# Geometric sliding mode control of mechanical systems on Lie groups

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

###### Abstract

This paper presents a generalization of conventional sliding mode control
designs for systems in Euclidean spaces to fully-actuated simple mechanical
systems whose configuration space is a Lie group for the trajectory-tracking
problem. A generic kinematic control is first devised in the underlying Lie
algebra, which enables the construction of a Lie group on the tangent bundle
where the system state evolves. A sliding subgroup is then proposed on the
tangent bundle with the desired sliding properties, and a control law is
designed for the error dynamics trajectories to reach the sliding subgroup
globally exponentially. Tracking control is then composed of the reaching law
and sliding mode, and is applied for attitude tracking on the special
orthogonal group $SO(3)$ and the unit sphere $\mathcal{S}^{3}$. Numerical
simulations show the performance of the proposed geometric sliding-mode
controller (GSMC) in contrast with two control schemes of the literature.

###### keywords:

Geometric control; Lie groups; Mechanical systems; Sliding subgroups.

††thanks: This paper was not presented at any IFAC meeting. ∗Corresponding
author Yu Tang, on leave from the National Autonomous University of Mexico,
Mexico City, MEXICO.

,

##  1 Introduction

Sliding mode control (SMC) (Utkin, 1977) has been proven to be a very powerful
control design method for systems evolving in Euclidean spaces. Its design
usually consists of two stages: the reaching stage where the controller drives
the system trajectories to a sliding surface, a subspace embedded in the
Euclidean space designed to convoy some specific characteristics (e.g.,
convergence time, actuator saturation) in accordance with the given control
objectives, and a sliding stage where the system trajectories converge to the
origin according to the reduced-order dynamics constrained in the sliding
surface, achieving the control objectives. In the sliding stage, the reduced-
order dynamics is independent of the system dynamics, and therefore, this
control design method ensures its robustness against a certain class of
disturbances and has achieved great success in a wide range of applications.

When this method is extended to mechanical systems whose configuration space
is a general Lie group, care must be taken in the design of the sliding
surface. Unlike the Euclidean case, when the system configuration space is a
Lie group $G$, its time rate of change belongs to the tangent space $T_{g}G$
at the configuration $g$. Therefore, the state space is composed of the
tangent bundle $G\times T_{g}G$. The topological structure and the underlying
properties of the configuration space and the tangent space are very
different. Without taking this into account in the SMC design, the sliding
surface may not belong to the tangent bundle, and therefore no guarantee is
offered to ensure that the system trajectories reach the sliding surface and
the sliding mode may not exist at all (Gómez et al., 2019). The main problem
is thus how to devise a group operation such that the tangent bundle is a Lie
group and that the sliding subgroup is immersed in the tangent bundle so that
the salient features of SMC in the Euclidean space mentioned above may be
inherited by a general Lie group.

We present in this paper a general method of designing a sliding mode control,
a geometric sliding mode control (GSMC), for fully-actuated mechanical systems
whose configuration space is a Lie group. A generic kinematic control is first
devised in the underlying Lie algebra (the tangent space at the group identity
with a bilinear map), which enables us to build a Lie group on the tangent
bundle where the system state evolves. Then a sliding subgroup is proposed on
the tangent bundle, and the sliding mode is guaranteed to exist. The sliding
subgroup is designed to convoy control objectives, in particular, the almost
global asymptotic convergence of the trajectories of the reduced-order
dynamics to the identity of the tangent bundle is considered, which is the
strongest convergence that may be achieved by continuous time-invariant
feedback in a smooth Lie group (Bhat & Bernstein, 2000). The reaching control
law is then designed to drive the trajectories to the sliding subgroup
globally exponentially. Tracking is then composed of the reaching law and
sliding mode, as in the Euclidean case.

###  1.1 Related work

The geometric approach to control designs has achieved significant advances
for mechanical systems on nonlinear manifolds, for recent developments in this
topic, see, for instance, Bullo & Lewis (2005) and the references therein. As
recognized in Koditschek (1989); Bullo et al. (1995); Maithripala et al.
(2006), a key point in control design is how to define the tracking error. The
tracking error defined on a Riemannian manifold relying on an error function
and a transport map in Bullo & Murray (1999) may be simplified if the manifold
is endowed with a Lie group structure (Maithripala et al., 2006), where the
error notion can be globally defined explicitly and is easier to be
manipulated for stability analysis of the closed-loop system (Maithripala &
Berg, 2015; Saccon et al., 2013; De Marco et al., 2018; Lee, 2012; Sarlette et
al., 2010). A similar situation is encountered in observer designs using an
estimation error defined on a Riemannian manifold (Aghannan & Rouchon, 2003)
versus an estimation error defined by the group operation on Lie groups
bonnabel2009non. The ability to define a global error on Lie groups provides a
powerful tool for treating the error as an object in the state space globally
and controlling it as a physical system so that the tracking problem can be
reduced to stabilizing the error dynamics to the group identity (Bullo et al.,
1995; Maithripala et al., 2006; Spong & Bullo, 2005). Moreover, a separation
principle can be proved in the geometric approach to control designs
(Maithripala et al., 2006; Maithripala & Berg, 2015) when part of the state in
the control law is estimated by an exponentially convergent observer designed
on the Lie group (Bonnabel et al., 2009), similar to an LTI system. This opens
a wide field of applications for systems on Lie groups, such as rigid body
motion control and trajectory tracking in $2$D and $3$D spaces, given the
significant advances in both geometric control designs (Bullo & Lewis, 2005;
Spong & Bullo, 2005; Lee, 2011; Akhtar & Waslander, 2020; Rodríguez-Cortés &
Velasco-Villa, 2022) and observer designs (Aghannan & Rouchon, 2003; Bonnabel
et al., 2009; Mahony et al., 2008; Lageman et al., 2009; Zlotnik & Forbes,
2018).

GSMC on Lie groups has been considered using two main approaches: developing
the SMC in the underlying Lie algebra or developing it on the Lie group
itself. The main idea in the former approach is first expressing the tracking
error defined on the Lie group in its Lie algebra through the locally
diffeomorphic logarithmic map (Bullo et al., 1995). Since the Lie algebra is a
vector space, a sliding surface can be designed as in the Euclidean case
(Culbertson et al., 2021; Liang et al., 2021; Espíndola & Tang, 2022). In the
latter approach, the sliding subgroup is designed directly on the Lie group.
Since the topological structures of the configuration space (a Lie group) and
the tangent space (a vector space) are very different when the underlying Lie
group is not diffeomorphic to an Euclidean space, an important question arises
as to how to ensure the sliding surface to be indeed a subgroup of the state
space formed by the tangent bundle to guarantee the existence of the sliding
mode and thus to inherit the salient features of SMC in the Euclidean space.

SMC designs using the second approach have been reported for Lie groups such
as $SO(3)$, $\mathcal{S}^{3}$ for attitude control, and $SE(3)$ for motion
controls (Ghasemi et al., 2020; Lopez & Slotine, 2021). However, the issue of
whether the tangent bundle is a Lie group and whether the sliding subgroup is
properly immersed on the tangent bundle was not addressed in these works.
Therefore, the potential problem of lack of robustness due to the nonexistence
of the sliding mode might appear. Recently, Gómez et al. (2019) brought this
issue to the attention of the control community, and proposed an SMC on the
rotation group $SO(3)$ with a sliding surface which was ensured to be a Lie
subgroup immersed in the tangent bundle $SO(3)\times\mathbb{R}^{3}$, and a
finite-time convergent controller was devised for attitude control. This
design method was applied in Meng et al. (2023) to design a second-order SMC
for fault-tolerant control designs.

###  1.2 Contributions

We generalize the conventional sliding mode control designs for systems in
Euclidean spaces to fully-actuated simple mechanical systems whose
configuration space is a Lie group for the trajectory-tracking problem. The
main contributions can be summarized as follows: (1) we endow the state space
formed by the tangent bundle of the error dynamics with a Lie group structure
by defining a group operation that is based on a generic kinematic control
designed in the Lie algebra of the configuration Lie group; (2) we design a
smooth sliding subgroup and show it to be a Lie subgroup of the tangent
bundle, therefore, inheriting the Lie group structure of the state space; and
(3) we design a coordinate-free geometric sliding mode controller for a fully-
actuated mechanical system on a Lie group which drives the error dynamics to
the sliding subgroup globally exponential at the reaching stage, the error
dynamics then converges to the identity of the tangent bundle almost globally
asymptotically at the sliding stage. In addition, rigid body tracking in $3$D
space is addressed on the special orthogonal groups $SO(3)$ and on the unit
sphere $\mathcal{S}^{3}$, respectively, by applying the proposed geometric
sliding mode control.

###  1.3 Organization

The rest of the paper is organized as follows. Section 2 presents the notation
and background materials for simple mechanical systems with Lie groups as the
configuration space. Section 3 first endows the state space formed by the
tangent bundle with a Lie group structure under a group operation, which is
defined based on a generic kinematic control law in the Lie algebra of the
configuration space. Then, a smooth sliding subgroup is defined, which is a
Lie subgroup immersed in the tangent bundle. The convergence to the identity
of the tangent bundle of the reduced-order dynamics constrained on the sliding
surface is analyzed based on Lyapunov stability. Section 4 gives the design of
the GSMC, composed of a reaching law to the sliding subgroup and the
convergence property of the sliding subgroup. Attitude tracking of a rigid
body in $3$D space is addressed in Section 5 respectively on the rotational
group $SO(3)$ and the unit sphere $\mathcal{S}^{3}$, and simulation results
under the GSMC developed on $SO(3)$ are presented in Section 6 for
illustration and comparison. Conclusions are drawn in Section 7.

