold of the Lagrangian system (36) as
$\bar{g}_{e}=(q_{e},\dot{q}_{e})\in\mathcal{M}$. The sliding variable
therefore, according to (26), is given by

$s\left(\bar{g}_{e}\right)=\dot{q}_{e}+\lambda(q_{0,e}q_{e}-\bar{1}),$ (42)

$\lambda>0$ being a design parameter. The following control law will render
the sliding subgroup $\bar{H}$ attractive

$\displaystyle\bar{\tau}_{c}$
$\displaystyle=-\lambda\left(D(q_{e})\left(q_{0,e}\dot{q}_{e}+\dot{q}_{0,e}q_{e}\right)+C(q_{e},\dot{q}_{e})\left(q_{0,e}q_{e}-\bar{1}\right)\right)$
$\displaystyle\quad-K_{r}s\left(\bar{g}_{e}\right),$ (43)

where $K_{r}=K^{T}_{r}\in\mathbb{R}^{4\times 4}$ is a positive definite gain
matrix.

######  Theorem 7 (Almost-global asymptotic stability).

The control law (43) in closed loop with the Lagrangian dynamics (36)
stabilizes the equilibrium point $\bar{g}_{e}=e$ almost globally
asymptotically.

It follows directly of the sliding variable $s(\bar{g}_{e})$ as a consequence
of Lemma 3, Lemma 4 and Lemma 5.

##  5 SIMULATIONS

To illustrate the theoretical results and the performance of the GSMC, the
controller (43) was compared with the controller (37) in Espíndola & Tang
(2023a) developed in Euclidean space $\mathbb{R}^{4}$. It was shown in
Espíndola & Tang (2023a) that the sliding surface

$s_{0}=\dot{q}-\dot{q}_{d}+\Lambda\left(q-q_{d}\right)\in\mathbb{R}^{4}$ (44)

is exponentially stable, $\Lambda=\Lambda^{T}>0$ being a gain matrix.

The desired trajectory was given by $\omega_{d}(t)=[0,0,0.1]^{T}$
($\mathrm{rad}/s$). Initial conditions and design parameters are shown in
Table 1. The inertia matrix considered is given by

$M=\left[\begin{array}[]{ccc}3.6046&-0.0706&0.1491\\\ -0.0706&8.6868&0.0449\\\
0.1491&0.0449&9.3484\end{array}\right]\;\left(\mathrm{Kg}\mathrm{m}^{2}\right).$
(45)

For comparison purposes, the same controller gains were chosen for both
controllers under the ideal scenario, where the measurements were noise-free
and the inertia matrix (45) was known. For further comparisons under non-ideal
situations, simulations under uncertainties of $\pm 30\%$ in the inertia
matrix $M$, and in the presence of measurement noises were carried out,
respectively.

Table 1: Parameters and initial conditions.

Initial Condition | Value | Units
---|---|---
$q(0)$ |  $\left[0,\bar{u}^{T}\right]^{T}$$\dagger$ |
$\omega(0)$ | $\left[0,0,0\right]^{T}$ | $\mathrm{rad}/s$
$q_{d}(0)$ | $\left[1,0,0,0\right]^{T}$ |
$\omega_{d}(0)$ | $\left[0,0,0.1\right]^{T}$ | $\mathrm{rad}/s$
Parameter | Value | Units
$m_{0}$ | $6$ | $\mathrm{Kg}\mathrm{m}^{2}$
$K_{r}$ | $3I_{4}$ |
$\lambda$ | $0.1$ |
Espíndola & Tang (2023a) | Value | Units
$K_{s}$ | $3I_{4}$ |
$\Lambda$ | $0.1I_{4}$ |

  * $\dagger$

$u\vcentcolon=[1,2,3]^{T}$, and $\bar{u}=u/\|u\|$.

Fig. 1 illustrates the behavior of the controllers in the ideal situation.
Fig. 1(a) shows the norm $\|q_{e}-\bar{1}\|$. Note that both controllers
achieved the convergence at $50$ $(s)$. Fig. 1(b) shows the norm of the
sliding variables $\|s\left(\bar{g}\right)\|$ and $\|s_{0}\|$. A large
transient response in $s_{0}$ of the non-geometric controller is observed,
which caused it to take a longer time ($40$ (s)) to converge, while the
geometric controller stabilizes $s\left(\bar{g}\right)$ at $15$ ($s$). Fig.
1(c) draws the norm of the torque $\|\tau\|$. Note that the non-geometric
controller requires slightly higher torque input than the geometric controller
at the transient. Finally, Fig. 1(d) illustrates the energy consumption of
each controller in terms of $\sqrt{\int^{t}_{0}\tau^{T}\tau d\mathrm{t}}$. The
lower energy consumption in the geometric controller can be noticed.

Figure 1: The ideal case: Performance comparison of the non-geometric
controller in Espíndola & Tang (2023a) (blue line) and the geometric
controller (43) (red line).

