2023 1

Yogesh et al. Geometric Tracking Control of a Multi-rotor UAV for Partially
Known Trajectories

Corresponding author Yogesh Kumar,

# Geometric Tracking Control of a Multi-rotor UAV for Partially Known
Trajectories

Yogesh Kumar  S.B. Roy  P.B. Sujit  Department of Electronics and
Communication Engineering, Indraprastha Institute of Information Technology
Delhi, Delhi, India  Department of Electrical Engineering and Computer
Science, Indian Institute of Science Education and Research Bhopal, Madhya
Pradesh, India<EMAIL_ADDRESS>Kumar Y  Roy SB  Sujit PB

(Date Month Year; Date Month Year; Date Month Year)

###### Abstract

[Abstract]This paper presents a trajectory-tracking controller for multi-rotor
unmanned aerial vehicles (UAVs) in scenarios where only the desired position
and heading are known without the higher-order derivatives. The proposed
solution modifies the state-of-the-art geometric controller, effectively
addressing challenges related to the non-existence of the desired attitude and
ensuring positive total thrust input for all time. We tackle the additional
challenge of the non-availability of the higher derivatives of the trajectory
by introducing novel nonlinear filter structures. We formalize theoretically
the effect of these filter structures on the system error dynamics.
Subsequently, through a rigorous theoretical analysis, we demonstrate that the
proposed controller leads to uniformly ultimately bounded system error
dynamics.

, , and . Geometric Tracking Control of a Multi-rotor UAV for Partially Known
Trajectories. Int J Robust Nonlinear Control. 2023;xx(xx):x–xx.

###### keywords:

Nonlinear control, geometric control, multi-rotor UAV, quadrotors, nonlinear
filters on $\mathrm{SO(3)}$.

††articletype: Article Type††journal: Journal††volume: 00

##  1 Introduction

Multi-rotor unmanned aerial vehicles (UAVs) are becoming a commonplace
operational necessity due to their dynamic capabilities, such as vertical
take-off, landing, and hovering. Over the years, many developments in multi-
rotor controller synthesis have occurred, especially for quadrotors
considering various factors, including implementation complexity, required
performance, and available computational resources 1. The commonly used multi-
rotor UAVs are underactuated with highly non-linear and coupled dynamics,
which demands special consideration when designing globally stable trajectory
tracking controller 2.

While hardware and software advances are shrinking the gaps between theory and
practice, the ever-increasing application domain introduces new challenges.
One such challenge is to solve a tracking control problem where the desired
trajectory is not fully known, especially for cases that require non-
cooperative target tracking or coordination with other vehicles 3, 4. Due to
hardware or security considerations, obtaining complete information about the
motion of the other vehicle other than the pose measurement is often
challenging 5. The non-cooperative tracking problem is usually tackled using
the augmentation of vision sensors, such as - position-based visual servoing
(PBVS) 6, 3, 7 or image-based visual servoing (IBVS) 8, 9, 10, 11.

Compensating for unknown target dynamics is a challenging issue that is often
tackled by either introducing a robust conservative component in the
controller or deriving feedback compensation for the target velocity based on
ad-hoc techniques such as optical flow 9, 11, 12 and Kalman filter 3, 13, 8,
etc. while considering restrictive assumptions on the target motion. In our
earlier work 14, we proposed an IBVS method for quadrotors to overcome these
challenges at the kinematic level. However, the internal attitude and
translational controllers further require higher derivatives of the target
dynamics. The knowledge of higher derivatives of target dynamics is thus
necessary, and solutions such as high controller gain 10 or command filter-
based 15 techniques 16 are proposed in the literature. Recently, the authors
in 17 introduced a high gain controller based on the Euler angle formulation
to address the requirement of higher derivatives of the target dynamics.
However, Euler angle-based methods show singularities 18, which restricts the
UAV performance for nontrivial trajectories.

This paper proposes a solution to the general problem of designing a
trajectory-tracking controller for the multi-rotor UAV when only the desired
position and heading are available. We will design our controller by suitably
modifying the state-of-the-art geometric controller proposed in 18, 19.
Developed on the special Euclidean group $SE(3)$, the geometric controller
enables us to design an almost global trajectory tracking controller for
multi-rotor UAVs. However, two major limitations of this controller are (i)
the thrust is not saturated, hence no guarantee of the existence of the
desired angular orientation, and (ii) the designed thrust can be negative when
the angle between the third body axis of a multi-rotor UAV and desired thrust
direction is greater than $90^{\circ}$ 20. Our proposed solution addresses
these limitations in the current context of designing the controller for
partially known trajectories.