##  2 Mechanical systems on Lie groups

This section provides the notation and introduces the motion equations for a
fully-actuated simple mechanical system on Lie groups. More details can be
found in Bullo & Lewis (2005) and Abraham et al. (2012).

Given a finite-dimension Lie group $G$, the identity of the group is denoted
by $e\in G$. $T_{e}G$ denotes the tangent space in the identity, which also
defines its Lie algebra $\mathfrak{g}\triangleq T_{e}G$ in the Lie bracket
$[\cdot,\cdot]\in\mathfrak{g}$. Let $L_{g}(h)=gh\in G$ and $R_{g}(h)=hg\in G$
be the left and right translation maps, respectively, $\forall g,h\in G$, and
denote its corresponding tangent maps $T_{e}L_{g}(\nu)=g\cdot\nu\in T_{g}G$
and $T_{e}R_{g}(\nu)=\nu\cdot g\in T_{g}G$, $\forall\nu\in T_{e}G$, it
describes the natural isomorphism $T_{e}G\simeq T_{g}G$, which induces the
equivalence $TG\simeq G\times T_{e}G$ for the tangent bundle $TG=G\times
T_{g}G$. The inverse tangent map from $T_{g}G$ to $T_{e}G$ is denoted by
$\nu=g^{-1}\cdot v^{L}_{g}$, where $v^{L}_{g}=\nu_{L}(g)\in T_{g}G$, being
$\nu_{L}\in\Gamma^{\infty}(TG)$ a left-invariant vector field, with
$\Gamma^{\infty}(TG)$ denoting the set of $C^{\infty}$-sections of $TG$, and
respectively for a right-invariant vector field
$\nu_{R}\in\Gamma^{\infty}(TG)$, it follows that $v^{R}_{g}=\nu_{R}(g)\in
T_{g}G$, and accordingly $\nu=v^{R}_{g}\cdot g^{-1}$.

The cotangent space at $g\in G$ is denoted by $T^{*}_{g}G$, while
$\mathfrak{g}^{*}$ describes the dual space of the Lie algebra $\mathfrak{g}$.
Likewise, the cotangent bundle is denoted by $T^{*}G\simeq
G\times\mathfrak{g}^{*}$. Given a $\mathbb{R}$-vector space $V$, its dual
space $V^{*}$, and a bilinear map $B\vcentcolon V\times V\to\mathbb{R}$, the
flat map $B^{\flat}\vcentcolon V\to V^{*}$ is defined as $\langle
B^{\flat}(v);u\rangle=B(u,v)$, $\forall u,v\in V$, $B^{\flat}(v)\in V^{*}$,
where $\langle\alpha;v\rangle=\alpha(u)$ denotes the image in $\mathbb{R}$ of
$v\in V$ under the covector $\alpha\in V^{*}$. If the flat map is invertible,
then the inverse, known as the sharp map, is denoted by $B^{\sharp}\vcentcolon
V^{*}\to V$

The inner product on a smooth manifold $\mathcal{M}$ is denoted by
$\langle\langle\cdot,\cdot\rangle\rangle\in\mathbb{R}$. A Riemannian metric
$\mathbb{G}$ on a Lie group $G$ assigns the inner product
$\mathbb{G}(g)\cdot\left(X_{g},Y_{g}\right)$ on each $T_{g}G$, $\forall
X_{g},Y_{g}\in T_{g}G$. Moreover, when $\mathbb{G}$ is left-invariant (resp.
right-invariant), it induces an inner product in the Lie algebra
$\mathfrak{g}$ by
$\mathbb{I}\left(\xi,\zeta\right)=\mathbb{G}(g)\cdot\left(\xi_{L}(g),\zeta_{L}(g)\right)$,
$\forall\xi,\zeta\in\mathfrak{g}$. The kinetic energy is given by
$\mathrm{KE}(v_{g})=(1/2)\mathbb{G}(g)\cdot\left(v_{g},v_{g}\right)=(1/2)\mathbb{I}(\nu,\nu)$,
where $\mathbb{I}$ is the kinetic energy tensor, which induces a kinetic
energy metric $\mathbb{G}$ on $G$. In the rotational motion of a rigid body,
$\mathbb{I}$ also represents the inertia tensor.

In the sequel, only the left invariance will be used. The proposed control
methodology can be developed similarly for the right invariance. Also,
subscripts and superscripts $L$ will be dropped when the meaning is clear. A
left-invariant covariant derivative (affine connection) on a Lie group is
denoted by $\nabla_{\xi_{L}}\zeta_{L}\in\Gamma^{\infty}(TG)$ for any vector
fields $\xi_{L},\zeta_{L}\in\Gamma^{\infty}(TG)$. In addition, the Levi-Civita
connection associated with the Riemannian metric $\mathbb{G}$ is denoted by
$\overset{\mathbb{G}}{\nabla}$, which is unique and torsion-free. A left-
invariant affine connection on a Lie group is uniquely determined by a
bilinear map $B:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ called the
restriction of the left-invariant connection. In particular, the restriction
for the left-invariant Levi-Civita connection $\overset{\mathbb{G}}{\nabla}$
is defined as

$\overset{\mathfrak{g}}{\nabla}_{\xi}\zeta\triangleq\frac{1}{2}\left[\xi,\zeta\right]-\frac{1}{2}\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\xi}\mathbb{I}^{\flat}(\zeta)+\mathrm{ad}^{*}_{\zeta}\mathbb{I}^{\flat}(\xi)\right),$
(1)

where the adjoint map
$\mathrm{ad}\vcentcolon\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ is
defined as $\mathrm{ad}_{\xi}\zeta=\left[\xi,\zeta\right]$, and
$\mathrm{ad}^{*}_{\xi}\vcentcolon\mathfrak{g}^{*}\to\mathfrak{g}^{*}$ is the
dual map defined as
$\langle\mathrm{ad}^{*}_{\xi}\alpha;\zeta\rangle=\langle\alpha;\left[\xi,\zeta\right]\rangle$.
Furthermore, the adjoint action $\mathrm{Ad}\vcentcolon
G\times\mathfrak{g}\to\mathfrak{g}$ is $\mathrm{Ad}_{g}\zeta=g\cdot\zeta\cdot
g^{-1}$, $\forall g\in G$. So, the left-invariant Levi-Civita connection is
explicitly expressed as

$\overset{\mathbb{G}}{\nabla}_{\xi_{L}}\zeta_{L}\triangleq\left(\mathrm{d}\zeta(\xi)+\overset{\mathfrak{g}}{\nabla}_{\xi}\zeta\right)_{L},$
(2)

where
$\mathrm{d}\zeta(\xi)\triangleq\frac{d}{dt}|_{t=0}\;\zeta\left(g\;\mathrm{exp}(\xi
t)\right)$, being $\mathrm{exp}\vcentcolon\mathfrak{g}\to G$ the exponential
map on $G$, which is a local $C^{\infty}$-diffeomorphism, and whose inverse is
called the logarithmic map denoted by $\mathrm{log}\vcentcolon
G\to\mathfrak{g}$. By the left-invariance of vector fields
$\xi_{L},\zeta_{L}\in\Gamma^{\infty}(TG)$, the covariant derivative (2) is
expressed in terms of $\xi,\zeta\in\mathfrak{g}$ as follows

$\nabla_{\xi}\zeta\triangleq\mathrm{d}\zeta(\xi)+\overset{\mathfrak{g}}{\nabla}_{\xi}\zeta.$

Consider a differentiable curve $g:I\to G$, where $I$ is the set of all
intervals. Then a body velocity $\nu:I\to\mathfrak{g}$ is defined as $t\mapsto
T_{g(t)}L_{g^{-1}(t)}\left(\dot{g}(t)\right)$, for all $t\in I$, and therefore

$\dot{g}(t)=g(t)\cdot\nu(t).$ (3)

A forced mechanical system is governed by the intrinsic Euler-Lagrange
equations

$\overset{\mathbb{G}}{\nabla}_{\dot{g}(t)}\dot{g}(t)=F_{u}+\Delta_{d},$ (4)

where
$F_{u}=\sum^{m}_{a=1}u^{a}(t)\mathbb{G}^{\sharp}\left(T^{*}_{g(t)}L_{g^{-1}(t)}\left(f^{a}\right)\right)$
is the control force applied to the system on $T_{g}G$, being
$u^{a}:I\to\mathbb{R}$ the control inputs, and $f^{a}(g)\in\mathfrak{g}^{*}$
the control forces. Furthermore, $\Delta_{d}\in T_{g}G$ represents the vector
field version of constraint forces, such as potential external forces,
uncontrolled conservative plus dissipative forces, and unmodeled disturbances.

In view of (3) and the left-invariance of $\dot{g}$, the Levi-Civita
connection in (4) can be explicitly expressed using (1)-(2) as

$\overset{\mathbb{G}}{\nabla}_{\dot{g}(t)}\dot{g}(t)=g(t)\cdot\left(\dot{\nu}(t)+\overset{\mathfrak{g}}{\nabla}_{\nu(t)}\nu(t)\right),$

resulting in the controlled Euler-Poincaré equation

$\dot{\nu}(t)+\overset{\mathfrak{g}}{\nabla}_{\nu(t)}\nu(t)=f_{u}+\delta_{d},$
(5)

with
$f_{u}=\sum^{m}_{a=1}u^{a}(t)\mathbb{I}^{\sharp}\left(f^{a}\right)\in\mathfrak{g}$,
and $\delta_{d}=g^{-1}\cdot\Delta_{d}\in\mathfrak{g}$.