To better appreciate the convergence behavior of these controllers, the
sliding variable of both controllers is shown in Fig. 2. Note that the
behavior of the sliding surface $s\left(\bar{g}_{e}\right)$ of the geometric
controller first exhibits the convergence to the siding surface $s=0_{4\times
1}$ and later to the equilibrium
$\left(s\left(\bar{g}_{e}\right),q_{e}\right)=\left(0_{4\times
1},\bar{1}\right)$. On the other hand, the sliding surface $s_{0}$ of the non-
geometric controller oscillates around the surface $s_{0}=0_{4\times 1}$ and
then convergences to the equilibrium $\left(s_{0},\
q_{e}\right)=\left(0_{4\times 1},\bar{1}\right)$ with an oscillating behavior,
leading to more energy consumption than the geometric controller.

Figure 2: The ideal case: behaviour of the sliding variables in both
controllers.

In the presence of uncertainties of $\pm 30\%$ in the inertia matrix, Fig. 3
shows the performance of both controllers. Notice from Fig. 3(a)-(d) that the
performance of both controllers is similar to that in the ideal case, being
the geometric controller (43) more energy efficient than the non-geometric
controller (Fig. 3(d)).

Figure 3: Uncertain inertia matrix: Performance of the non-geometric
controller in Espíndola & Tang (2023a) (blue line) and the geometric
controller (43) (red line).

In the last simulation, noise in the quaternion and the angular velocity
measurements were added by $q_{m}=(q+n_{1}\nu)/\|q+n_{1}\nu\|$ and
$\omega_{m}=\omega+n_{2}u$, where $\nu\in\mathbb{R}^{4}$ and
$u\in\mathbb{R}^{3}$ are vectors with Gaussian distribution entries of zero
mean and unit variance, $n_{1}\in(0,0.1)$ and $n_{2}\in(0,0.1)$ are uniform
distributions. Fig. 4 illustrates the behavior of both controllers under these
noisy measurements. Note that the norm $\|q_{e}-\bar{1}\|$ oscillates inside
$[0,0.1]$ in the steady state (Fig. 4(a)), which corresponds to the noise
variance of $q_{m}$. Likewise, the norm of the sliding variable was kept
inside $[0,0.05]$ after the transient in both controllers (Fig. 4(b)), which
is smaller than the noise variance of the angular velocity measurement
$\omega_{m}$. However, a larger impact of the noise on the behavior was
observed in the non-geometric controller of Espíndola & Tang (2023a), which
caused a larger torque input and more energy consumption than in the geometric
(Figs. 4(c)-(d)).

Figure 4: Noisy measurements: Performance of the non-geometric controller in
Espíndola & Tang (2023a) (blue line) and the geometric controller (43) (red
line).

##  6 Conclusions

A geometric sliding mode controller on the quaternion group was developed in
this paper for attitude tracking. The state manifold of the Lagrangian
dynamics that describes the attitude using the unit quaternion was shown to be
a Lie group. A sliding subgroup that heritages the Lie group structure was
designed, which allows us to design a sliding mode control on the Lie group.
The almost global asymptotic stability of the closed-loop system was achieved.
The performance of the proposed geometric sliding mode controller was
illustrated and compared with a non-geometric sliding mode controller in the
simulations, showing that the geometric controller performed better than the
non-geometric controller in the presence of uncertain inertia matrix and noise
in the measurement. Furthermore, more energy efficiency was observed in the
geometric controller.

## Acknowledgments

This work was partially supported by PAPIIT-UNAM IN112421.

##  Appendix A Properties of the Maps

Some properties of map $J$ are listed below.

_Properties of map $J$_ (Espíndola & Tang, 2023a): For all
$x,y\in\mathbb{R}^{4}$, the following properties hold for $J(\cdot)$:

  1. 1.

$J^{T}(x)y=-J^{T}(y)x$,

  2. 2.

$J^{T}(x)y=0_{3\times 1}$, for all $y=kx,\;\forall k\in\mathbb{R}$,

  3. 3.

$J^{T}(x)J(x)=||x||^{2}I_{3}$.

These same properties also hold for map $L$.

Likewise, some properties of maps $Q$ and $W$ are summarized in the following
lemma.

######  Lemma 8 (Properties of maps $Q(x)$ and $W(x)$).

Maps $Q(x),W(x):\mathbb{R}^{4}\to\mathbb{R}^{4\times 4}$ defined in (15)
verify the following properties $\forall x,y\in\mathbb{R}^{4}$

  1. 1.

$Q(x)\in SO(4),\;\forall x\in\mathcal{S}^{3}$,

  2. 2.

$Q(y)Q^{T}(x)=J(y)J^{T}(x)+yx^{T}$,

  3. 3.

$u^{T}Q(y)Q^{T}(x)u=0\iff y^{T}x=0$, $\forall u\in\mathbb{R}^{4}$,

  4. 4.

$Q(Q(x)y)=Q(x)Q(y)$,

  5. 5.

$Q\left(Q^{T}(x)y\right)=Q^{T}(x)Q(y)$,

  6. 6.

$Q(\bar{w})\in\mathfrak{so}(4)$,
$\forall\bar{w}=[0,w^{T}]^{T}\in\mathbb{R}^{4}$, $w\in\mathbb{R}^{3}$,

  7. 7.

$Q(x)y=W(y)x$,

  8. 8.

$Q(x)W(y)=W(y)Q(x)$,

  9. 9.

$W\left(Q(x)y\right)=W(y)W(x)$.

Properties 1-3 are proved in Espíndola & Tang (2023a). Properties 4-5 and 7-9
can be verified by direct substitution. Finally, Property 6 is trivial by
considering $x_{0}=0$ in (15).

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