The main contributions of the paper are as follows

  * •

We consider the problem of multi-rotor tracking a partially known trajectory
and develop a trajectory-tracking controller by suitably modifying the state-
of-the-art geometric controller 18.

  * •

Unlike 21, an auxiliary filter dynamics in $SO(3)$ space is proposed for the
UAV attitude system, which tackle the requirement of higher-order derivatives
of the desired position in desired attitude derivatives. A similar approach
using command filter is presented in 22; however, considering a quaternion-
based formulation. To the best of our knowledge, this is the first time the
consequences of such filter dynamics for attitude representations in $SO(3)$
are quantified on the overall system through a Lyapunov-based analysis.

  * •

To ensure the non-zero desired thrust requirements for the existence of the
desired attitude 10, 23, we propose a projection-based filter dynamics to
yield a bounded second-order derivative of the desired position. Further, we
discuss the effect of large attitude errors on the thrust control inputs
proposed in 18, 20 in detail and introduce a novel control strategy.

  * •

The effect of the proposed control scheme, along with the auxiliary filter
dynamics, is theoretically analyzed. The overall system consisting of
translational, angular, and auxiliary dynamics is shown to be uniformly
ultimately bounded through rigorous mathematical arguments. The careful design
and extensive theoretical analysis enable us with an ultimate bound that can
be made arbitrarily small by choosing high enough auxiliary dynamics gains
without burdening the controller gains.

The rest of the paper is organized as follows. Section 2 describes the problem
formulation with the control objective. The controller design is presented in
Section 3 followed by the theoretical analysis in Section 4. Finally, section
5 draws the conclusions.

##  2 Problem Formulation and Preliminaries

###  2.1 Notations

The set of real numbers is represented as $\mathbb{R}$, while the set of
positive real numbers as $\mathbb{R}_{+}$. The identity matrix of size $3$ is
represented as $I$. Operators $\lambda_{max}(.)$ and $\lambda_{min}(.)$
represents the maximum and minimum eigenvalue of input matrix, respectively.
The special orthogonal group $\mathrm{SO(3)}$ which contains the orthogonal
matrices is defined as

$\displaystyle\mathrm{SO(3)}=\\{R\in\mathbb{R}^{3\times
3}|R^{T}R=RR^{T}=I,det(R)=1\\}.$ (1)

The group of skew-symmetric matrices $\mathfrak{so}(3)$ is defined as

$\displaystyle\mathfrak{so}(3)=\\{S\in\mathbb{R}^{3\times 3}|S=-S^{T}\\}.$ (2)

Map $(\hat{.}):\mathbb{R}^{3}\rightarrow\mathfrak{so}(3)$ is the skew-
symmetric map whereas it’s inverse is denoted by
$(.)^{\vee}:\mathfrak{so}(3)\rightarrow\mathbb{R}^{3}$ and
$e_{3}=[0,0,1]^{T}$. For a vector $v$, $\|v\|$ denotes the 2-norm whereas for
a matrix $M$, $\|M\|$ denotes the 2-norm until unless specified. $\ln()$
represents the natural logarithmic operator. For a vector
$v=[v_{1},v_{2},v_{3}]^{T}$,

$\displaystyle\tanh(v)=$
$\displaystyle[\tanh(v_{1}),\tanh(v_{2}),\tanh(v_{3})]^{T}\in\mathbb{R}^{3},$
(3) $\displaystyle Cosh(v)=$ $\displaystyle
daig\\{\cosh(v_{1}),\cosh(v_{2}),\cosh(v_{3})\\}\in\mathbb{R}^{3\times 3},$
(4) $\displaystyle Tanh(v)=$ $\displaystyle
daig\\{\tanh(v_{1}),\tanh(v_{2}),\tanh(v_{3})\\}\in\mathbb{R}^{3\times 3}.$
(5) $\displaystyle Sech(v)=$ $\displaystyle Cosh^{-1}(v).$ (6)

###  2.2 Multi-rotor Dynamics

Consider a multi-rotor UAV illustrated in Figure 1. Let {I} and {B} be the
inertial and the vehicle body frames, with origins $O_{I}$ and $O_{B}$,
respectively. The configuration of UAV is defined by the position
$x\in\mathbb{R}^{3}$ of the center of mass and attitude
$R=[{x}_{B},{y}_{B},{z}_{B}]\in\mathrm{SO(3)}$ in the inertial frame.