The underlying mechanical system on the Lie group $G$ is then defined by the
configuration Lie group $G$, the inertia tensor $\mathbb{I}$, and the external
forces $f_{u}+\delta_{d}$.

##  3 Lie Group Structure of the State Space and the Sliding Subgroup

In this section, we will endow the tangent bundle $TG\simeq
G\times\mathfrak{g}$ with a Lie group structure by a properly designed group
operation. For this purpose, an intrinsic control for kinematics is first
proposed (3). Then we design a smooth sliding subgroup that is immersed in the
tangent bundle so that it inherits the Lie group structure of the state space.

###  3.1 Intrinsic kinematic control

The purpose of this subsection is to design a control law
$\nu(t)\in\mathfrak{g}$ for the kinematics (3) to render $g(t)\to e$, the
group identity. Let $V:G\to\mathbb{R}_{\geq 0}$ be an infinitely
differentiable proper Morse function, which satisfies $V(g)>0$, $\forall g\in
G\backslash\\{e\\}$, $\mathrm{d}V(g)=0$ and $V(g)=0\iff g=e$. Morse functions,
a class of error functions (Koditschek, 1989; Bullo et al., 1995), are
guaranteed to exist on many Lie groups of practical interest considered in
this paper (Maithripala & Berg, 2015; Bullo & Lewis, 2005). They represent
potential energy that can be used to measure the distance between the
configuration $g$ and the identity $e$ on $G$. The following definition
specifies the class of kinematic controls considered in the paper.

######  Definition 3.1 (Kinematic control law).

Let $g:I\to G$ be a differentiable curve governed by (3), for all $t\in I$. A
kinematic control law is a map $\nu_{u}:G\to\mathfrak{g}$ that satisfies the
following properties.

  1. i.

$\nu_{u}(e)=0$,

  2. ii.

$\nu_{u}\left(g^{-1}\right)=-\nu_{u}\left(g\right)$,

  3. iii.

$\langle\mathrm{d}V(g(t));-g(t)\cdot\nu_{u}(g(t))\rangle<0$, $\forall g(t)\in
G\backslash\mathcal{O}_{u}$, where $\mathcal{O}_{u}\triangleq\left\\{g\in
G\backslash\\{e\\}\;|\;g\cdot\nu_{u}(g)=0\right\\}$,

  4. iv.

$\langle\mathrm{d}V(g(t));-g(t)\cdot\nu_{u}(g(t))\rangle=-y\left(g(t)\right)V\left(g(t)\right)$,
$\forall g(t)\in\mathcal{U}$, where $y:\mathcal{U}\to\mathbb{R}_{>0}$, and
$\mathcal{U}\subset G\backslash\mathcal{O}_{u}$ is a neighborhood of $e$.

Some comments on the class of kinematic controls are in order. Properties
(i)-(ii) are instrumental to building a particular Lie subgroup on the tangent
bundle. Properties (iii)-(iv) represent the sliding (convergence) property of
the reduced-order dynamics on the sliding subgroup (Lemma 5 below). In
particular, Property (iii) states the almost-global asymptotic stability for
system (3) in closed loop with the kinematic control law
$\nu(t)=-\nu_{u}(g(t))$. Note that since $\mathcal{O}_{u}$ is the set of
closed-loop equilibria other than $g(t)=e$, they are critical points of
$V(g)$. Since $V(g)$ is a Morse function, the set $\mathcal{O}_{u}$ consists
of a finite number of isolated points. In addition, this set is nowhere dense,
which means that it cannot separate the configuration space. Therefore, the
complement $G\backslash\mathcal{O}_{u}$ is open and dense, i.e.,
$G\backslash\mathcal{O}_{u}$ is a submanifold of $G$ (Maithripala et al.,
2006). Finally, Property (iv) establishes the local exponential stability of
the closed-loop system, where the existence of the neighborhood $\mathcal{U}$
is immediate because $V(g)$ is a Morse function, which has a unique minimum at
$e\in G$ by definition.

Note that both $V(g)$ and $\nu_{u}(g)$ are of free design, provided that the
properties in Definition 3.1 hold. However, it is worth considering the
kinematic control law in the logarithmic coordinate, that is,
$\nu_{u}(g)=\mathrm{log}(g)$, or some parallel vectors to $\mathrm{log}(g)$
(Akhtar & Waslander, 2020), as this map has been found to provide the
strongest stability results, for example, almost global and local exponential
convergence to the identity through a geodesic path (Bullo et al., 1995).

###  3.2 Lie Group structure for the state space

For systems described in (3) and (5), the state space is the tangent bundle
$TG\simeq G\times\mathfrak{g}$. To endow it with a Lie group structure, we
consider the binary operation $\star:\ TG\times TG\mapsto TG$ defined in the
following

$\displaystyle h_{1}\star h_{2}\triangleq$
$\displaystyle\big{(}g_{1}g_{2},\;\nu_{1}+\nu_{2}+\lambda\nu_{u}(g_{1})+\lambda\nu_{u}(g_{2})-\lambda\nu_{u}(g_{1}g_{2})\big{)},$
(6)

$\forall h_{1}=(g_{1},\nu_{1}),\;h_{2}=(g_{2},\nu_{2})\in TG$, and
$\lambda\in\mathbb{R}_{>0}$.

######  Lemma 1 (The state space $TG$ as a Lie group).

The tangent bundle $TG\equiv G\times\mathfrak{g}$ endowed with the binary
operation (3.2) is a Lie group, with

  1. i.

Identity element: $f\triangleq(e,0)\in TG$,

  2. ii.

Inverse element: $h^{-1}\triangleq\left(g^{-1},-\nu\right)\in TG$, $\forall
h=(g,\nu)\in TG$.

Being $TG$ a smooth manifold with (3.2) a smooth operation, it only remains to
verify the group axioms as follows.

  1. 1.

$\forall h=(g,\nu)\in TG$, it satisfies

$\displaystyle h\star f$
$\displaystyle=\left(ge,\;\nu+0+\lambda\nu_{u}(g)+\lambda\nu_{u}(e)-\lambda\nu_{u}(ge)\right)$
$\displaystyle=f\star h=h,$

where Property of Definition 3.1(i) is used.

  2. 2.

The group operation between $h=(g,\nu)\in TG$ and its inverse
$h^{-1}=\left(g^{-1},-\nu\right)\in TG$ verifies

$\displaystyle h^{-1}\star h$
$\displaystyle=\big{(}g^{-1}g,-\nu+\nu+\lambda\nu_{u}(g^{-1})+\lambda\nu_{u}(g)-\lambda\nu_{u}(g^{-1}g)\big{)}$
$\displaystyle=\big{(}gg^{-1},\nu-\nu+\lambda\nu_{u}(g)+\lambda\nu_{u}(g^{-1})-\lambda\nu_{u}(gg^{-1})\big{)}$
$\displaystyle=h\star h^{-1}=f,$

where Properties (i)-(ii) of Definition 3.1 are used.

  3. 3.

The associativity $h_{1}\star\left(h_{2}\star h_{3}\right)=\left(h_{1}\star
h_{2}\right)\star h_{3}$ is proved straightforwardly by substitution, using
the properties of Definition 3.1.

$\blacksquare$

######  Remark 2 (Tangent bundle $TG$).

The definition of the group operation (3.2) relying on the kinematic control
$\nu_{u}(g)$ in Definition 3.1 is crucial to define a sliding Lie subgroup
immersed in $TG$ in the next subsection. In fact, a group operation to endow
$TG$ to be a Lie group may simply be $h_{1}\star
h_{2}=\left(g_{1}g_{2},\;\nu_{1}+\nu_{2}\right)$. However, this operation does
not allow to design of a useful sliding subgroup, in particular, it fails to
prove closure under the group operation, as will be seen below.

######  Remark 3 (Associativity).

The associativity proved in Lemma 1 ensures the proposed Lie group $TG$ to be
globalizable (Olver et al., 1996), that is, the local Lie group $TG$ can be
extended to be a global topological group. This fact allows us to develop a
sliding mode control defined globally on the state space in contrast to the
Lie groups defined locally in Gómez et al. (2019) and Meng et al. (2023).

###  3.3 Sliding Subgroup on $TG$

In this subsection, we define a smooth sliding subgroup on the tangent bundle.
The following lemma shows that $H\subset TG$ is an immersed submanifold of
$TG$ that inherits the topology and smooth structure of the tangent bundle
$TG$ (Lee, 2013).

######  Lemma 4 (Sliding Lie subgroup).

Define

$H\triangleq\left\\{h=(g,\nu)\in TG\;|\;s(h)=0\right\\}\subset TG,$ (7)

where $\forall h=(g,\nu)\in TG$, the map $s:TG\mapsto\mathfrak{g}$ is defined
as

$s(h)=\nu+\lambda\nu_{u}(g).$ (8)

Then $H\subset TG$ is a Lie subgroup under the group operation (3.2).

The smoothness of $H$ is immediate, because the map defined in (8) is smooth.
The proof consists thus in showing that subset $H$ inherits the group
structure of the Lie group $TG$, by verifying the following:

  1. i.

Identity: The identity of the tangent bundle $f=(e,0)\in H$. This is immediate
by Definition 3.1(i) since $s(f)=0+\lambda\nu_{u}(e)=0$.

  2. ii.