We model the multi-rotor dynamics as

$\displaystyle\dot{x}=$ $\displaystyle v,$ (7) $\displaystyle m\dot{v}=$
$\displaystyle mge_{3}-fRe_{3},$ (8) $\displaystyle\dot{R}=$ $\displaystyle
R\hat{\Omega},$ (9) $\displaystyle J\dot{\Omega}=$ $\displaystyle-\Omega\times
J\Omega+M,$ (10)

where inertial frame position $x(t)\in\mathbb{R}^{3}$, inertial frame velocity
$v(t)\in\mathbb{R}^{3}$, the rotation matrix from {B} to {I}, $R(t)\in SO(3)$
and the body-fixed angular velocity $\Omega(t)\in\mathbb{R}^{3}$ represents
the multi-rotor states. The total thrust $f\in\mathbb{R}$ and moment
$M=[M_{1},M_{2},M_{3}]^{T}\in\mathbb{R}^{3}$ expressed in body frame, are
considered as the control inputs to the multi-rotor model. For a given pair of
total thrust and moment, the rotor speeds can be calculated using the
following relation

$\displaystyle\begin{bmatrix}f\\\ M\end{bmatrix}=\Gamma\omega_{T},\
\omega_{T}=[\omega_{1}^{2},...,\omega_{n}^{2}]^{T},$ (11)

where n is the number of rotors and $\Gamma\in\mathbb{R}^{4\times n}$ is a
constant mixing matrix constructed based on the rotor configuration parameters
such as placement of rotors, thrust, and moment coefficient of rotors etc 2.

Figure 1: Coordinate frames illustration. The $x$, $y$, $z$ axes of a
coordinate frame {A} are denoted by ${x}_{A}$, ${y}_{A}$ and ${z}_{A}$,
respectively.

###  2.3 Control Objective

The objective is to design a nonlinear trajectory tracking controller for a
multi-rotor UAV to track a position reference $x_{d}(t)$ and a heading
direction $x_{B_{d}}(t)$ or a heading angle $\psi_{d}(t)$, where
$x_{B_{d}}(t)=[cos\psi_{d}(t),sin\psi_{d}(t),0]^{T}$. Moreover, we consider
that the higher derivatives of the position reference and heading direction (
i.e. $\frac{d^{n}x_{d}(t)}{dt^{n}}\ \text{and}\
\frac{d^{n}x_{B_{d}}(t)}{dt^{n}},\ n\geq 1$) are not available, which are
essential for an accurate and efficient trajectory tracking controller.

We assume that the reference trajectory has bounded derivatives as

$\displaystyle\sup_{t\geq 0}||\frac{d^{n}{x}_{d}(t)}{dt^{n}}||={h}_{n},\quad
n=1,2,3,4,\quad\sup_{t\geq 0}||\frac{dx_{B_{d}}(t)}{dt}||={h}_{5},\ \ \
\sup_{t\geq 0}||\frac{d^{2}x_{B_{d}}(t)}{dt^{2}}||={h}_{6}.$ (12)

where $h_{1},h_{2},h_{3},h_{4},h_{5},h_{6}\in\mathbb{R}_{+}$ are finite
constants.

##  3 Control Design

We follow a multistage procedure illustrated in Figure 2 to design the
trajectory tracking controller. We first design the position tracking
controller to obtain the control input $f(t)$. Subsequently, a desired
rotation matrix $R_{c}(t)\in\mathrm{SO(3)}$ constructed exploiting unit vector
direction of $f$ and desired heading direction $x_{B_{d}}(t)$. We then pass
$R_{c}(t)$ through an auxiliary filter dynamics to obtain the filtered desired
attitude reference, which the proposed attitude tracking controller would
track.