Inverse. $\forall h=(g,\nu)\in H$, $s(h)=0\implies\nu=-\lambda\nu_{u}(g)$. By
Definition 3.1(ii) it follows that

$\displaystyle s\left(h^{-1}\right)$
$\displaystyle=-\nu+\lambda\nu_{u}\left(g^{-1}\right)$
$\displaystyle=-\left(-\lambda\nu_{u}(g)\right)+\lambda\nu_{u}\left(g^{-1}\right)=0.$

This proves that $h^{-1}\in H$ for all $h\in H$.

  3. iii.

Closure. Given $h_{1}=(g_{1},\nu_{1}),\ h_{2}=(g_{2},\nu_{2})\in H$, then
$s(h_{1})=0\implies\nu_{1}=-\lambda\nu_{u}(g_{1})$ and
$s(h_{2})=0\implies\nu_{2}=-\lambda\nu_{u}(g_{2})$. By (3.2) $h_{1}\star
h_{2}=\big{(}g_{1}g_{2},-\lambda\nu_{u}(g_{1}g_{2})\big{)}$. Thus,
$s\left(h_{1}\star
h_{2}\right)=-\lambda\nu_{u}(g_{1}g_{2})+\lambda\nu_{u}(g_{1}g_{2})=0$. That
is, $H$ is closed under the group operation.

$\blacksquare$

The following lemma shows that once a trajectory reaches the sliding subgroup
it will stay on it and converges to the group identity.

######  Lemma 5 (Properties of the sliding subgroup $H$).

Consider the sliding Lie subgroup $H\subset TG$ in (7). Then $H$ is forward
invariant, i.e., $h(t_{r})\in H$ for some $t_{r}\in I$ $\implies h(t)\in
H,\forall t\geq t_{r}$. Moreover, $h(t)\to(e,\ 0)$ almost globally
asymptotically.

Consider a differentiable curve $g:I\to G$ of the dynamics (3). Let
$V:G\to\mathbb{R}$ be a proper Morse function with the unique minimum at $e\in
G$. Then, along the trajectory $g(t)$ and $\forall t\in I$, it yields

$\frac{\mathrm{d}}{\mathrm{d}t}V\left(g(t)\right)=\langle\mathrm{d}V\left(g(t)\right);\dot{g}(t)\rangle=\langle\mathrm{d}V\left(g(t)\right);g(t)\cdot\nu(t)\rangle.$

Assume that $h(t_{r})\in H$, for some $t_{r}\in I$. Then $s(h)=0$ gives
$\nu(t)=-\lambda\nu_{u}(g(t))$. Therefore,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}V\left(g(t)\right)$
$\displaystyle=\langle\mathrm{d}V\left(g(t)\right);g(t)\cdot\nu(t)\rangle$
$\displaystyle=\langle\mathrm{d}V\left(g(t)\right);-\lambda
g(t)\cdot\nu_{u}\left(g(t)\right)\rangle,$

In light of Definition 3.1(iii), it follows that
$\frac{\mathrm{d}}{\mathrm{d}t}V\left(g(t)\right)<0$, for all $g(t_{r})\in
G\backslash\mathcal{O}_{u}$, and
$\frac{\mathrm{d}}{\mathrm{d}t}V\left(g(t)\right)=0\iff g=e$, where
$\mathcal{O}_{u}$ is a nowhere-dense set with a finite number of points given
in Definition 3.1(iii). Therefore, $h(t)$ will remain on $H$ for all $t\geq
t_{r}$, and the equilibrium $g(t)=e$ of (3) is almost globally asymptotically
stable for all $g(t_{r})\in G\backslash\mathcal{O}_{u}$ and locally
exponentially stable $\forall g(0)\in\mathcal{U}$, according to Definition
3.1(iv).

$\blacksquare$

##  4 Geometric Sliding Mode Control (GSMC)

In this section, we design a control law, called the reaching law, for $f_{u}$
in the Euler-Poincaré equation (5) to drive the trajectory
$h(t)=(g(t),\nu(t))\in TG$ to the sliding subgroup $H$. Then the tracking
control objective will be achieved as a consequence of Lemma 5.

###  4.1 Reaching Law

The Euler-Lagrange dynamics (4), ignoring disturbance forces $\Delta_{d}$, is
expressed as

$\nabla_{\nu(t)}\nu(t)=f_{u}.$ (9)

which is defined on $TG$, being $h(t)=\left(g(t),\nu(t)\right)$ the state
variable.

The intrinsic acceleration for the sliding variable (8) is calculated, by
using (9), through the covariant derivative of $s(h)\in\mathfrak{g}$ with
respect to itself as

$\displaystyle\nabla_{s(h)}s(h)$
$\displaystyle=\frac{d}{dt}s(h)+\overset{\mathfrak{g}}{\nabla}_{s(h)}s(h)$
$\displaystyle=\dot{\nu}+\lambda\dot{\nu}_{u}(g)+\overset{\mathfrak{g}}{\nabla}_{s(h)}s(h).$

Substituting the Euler-Poincaré equation (5) yields

$\displaystyle\nabla_{s(h)}s(h)$ $\displaystyle=$
$\displaystyle-\overset{\mathfrak{g}}{\nabla}_{\nu(t)}\nu(t)+\lambda\dot{\nu}_{u}(g)+\overset{\mathfrak{g}}{\nabla}_{s(h)}s(h)+f_{u}$
$\displaystyle=$
$\displaystyle\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\nu(t)}\mathbb{I}^{\flat}(\nu(t))\right)-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(s(h))\right)+\lambda\dot{\nu}_{u}(g)+f_{u},$
(10)

where the skew-symmetry of the Lie bracket $[\cdot,\cdot]\in\mathfrak{g}$ in
(1) is used. The reaching law is then proposed as follows

$f_{u}=\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\lambda\nu_{u}(g)}\mathbb{I}^{\flat}(\nu(t))\right)-\lambda\dot{\nu}_{u}(g)-k_{s}s(h),$
(11)

with $k_{s}>0$ a design parameter.

######  Theorem 6 (Reaching Controller).

The reaching law (11) drives exponentially the trajectories of the closed-loop
system (4.1) to the sliding subgroup $H$ $\forall h(0)\in TG$, i.e.,
$s(h(t))\to 0$ globally exponentially.

Consider the function $W:TG\to\mathbb{R}$ defined below

$W(h)=\frac{1}{2}\mathbb{I}(s(h),s(h)).$ (12)

Its time evolution along trajectories of (4.1) is given by

$\displaystyle\dot{W}(h)$
$\displaystyle=\mathbb{I}\left(\nabla_{s(h)}s(h),\;s(h)\right)$
$\displaystyle=\mathbb{I}\left(\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\nu(t)}\mathbb{I}^{\flat}(\nu(t))\right)-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(s(h))\right)\right.$
$\displaystyle\quad\quad+\lambda\dot{\nu}_{u}(g)+f_{u},\;s(h)\Bigl{)},$

which in closed loop with the controller (11) yields

$\displaystyle\dot{W}(h)$
$\displaystyle=\mathbb{I}\left(\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\nu(t)}\mathbb{I}^{\flat}(\nu(t))\right)-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(s(h))\right)\right.$
$\displaystyle\quad\quad+\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\lambda\nu_{u}(g)}\mathbb{I}^{\flat}(\nu(t))\right)-k_{s}s(h),\;s(h)\Bigl{)},$
$\displaystyle=\mathbb{I}\left(\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(\nu(t))\right)-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(s(h))\right)\right.$
$\displaystyle\quad\quad-k_{s}s(h),\;s(h)\Bigl{)}.$

By Lemma 12 in Appendix A the term
$\mathbb{I}\left(\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\zeta}\mathbb{I}^{\flat}(\eta)\right),\;\zeta\right)=0$,
for any $\zeta,\eta\in\mathfrak{g}$. Therefore,

$\dot{W}(h)=-k_{s}\mathbb{I}\left(s(h),\;s(h)\right)=-2k_{s}W(h).$

It follows from Proposition 6.26 of Bullo & Lewis (2005) that $W(h(t))\to 0$
exponentially. $\blacksquare$

######  Remark 7 (Passivity of the Lagrangian dynamics).

Note that the first two right-hand terms of the control law (11) complete the
terms
$\mathbb{I}^{\sharp}\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(\nu(t))-\mathbb{I}^{\sharp}\mathrm{ad}^{*}_{s(h)}\mathbb{I}^{\flat}(s(h))$.
By exploring the intrinsic passivity properties in Lemma 12 in Appendix A,
these terms were not canceled in the above stability analysis. This result was
first given for the Lie group $SO(3)$ in Koditschek (1989). The lemma 12
extends this result to coordinate-free Lagrangian dynamics on a general Lie
group, which has not been explored, to the authors’ knowledge, in the
literature for stability analysis.

######  Remark 8 (The reaching controller ).

The reaching law (11) achieves the convergence of $s(h(t))\to 0$ for the
Euler-Lagrange dynamics (9), which implies that $h(t)\in TG$ reaches the
sliding subgroup $H$ exponentially. Note that the result of Theorem 6 holds
when the external constraint forces $\delta_{d}$ can be compensated for by the
controller $f_{u}$, which was omitted from the control design. Otherwise, in
the presence of bounded $\delta_{d}$, $h(t)\in TG$ will remain bounded and
close to $H$.

###  4.2 Tracking Control

Let $g_{r}:I\to G$ be a twice differentiable configuration reference, with the
corresponding reference body velocity $\nu_{r}:I\to\mathfrak{g}$ given by
$\nu_{r}(t)\triangleq g^{-1}_{r}(t)\cdot\dot{g}_{r}(t)$. The problem is to
design a control law $f_{u}$ to track the reference. The Lie group structure
of the configuration space $G$ enables to define the following intrinsic
configuration error

$g_{e}(t)\triangleq g^{-1}_{r}(t)g(t).$

By left invariance the body velocity error is defined as

$\displaystyle\nu_{e}(t)$ $\displaystyle\triangleq
g^{-1}_{e}(t)\cdot\dot{g}_{e}(t)=\nu(t)-\eta_{r}(t),$ (13)

with $\eta_{r}(t)=\mathrm{Ad}_{g^{-1}_{e}}\nu_{r}(t)$. Then, the error
dynamics evolving on $TG$ is described by

$\nabla_{\nu_{e}(t)}\nu_{e}(t)=f_{u},$ (14)

being the state variable $h_{e}(t)=\left(g_{e}(t),\nu_{e}(t)\right)\in TG$.

The tracking problem, therefore, boils down to stabilizing the identity
$f=(e,0)$ on $TG$. By using the sliding-model control strategy, the error
state is first driven to the sliding subgroup in the reaching stage, and then
on the sliding subgroup, the reduced-order dynamics converges to the identity
$f$ ensured by Lemma 5.

In terms of the error state $h_{e}$ the sliding variable (8) is given by

$s(h_{e})=\nu_{e}(t)+\lambda\nu_{u}(g_{e}),$ (15)

and, its covariant derivative, by using (1)-(2), is

$\displaystyle\nabla_{s(h_{e})}s(h_{e})$
$\displaystyle=\frac{d}{dt}s(h_{e})+\overset{\mathfrak{g}}{\nabla}_{s(h_{e})}s(h_{e})$
$\displaystyle=\dot{\nu}_{e}(t)+\lambda\dot{\nu}_{u}(g_{e})-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h_{e})}\mathbb{I}^{\flat}\left(s(h_{e})\right)\right)$
$\displaystyle=\dot{\nu}(t)-\dot{\eta}_{r}(t)+\lambda\dot{\nu}_{u}(g_{e})-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h_{e})}\mathbb{I}^{\flat}\left(s(h_{e})\right)\right).$

Ignoring disturbance $\delta_{d}$ it follows from the Euler-Poincaré equation
(5) that

$\displaystyle\nabla_{s(h_{e})}s(h_{e})=$
$\displaystyle\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\nu(t)}\mathbb{I}^{\flat}\left(\nu(t)\right)\right)+f_{u}-\dot{\eta}_{r}(t)$
(16)
$\displaystyle+\lambda\dot{\nu}_{u}(g_{e})-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h_{e})}\mathbb{I}^{\flat}\left(s(h_{e})\right)\right).$

We proposed the following tracking controller

$\displaystyle f_{u}$
$\displaystyle=\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\lambda\nu_{u}(g_{e})-\eta_{r}(t)}\mathbb{I}^{\flat}(\nu(t))\right)-\lambda\dot{\nu}_{u}(g_{e})+\dot{\eta}_{r}(t)$
$\displaystyle\quad-k_{s}s(h_{e}),$ (17)

where $k_{s}>0$ is a design parameter. The following theorem establishes the
stability of the equilibrium $h_{e}=f$ in the closed-loop system (16)-(4.2).

######  Theorem 9 (Tracking Controller).

Consider the error dynamics (16) in closed loop with the controller (4.2).
Then, the equilibrium $h_{e}(t)=f$ is

  1. i.

almost-globally asymptotically stable, for all
$h_{e}(0)\in\overline{TG}\triangleq
G\backslash\mathcal{O}_{u}\times\mathfrak{g}$,

  2. ii.

locally exponentially stable for all
$h_{e}(0)\in\overline{TU}\triangleq\mathcal{U}\times\mathfrak{g}$, where
$\mathcal{O}_{u}$ and $\mathcal{U}$ are given in Definition 3.1(iii)-(iv).

Substituting the controller (4.2) in the error dynamics (16) yields the
closed-loop dynamics

$\displaystyle\nabla_{s(h_{e})}s(h_{e})$
$\displaystyle=\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h_{e})}\mathbb{I}^{\flat}\left(\nu(t)\right)\right)-k_{s}s(h_{e})$
$\displaystyle\quad-\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{s(h_{e})}\mathbb{I}^{\flat}\left(s(h_{e})\right)\right),$

which has an equilibrium point at $s(h_{e})=0$. The results follow as a
consequence of Theorem 6 (the reaching stage) and Lemma 5 (the sliding mode).
$\blacksquare$

######  Remark 10 (The tracking controller).

Theorem 9 gives a coordinate-free sliding mode control for a mechanical system
whose configuration space is a general Lie group. The group structure allows
defining globally a tracking error, whose dynamics evolves on the tangent
bundle. The Lie subgroup of the sliding subgroup immersed on the tangent
bundle ensures the existence of the sliding mode and thus inherits the salient
features of the SMC in Euclidean spaces.

Similarly to the Euclidean case, the design of the sliding subgroup and the
reaching law may incorporate other control objectives, such as finite-time
convergence and controller saturation, which are, however, beyond the scope of
the main purposes of this paper.

##  5 Attitude Tracking of a Rigid Body

In this section, we present the attitude tracking of a rigid body in the $3$D
space using the proposed GSMC. To illustrate the theoretic development, the
problem is addressed using attitude representation by first the rotation
matrix on $SO(3)$ and then by the unit quaternion $\mathcal{S}^{3}$.

###  5.1 GSMC for Attitude Tracking on $SO(3)$

The group of rotations on $\mathbb{R}^{3}$ is the Lie group
$SO(3)=\left\\{R\in\mathbb{R}^{3\times
3}\;|\;RR^{T}=R^{T}R=I_{3},\;\mathrm{det}(R)=+1\right\\}$, with the usual
multiplication of matrices as the group operation. The identity of the group
is the identity matrix $I_{3}$ of $3\times 3$, and the inverse is the
transpose $R^{T}\in SO(3)$ for any $R\in SO(3)$. The Lie algebra is given by
the set of skew-symmetric matrices
$\mathfrak{so}(3)=\left\\{S\in\mathbb{R}^{3\times 3}\;|\;S^{T}=-S\right\\}$,
which is isomorphic to $\mathbb{R}^{3}$, i.e.,
$\mathfrak{so}(3)\simeq\mathbb{R}^{3}$. The Lie bracket in $\mathbb{R}^{3}$ is
defined by the cross product
$[\zeta,\eta]=\mathrm{ad}_{\zeta}\eta\triangleq\zeta\times\eta$,
$\forall\zeta,\eta\in\mathbb{R}^{3}$. Denote the isomorphism
$\cdot^{\wedge}\vcentcolon\mathbb{R}^{3}\to\mathfrak{so}(3)$, and respectively
the inverse map $\cdot^{\vee}\vcentcolon\mathfrak{so}(3)\to\mathbb{R}^{3}$.
Then for a differentiable curve $R\vcentcolon I\to SO(3)$ with left-invariant
dynamics $\dot{R}(t)\in T_{R}SO(3)$, the body angular velocity is given by

$\Omega^{\wedge}(t)=R^{T}(t)\dot{R}(t)=\left[\begin{array}[]{ccc}0&-\Omega_{3}(t)&\Omega_{2}(t)\\\
\Omega_{3}(t)&0&-\Omega_{1}(t)\\\
-\Omega_{2}(t)&\Omega_{1}(t)&0\end{array}\right],$

for all $t\in I$. The kinetic energy of the rotational motion of a rigid body
is calculated as
$\mathrm{KE}(\Omega)=\frac{1}{2}\mathbb{J}(\Omega,\Omega)\triangleq\frac{1}{2}\langle\langle\mathbb{J}\Omega,\Omega\rangle\rangle$,
where $\mathbb{J}=\mathbb{J}^{T}\in\mathbb{R}^{3\times 3}$ is the positive-
definite inertia tensor. Therefore,
$\mathrm{ad}^{*}_{\zeta}\mathbb{J}^{\flat}(\eta)=\left(\mathbb{J}\eta\right)^{\wedge}\zeta$,
and $\mathbb{J}^{\sharp}(\zeta)=\mathbb{J}^{-1}\zeta$. Hence, the rotational
motion described by the Euler-Lagrange equation (4) is

$\nabla_{\Omega(t)}\Omega(t)=\tau_{u}.$ (18)

The state $(R,\omega)$ evolves on the tangent bundle $TSO(3)\simeq
SO(3)\times\mathbb{R}^{3}$, and the control torque
$\tau_{u}=\mathbb{J}^{-1}\tau\in\mathbb{R}^{3}$ is expressed in the body
frame. Furthermore, (18) is explicitly expressed, by using the Euler-Poincaré
equation (5) and restriction (1), as

$\dot{\Omega}(t)-\mathbb{J}^{-1}\left(\mathbb{J}\Omega(t)\right)^{\wedge}\Omega(t)=\tau_{u}.$
(19)

Let $R_{r}\vcentcolon I\to SO(3)$ be a twice differentiable attitude
reference, and $\Omega_{r}:I\to\mathbb{R}^{3}$, the reference angular velocity
expressed in the body frame, which holds
$\Omega_{r}(t)=\left(R^{T}_{r}(t)\dot{R}_{r}(t)\right)^{\vee}$. Then, the
intrinsic attitude error is

$R_{e}(t)\triangleq R^{T}_{r}(t)R(t).$

In view of (13) the (left-invariant) velocity error is

$\displaystyle\Omega_{e}(t)$
$\displaystyle\triangleq\left(R^{T}_{e}(t)\dot{R}_{e}(t)\right)^{\vee}=\Omega(t)-\sigma(t),$
(20) $\displaystyle\sigma(t)$
$\displaystyle\triangleq\mathrm{Ad}_{R^{-1}_{e}}\Omega_{r}(t)=R^{T}_{e}(t)\Omega_{r}(t).$

Therefore, the distance between $R_{e}(t)$ and $I_{3}$ is properly measured
with the Morse function $V_{1}(R_{e})\triangleq
2-\sqrt{1+\mathrm{tr}(R_{e}(t))}$, proposed by Lee (2012). In fact,
$V_{1}(R_{e})=0\iff R_{e}=I_{3}$ and is positive for all $R_{e}\in
SO(3)\backslash\\{I_{3}\\}$. Moreover, along the trajectories of (20), it
satisfies

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}V_{1}(R_{e})$
$\displaystyle=\left\langle\left\langle\psi(R_{e})\left(R_{e}(t)-R^{T}_{e}(t)\right)^{\vee},\;\Omega_{e}(t)\right\rangle\right\rangle,$
$\displaystyle\psi(R_{e})$
$\displaystyle\triangleq\frac{1}{2\sqrt{1+\mathrm{tr}(R_{e})}},$

for all $R_{e}\in SO(3)\backslash\mathcal{O}_{R}$, where
$\mathcal{O}_{R}\triangleq\\{R\in SO(3)|\mathrm{tr}(R)$ $=-1\\}$. Furthermore,
given $\mathcal{U}_{R}$ $\triangleq$ $\\{R_{e}\in SO(3)\backslash$
$\mathcal{O}_{R}\;|\;V_{1}(R_{e})<2-\epsilon\\}$, for some $\epsilon>0$
arbitrarily small, $V_{1}(R_{e})$ verifies (Lee, 2012)

$\displaystyle\left\|\psi(R_{e})\left(R_{e}-R^{T}_{e}\right)^{\vee}\right\|^{2}$
$\displaystyle\leq V_{1}(R_{e})\leq
2\left\|\psi(R_{e})\left(R_{e}-R^{T}_{e}\right)^{\vee}\right\|^{2},$

for all $R_{e}\in\mathcal{U}_{R}$.

Consider the kinematic control law

$\displaystyle\Omega_{u}(R_{e})$
$\displaystyle\equiv\mathrm{log}(R_{e})^{\vee},$ (21)
$\displaystyle\mathrm{log}(R_{e})$
$\displaystyle\triangleq\left\\{\begin{array}[]{ll}0_{3\times
3},&R_{e}=I_{3},\\\
\frac{\phi(R_{e})}{2\sin\left(\phi(R_{e})\right)}\left(R_{e}-R^{T}_{e}\right),&R_{e}\neq
I_{3},\end{array}\right.$ (24)

where
$\phi(R_{e})\triangleq\arccos{\left(\frac{1}{2}\left(\mathrm{tr}(R_{e})-1\right)\right)}\in(-\pi,\pi)$,
and $0_{n\times m}$ is a matrix of size $n\times m$ with zero-entries. It can
verify readily Definition 3.1(i)-(ii) by (21). To verify Definition
3.1(iii)-(iv) under the kinematic control $\Omega_{e}(t)=-\Omega_{u}(R_{e})$,
consider the derivative of the Morse function $V_{1}(R_{e})$ along the error
kinematics $\dot{R}_{e}=R_{e}\Omega_{e}^{\wedge}$:

$\dot{V}_{1}(R_{e})=\left\langle\left\langle\psi(R_{e})\left(R_{e}-R^{T}_{e}\right)^{\vee},\;-\Omega_{u}(R_{e})\right\rangle\right\rangle<0,$

for all $R_{e}\in SO(3)\backslash\mathcal{O}_{R}$ and $\dot{V}_{1}(R_{e})\leq-
y_{1}(R_{e})V_{1}(R_{e})$ for all $R_{e}\in\mathcal{U}_{R}$, where

$y_{1}(R_{e})\triangleq\frac{\phi(R_{e})}{4\psi(R_{e})\sin\phi(R_{e})}>0,\;\forall
R_{e}\in\mathcal{U}_{R}.$

This proves that the kinematic control (21) also holds Definition
3.1(iii)-(iv).

Therefore, based on the kinematic control law (21) the following group
operation is defined

$\displaystyle r_{1}\star r_{2}$ (25) $\displaystyle=$
$\displaystyle\bigg{(}R_{1}R_{2},\;\Omega_{1}+\Omega_{2}+\lambda\Omega_{u}(R_{1})+\lambda\Omega_{u}(R_{2})-\lambda\Omega_{u}(R_{1}R_{2})\Big{)},$

for any $r_{1}=(R_{1},\Omega_{1})$, $r_{2}=(R_{2},\Omega_{2})\in TSO(3)$.
Thus, the tangent bundle $TSO(3)\simeq SO(3)\times\mathbb{R}^{3}$ is endowed
with a Lie group structure with identity $(I_{3},0_{3\times 1})\in TSO(3)$ and
inverse $r^{-1}=(R^{T},-\Omega)\in TSO(3)$, $\forall r=(R,\Omega)\in TSO(3)$.
Likewise, given $r_{e}(t)=\left(R_{e}(t),\Omega_{e}(t)\right)\in TSO(3)$ and
in view of (15), the map $s:TSO(3)\to\mathbb{R}^{3}$

$s(r_{e})=\Omega_{e}(t)+\lambda\Omega_{u}(R_{e}),$ (26)

for some scalar $\lambda>0$, defines a Lie subgroup

$H_{R}=\\{r_{e}(t)=\left(R_{e}(t),\Omega_{e}(t)\right)\in
TSO(3)\;|\;s(r_{e})=0_{3\times 1}\\},$ (27)

under the group operation (25).

Thus, the tracking controller on $SO(3)$ is obtained from (4.2) and (26) as

$\displaystyle\tau_{u}$
$\displaystyle=\mathbb{J}^{-1}\left(\left(\mathbb{J}\Omega(t)\right)^{\wedge}\left(\lambda\Omega_{u}(R_{e})-\sigma(t)\right)\right)-\lambda\dot{\Omega}_{u}(R_{e})+\dot{\sigma}(t)$
$\displaystyle\quad-k_{s}s(r_{e}),$ (28)

where $k_{s}>0$ is a controller gain. Theorem 9 proves that controller (28) in
closed loop with the system (19) renders the equilibrium point
$r_{e}(t)=\left(I_{3},0_{3\times 1}\right)$ almost globally asymptotically
stable for all $r_{e}(0)\in
SO(3)\backslash\mathcal{O}_{r}\times\mathbb{R}^{3}$, and exponentially stable
for all $r_{e}(0)\in\mathcal{U}_{r}\times\mathbb{R}^{3}$.

######  Remark 11.

Note that in applying Theorem 9 it should define first a tracking error using
the group operation on the configuration manifold, and then treat the error
dynamics as a physical system. Otherwise, the sliding surface may not be a Lie
subgroup. To see this more clear, consider $r=\left(R,\Omega\right)$,
$r^{-1}_{d}=\left(R^{T}_{r},-\Omega_{r}\right)\in TSO(3)$, then in the
following tracking error may be defined by the group operation (25)

$\displaystyle r^{\prime}_{e}$ $\displaystyle=r^{-1}_{d}\star
r=\left(R^{T}_{r}R,\;-\Omega_{r}+\Omega+\lambda\Omega_{u}(R^{T}_{r})+\lambda\Omega_{u}(R)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.-\lambda\Omega_{u}(R^{T}_{r}R)\right)$
$\displaystyle=\left(R_{e},\;-\Omega_{r}+\Omega+\lambda\Omega_{u}(R^{T}_{r})+\lambda\Omega_{u}(R)-\lambda\Omega_{u}(R_{e})\right)$
$\displaystyle=\left(R_{e},\bar{\Omega}_{e}\right).$

However, $H^{\prime}_{R}=\\{r^{\prime}_{e}(t)\in
TSO(3)\;|\;s(r^{\prime}_{e})=0_{3\times 1}\\}\subset TSO(3)$ is not a sliding
subgroup for the proposed Morse function $V_{1}(R_{e})$.

###  5.2 Attitude Tracking on $\mathcal{S}^{3}$

The set of $\mathbb{R}^{4}$-vectors evolving on the unit sphere
$\mathcal{S}^{3}=\left\\{q\in\mathbb{R}^{4}\;|\;q^{T}q=1\right\\}$, with
$q=\left[q_{0},\vec{q}^{T}\right]^{T}\in\mathcal{S}^{3}$, $q_{0}\in[-1,1]$,
and $\vec{q}\in\mathbb{R}^{3}$, is a Lie group with identity
$\imath=\left[1,0_{1\times 3}\right]^{T}\in\mathcal{S}^{3}$, and inverse
$q^{-1}=\left[q_{0},-\vec{q}^{T}\right]^{T}\in\mathcal{S}^{3}$, under the
group operation $(q_{1},q_{2})\mapsto q_{1}\otimes q_{2}\in\mathcal{S}^{3}$
defined as

$q_{1}\otimes q_{2}\triangleq
Q(q_{1})q_{2}=\left[\begin{array}[]{cc}q_{0,1}&-\vec{q}^{T}_{1}\\\
\vec{q}_{1}&q_{0,1}I_{3}+\vec{q}_{1}^{\wedge}\end{array}\right]\left[\begin{array}[]{c}q_{0,2}\\\
\vec{q}_{2}\end{array}\right],$

for any $q_{1}=\left[q_{0,1},\vec{q}^{T}_{1}\right]^{T}$,
$q_{2}=\left[q_{0,2},\vec{q}^{T}_{2}\right]^{T}\in\mathcal{S}^{3}$. The Lie
algebra is
$\mathfrak{s}^{3}=\left\\{\omega\in\mathbb{R}^{4}\;|\;\omega=\left[0,\Omega^{T}\right]^{T},\Omega\in\mathbb{R}^{3}\right\\}$,
which holds $\mathfrak{s}^{3}\simeq\mathbb{R}^{3}$. Its Lie bracket operation
corresponds to the cross product in $\mathbb{R}^{3}$. Thus, denote the
isomorphism $\overline{\cdot}\vcentcolon\mathbb{R}^{3}\to\mathfrak{s}^{3}$
with the inverse map
$\underline{\cdot}\vcentcolon\mathfrak{s}^{3}\to\mathbb{R}^{3}$.

Rodriguez formula $q\mapsto R(q)=I_{3}+2q_{0}\vec{q}^{\wedge}+2\vec{q}^{\wedge
2}\in SO(3)$ relates each antipodal point $\pm q$ with a physical rotation of
a rigid body, i.e., $\mathcal{S}^{3}$ double covers the group $SO(3)$. The
adjoint action in $\mathcal{S}^{3}$ is defined as
$\mathrm{Ad}_{q}\zeta\triangleq q\otimes\bar{\zeta}\otimes
q^{-1}=\overline{R(q)\zeta}$, for any $\zeta\in\mathbb{R}^{3}$.

Given a differentiable curve $q\vcentcolon I\to\mathcal{S}^{3}$ with a left-
invariant vector field $\dot{q}(t)\in T_{q}\mathcal{S}^{3}$, and a twice-
differentiable reference configuration $q_{r}\vcentcolon I\to\mathcal{S}^{3}$,
$\forall t\in I$, the body angular velocity $\overline{\Omega}(t)\triangleq
2q^{-1}(t)\otimes\dot{q}(t)=2Q^{T}(q(t))\dot{q}(t)\in\mathfrak{s}^{3}$ and the
reference angular velocity $\overline{\Omega}_{r}(t)\triangleq
2q^{-1}_{r}(t)\otimes\dot{q}_{r}(t)\in\mathfrak{s}^{3}$ can be defined. We
consider the following intrinsic tracking error $q_{e}(t)\triangleq
q^{-1}_{r}(t)\otimes q(t),$ and its left-invariant velocity error

$\displaystyle\overline{\Omega}_{e}(t)$ $\displaystyle\triangleq
2q^{-1}_{e}(t)\otimes\dot{q}_{e}(t)=\overline{\Omega}(t)-\overline{\zeta}(t),$
(29) $\displaystyle\overline{\zeta}(t)$
$\displaystyle=\mathrm{Ad}_{q^{-1}_{e}}\Omega_{r}(t),$

where $\dot{q}_{e}(t)\in T_{q_{e}}\mathcal{S}^{3}$ is left invariant. Propose
the Morse function $\mathcal{S}^{3}\ni q\mapsto
V_{2}(q)=\frac{1}{\sqrt{2}}\left\|\imath-q\right\|=\sqrt{1-q_{0}}$, which
satisfies $V_{2}(q)=0\iff q=\imath$, and $V_{2}(q)>0$ $\forall
q\in\mathcal{S}^{3}\backslash\\{\imath\\}$. That is, function $V_{2}(q)$ has a
unique minimum critical zero at identity $\imath\in\mathcal{S}^{3}$ and is
strictly positive for any other $q\in\mathcal{S}^{3}$. Moreover, it verifies
that

$\frac{\mathrm{d}}{\mathrm{d}t}V_{2}(q_{e})=\frac{-\dot{q}_{0,e}(t)}{2\sqrt{1-q_{0,e}(t)}}=\frac{1}{4\sqrt{1-q_{0,e}(t)}}\vec{q}^{T}_{e}(t)\Omega_{e}(t),$

which suggests the following kinematic control law

$\Omega_{u}(q_{e})\equiv\mathrm{log}(q_{e})\triangleq\left\\{\begin{array}[]{ll}0_{3\times
1},&q_{e}=\imath,\\\
\frac{\arccos(q_{0,e})}{\|\vec{q}_{e}\|}\vec{q}_{e},&q_{e}\neq\imath,\end{array}\right.$
(30)

for all $q_{e}(t)\in\mathcal{S}^{3}\backslash\\{-\imath\\}$. Indeed, when
$\Omega_{e}(t)=-\Omega_{u}(q_{e})$, it leads to

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}V_{2}(q_{e})$
$\displaystyle=-\frac{\arccos(q_{0,e})}{4\sqrt{1-q_{0,e}}}\|\vec{q}_{e}\|$
$\displaystyle=-\frac{\arccos(q_{0,e})}{4\sqrt{1-q_{0,e}}}\sqrt{1-q^{2}_{0,e}}$
$\displaystyle=-\frac{\arccos(q_{0,e})}{4\sqrt{1-q_{0,e}}}\sqrt{\left(1+q_{0,e}\right)\left(1-q_{0,e}\right)}$
$\displaystyle=-\frac{\arccos(q_{0,e})}{4\sqrt{1-q_{0,e}}}\sqrt{1+q_{0,e}}V_{2}(q_{e}),$
$\displaystyle=-y_{2}(q_{e})V_{2}(q_{e}),$

where $y_{2}(q_{e})>0$ for all
$q_{e}\in\mathcal{U}_{q}\triangleq\\{q_{e}\in\mathcal{S}^{3}\backslash\\{\imath\\}\;|\;V_{2}(q_{e})<2-\epsilon\\}$,
for some $\epsilon>0$ arbitrarily small. Consequently, the control law (30)
satisfies all properties of Definition 3.1 for the Morse function
$V_{2}(q_{e})$.

The kinematic control law (30) enables the definition of the tangent bundle
$T\mathcal{S}^{3}\simeq\mathcal{S}^{3}\times\mathbb{R}^{3}$ as a Lie group
under the group operation

$\displaystyle p_{1}\star p_{2}$ (31) $\displaystyle=$
$\displaystyle\left(q_{1}\otimes
q_{2},\;\Omega_{1}+\Omega_{2}+\lambda\Omega_{u}(q_{1})+\lambda\Omega_{u}(q_{2})-\lambda\Omega_{u}(q_{1}\otimes
q_{2})\right),$

$\forall p_{1}=(q_{1},\Omega_{1})$, $p_{2}=(q_{2},\Omega_{2})\in
T\mathcal{S}^{3}$. Note that the identity is $(\imath,0_{3\times 1})\in
T\mathcal{S}^{3}$, and inverse, $p^{-1}=(q^{-1},-\Omega)\in T\mathcal{S}^{3}$,
$\forall p=(q,\Omega)\in T\mathcal{S}^{3}$. Therefore, the map

$s(p_{e})=\Omega_{e}(t)+\lambda\Omega_{u}(q_{e}),$ (32)

where $p_{e}(t)=(q_{e}(t),\Omega_{e}(t))\in T\mathcal{S}^{3}$, defines the
sliding Lie subgroup

$H_{q}=\left\\{p_{e}\in T\mathcal{S}^{3}\;|\;s(p_{e})=0_{3\times 1}\right\\}.$
(33)

The attitude tracking controller on $\mathcal{S}^{3}$ is thus defined as (4.2)
using (29)-(32), which yields

$\displaystyle\tau_{u}$
$\displaystyle=\mathbb{J}^{-1}\left(\left(\mathbb{J}\Omega(t)\right)^{\wedge}\left(\lambda\Omega_{u}(q_{e})-\zeta(t)\right)\right)-\lambda\dot{\Omega}_{u}(q_{e})+\dot{\zeta}(t)$
$\displaystyle\quad-k_{s}s(p_{e}).$ (34)

By Theorem 9 controller (34) in closed loop with system (18) achieves the
asymptotic convergence of $p_{e}(t)\to(\imath,0_{3\times 1})$ for all
$p_{e}(0)\in\mathcal{S}^{3}\backslash\\{\imath\\}\times\mathbb{R}^{3}$, and
exponential convergence when $p_{e}(0)\in\mathcal{U}_{q}\times\mathbb{R}^{3}$.

##  6 Simulations

To illustrate the theoretical results and for comparison, the proposed GSMC
(28) was contrasted with two reported controllers: the ”linearization”-by-
state-feedback-like (LSF) controller Eq. (26) of Maithripala et al. (2006),
and the PD+ controller Eq. (23) of Lee (2012). For easy comparison, the
applied torque control for each controller is rewritten in terms of

$s_{i}=\tilde{\omega}_{i}+\gamma_{i}\tilde{\varphi}_{i},\quad\forall i=1,2,3,$
(35)

where $\tilde{\omega}_{i}\in\mathbb{R}^{3}$ is angular velocity error,
$\tilde{\varphi}_{i}\in\mathbb{R}^{3}$ is attitude error, and $\gamma_{i}>0$
is the control gain.

The proposed GSMC law (28) is expressed as

$\displaystyle\tau_{1}$ $\displaystyle=-k_{s}\mathbb{J}s_{1}+F_{1},$ (36)
$\displaystyle s_{1}$ $\displaystyle=\Omega_{e}(t)+\lambda\Omega_{u}(R_{e}),$
(37) $\displaystyle F_{1}$
$\displaystyle=\mathbb{J}\left(-\lambda\dot{\Omega}_{u}(R_{e})+\dot{\sigma}\right)+\left(\mathbb{J}\Omega\right)^{\wedge}\left(\lambda\Omega_{u}(R_{e})-\sigma\right).$

Likewise, the LSF controller (26) of Maithripala et al. (2006) is given by

$\displaystyle\tau_{2}$ $\displaystyle=-k\mathbb{J}s_{2}+F_{2},$ (38)
$\displaystyle s_{2}$
$\displaystyle=\Omega-\Omega_{r}+\frac{\kappa}{k}R^{T}\left(RR^{T}_{r}-R_{r}R^{T}\right)^{\vee},$
(39) $\displaystyle F_{2}$
$\displaystyle=\mathbb{J}\dot{\Omega}_{r}-\left(\mathbb{J}\Omega\right)^{\wedge}\left(\Omega\right)-\Omega^{\wedge}\Omega_{r},$

where $\tilde{I}=I_{3}$ and $K=\kappa I_{3}$, for some $\kappa>0$. Finally,
the PD+ controller (23) of Lee (2012) is rewritten as

$\displaystyle\tau_{3}$ $\displaystyle=-k_{\Omega}s_{3}+F_{3},$ (40)
$\displaystyle s_{3}$
$\displaystyle=\Omega_{e}+\frac{k_{R}}{k_{\Omega}}\psi(R_{e})\left(R_{e}-R^{T}_{e}\right)^{\vee},$
(41) $\displaystyle F_{3}$
$\displaystyle=\mathbb{J}R^{T}R_{r}\dot{\Omega}_{r}+\left(R^{T}R_{r}\Omega_{r}\right)^{\wedge}\mathbb{J}R^{T}R_{r}\Omega_{r}.$

The inertia tensor was given by

$\mathbb{J}=\left[\begin{array}[]{ccc}3.6046&-0.0706&0.1491\\\
-0.0706&8.6868&0.0449\\\ 0.1491&0.0449&9.3484\end{array}\right],$

while the reference trajectory was calculated as
$\Omega_{r}(t)=\left(R^{T}_{r}(t)\dot{R}_{r}(t)\right)^{\vee}=\left[0,0.1,0\right]^{T}$
(rad/s). Furthermore, the initial conditions were chosen as
$\Omega(0)=\left(1/\left(2\sqrt{14}\right)\right)\left[1,2,3\right]^{T}$
(rad/s), $R_{r}(0)=R_{312}(\pi/4,-\pi,\pi/4)$, where the expression
$R_{312}(\varphi,\vartheta,\psi)$ is a rotation matrix described by the
sequence 3-1-2 of Euler angles (Shuster et al., 1993), and the initial
attitude was calculated as $R(0)=R_{r}(0)R_{e}(0)$.

The simulations were carried out under three scenarios according to the
distance between $R_{e}(0)$ and the desired equilibrium $I_{3}$, and to the
undesired equilibrium $\mathrm{diag}(1,-1,-1)$ measured by the Morse function
$\Psi(R_{e})\triangleq\frac{1}{2}\mathrm{tr}(I_{3}-R_{e})$ used in Maithripala
et al. (2006). Therefore, the initial attitudes
$R_{e}(0)=R_{312}(0,-0.428\pi,0)$, $R_{e}(0)=R_{312}(0,-0.01\pi,0)\approx
I_{3}$, and $R_{e}(0)=R_{312}(0,-0.99\pi,0)\approx\mathrm{diag}(1,-1,-1)$ were
assigned.

Finally, the design parameters for each controller were tuned in such a way
that the energy-consumption level measured by
$\sqrt{\int^{t}_{0}\tau^{T}_{i}(t)\tau_{i}(t)\mathrm{d}t}$ in the first
scenario is the same. The resulting controller gains were $k_{s}=1$,
$\lambda=0.5$ for (36), $k=1$, $\kappa=0.5$ for (38), and $k_{\Omega}=18.5$,
$k_{R}=9.25$ for (40). With these design parameters the control gain of (35)
was $\gamma_{i}=0.5$ for all $i=1,2,3$.

###  6.1 Scenario 1. Intermediate case.

Figure 1 shows the performance of the controllers (36), (38), and (40) under
the initial condition $R_{e}(0)=R_{312}(0,-0.428\pi,0)$. Fig. 1(a) shows the
attitude error $\Psi(R_{e})\triangleq\frac{1}{2}\mathrm{tr}(I_{3}-R_{e})$, it
is observed that the proposed controller (36) and controller (38) achieve the
convergence $R_{e}\to I_{3}$ in $17$ (s), while controller (40) achieves it in
$30$ (s). Fig. 1(b) illustrates the norm $\|\Omega_{e}(t)\|$ for each
controller, where the angular velocity error $\Omega_{e}(t)$ is calculated as
(20), it can be seen that controller (40) takes $10$ (s) longer than the other
controllers to reach $\Omega_{e}(t)\to 0_{3\times 1}$. Furthermore, Figs. 1(c)
and (d) draw the control effort and the energy consumption respectively, it is
observed that, with the selected controller gains, all controllers consume the
same amount of energy. Finally, Fig. 2 shows the behavior of the sliding
variables (37), (39), and (41) compared to $s_{i}=0$ according to (35). It is
observed that the proposed controller (36) allows convergence $s_{1}\to
0_{3\times 1}$ to complete the reach phase, while the LSF control scheme (38)
presents an oscillatory behavior around the equilibrium point, in addition to
the PD + controller (40) that follows closely $s_{i}=0$ until it reaches
equilibrium.

Figure 1: Scenario 1: Behavior of controllers (36), (38), and (40) when the
initial attitude error is $R_{e}(0)=R_{312}(0,-0.428\pi,0)$. Figure 2:
Scenario 1: Behavior of the sliding variable (37), (39), and (41) when
$R_{e}(0)=R_{312}(0,-0.428\pi,0)$.

###  6.2 Scenario 2. Starting close to the desired equilibrium point $I_{3}$.

For this scenario, the initial condition was set to
$R_{e}(0)=R_{312}(0,-0.01\pi,0)$, which corresponds to an initial condition
close to the desired equilibrium $I_{3}$. Figs. 3(a) and (b) show that the
controllers (36) and (38) reach the desired equilibrium
$(R_{e},\Omega_{e})=(I_{3},0_{3\times 1})$ at the same time $20$ (s), while
the controller (40) takes $5$ (s) longer, which coincides with the previous
scenario. However, as illustrated in Fig. 3(d), the proposed controller uses
less energy than others to reach the desired equilibrium when the system
starts close to the desired equilibrium.

Figure 3: Scenario 2: Behavior of controllers (36), (38), and (40) when the
initial attitude error is close to $I_{3}$, i.e.,
$R_{e}(0)=R_{312}(0,-0.01\pi,0)$.

###  6.3 Scenario 3. Starting close to the undesired equilibrium point
$\mathrm{diag}(1,-1,-1)$.

Figure 4 displays the performance of the controllers starting close to the
undesired equilibrium point $\mathrm{diag}(1,-1,-1)$, i.e.,
$R_{e}(0)=R_{312}(0,-0.99\pi,0)$. It is observed in Fig. 4(a) that the
proposed controller (36) and the PD+ controller (36) present a delay of $1$
(s) before beginning the convergence of $R_{e}\to I_{3}$, however, the LSF
controller (38) has the longest delay of $2.5$ (s). Notice that the proposed
control scheme allows a faster convergence to the desired equilibrium point
(Figs. 4(a) and (b)) at a cost of more energy consumption (Figs. 4(c) and
(d)).

Figure 4: Scenario 3: Behavior of controllers (36), (38), and (40) when the
initial attitude error is close to $\mathrm{diag}(1,-1,-1)$), i.e.,
$R_{e}(0)=R_{312}(0,-0.99\pi,0)$.

##  7 Conclusions

This paper presented a geometric sliding mode control for fully actuated
mechanical systems evolving on Lie groups, generalizing the conventional
sliding mode control in Euclidean spaces. It was shown that the sliding
surface (a Lie subgroup) is immersed in the state space (a Lie group) of the
system dynamics, and the tracking is achieved by first driving the
trajectories of the system to the sliding subgroup and then converging to the
group identity of the reduced dynamics restricted on the sliding subgroup,
like sliding mode control designs for systems evolving on Euclidean spaces. An
application of the result to attitude control was presented for the rotation
group $SO(3)$ and the unit sphere $\mathcal{S}^{3}$. The simulation results
illustrated the scheme and compared it with similar control designs in the
literature.

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##  Appendix A A passivity-like lemma

######  Lemma 12.

Given that the inner product on $\mathfrak{g}$ is a symmetric bilinear map,
for any $\zeta,\eta\in\mathfrak{g}$, it holds

$\displaystyle\mathbb{I}\left(\zeta,\;\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\zeta}\mathbb{I}^{\flat}(\eta)\right)\right)$
$\displaystyle=\langle\mathbb{I}^{\flat}\left(\mathbb{I}^{\sharp}\left(\mathrm{ad}^{*}_{\zeta}\mathbb{I}^{\flat}(\eta)\right)\right);\zeta\rangle$
$\displaystyle=\langle\mathrm{ad}^{*}_{\zeta}\mathbb{I}^{\flat}(\eta);\zeta\rangle$
$\displaystyle=\langle\mathbb{I}^{\flat}(\eta);\left[\zeta,\zeta\right]\rangle$
$\displaystyle=0,$

because of the skew symmetry of the Lie bracket operation
$[\cdot,\cdot]\in\mathfrak{g}$.