Figure 2: Control architecture

###  3.1 Position Tracking Controller

To design a smooth, saturated position tracking controller, we consider an
error system 23 with position tracking error $e_{x}(t)$ and filter tracking
error $e_{\alpha}(t)$ as

$\displaystyle e_{x}=x-x_{d}=[e_{x_{1}},e_{x_{2}},e_{x_{3}}]^{T},$ (13)
$\displaystyle e_{\alpha}=\dot{e}_{x}+\alpha_{x}\tanh(e_{x})+\tanh(e_{f}),$
(14)

where $e_{f}(t)=[e_{f_{x}}(t),e_{f_{y}}(t),e_{f_{z}}(t)]^{T}$ is an auxiliary
filter variable with the dynamics

$\displaystyle\dot{e}_{f}=Cosh^{2}(e_{f})\left(-k_{\alpha}e_{\alpha}+\tanh(e_{x})-\alpha_{f}\tanh(e_{f})\right),$
(15)

and $\\{k_{\alpha},\alpha_{f},\alpha_{x}\in\mathbb{R}_{+}\\}$ are positive
constant control design parameters.

The thrust control input $f(t)$ is designed as

$\displaystyle f=\|f_{d}\|\sqrt{\frac{1+e_{3}^{T}R_{c}^{T}Re_{3}}{2}}$ (16)

where $f_{d}(t)$ is a virtual control input defined as

$\displaystyle f_{d}=mge_{3}-m{g}_{1}+2m\tanh(e_{x})-mk_{\alpha}\tanh(e_{f}),$
(17)

and $g_{1}(t)$ is an estimate of $\ddot{x}_{d}(t)$ which is not directly
available and $R_{c}$ is the desired attitude matrix carefully constructed in
the subsequent section. Note that the thrust expression requires term
$\dot{e}_{x}$, i.e., velocity feedback, in the $e_{f}$. Since we don’t have
the explicit knowledge of $\dot{x}_{d}$, one can circumvent the requirement of
unknown term $\dot{e}_{x}$ in thrust input calculation following the output-
feedback formulation in 24. This also makes the position dynamics loop output
feedback.

###  3.2 Estimate of $\ddot{x}_{d}(t)$

To obtain the smooth and bounded estimate $g_{1}(t)$ of $\ddot{x}_{d}(t)$, we
start by considering the following filter structure with filter input
$x_{d}(t)$ and output $x_{f_{d}}(t)$

$\displaystyle\gamma_{1}\dot{x}_{f_{d}}+x_{f_{d}}=x_{d},\ \ x_{f_{d}}(0)=0.$
(18)

Consider another filter with filter design parameter
$\gamma_{1}\in\mathbb{R}_{+}$ to filter the unknown variable $\dot{x}_{d}(t)$
as

$\displaystyle\gamma_{1}\dot{g}_{x_{d}}+{g}_{x_{d}}=\dot{x}_{d},\ \
{g}_{x_{d}}(0)=0.$ (19)

Integrating (19) for ${g}_{x_{d}}(t)$ yield

$\displaystyle{g}_{x_{d}}(t)=\frac{1}{\gamma_{1}}\left(x_{d}(t)-x_{f_{d}}(t)-exp(-\frac{1}{\gamma_{1}}t)x_{d}(0)\right).$
(20)

Since ${g}_{x_{d}}(t)$ represents the filtered output of unknown variable
$\dot{x}_{d}(t)$, we further pass it through a filter with design parameter
$\gamma_{2}\in\mathbb{R}_{+}$ as

$\displaystyle\gamma_{2}\dot{g}_{f_{d}}+g_{f_{d}}=g_{x_{d}},\ \
g_{f_{d}}(0)=0,$ (21)

to obtain the estimate of $\ddot{x}_{d}(t)$. Further following the filter
structure similar to (19) for the unknown variable $\dot{g}_{x_{d}}$ and
integrating it for its filter variable ${g}_{v_{d}}(t)$, we get

$\displaystyle{g}_{v_{d}}(t)=\frac{1}{\gamma_{2}}\left(g_{x_{d}}(t)-g_{f_{d}}(t)-exp(-\frac{1}{\gamma_{2}}t)g_{x_{d}}(0)\right).$
(22)

For clarity, the filter structures discussed above are illustrated in Figure
3. Filter block 1 includes structures given by equations (18) and (20), with
input $x_{d}(t)$ and output signal ${g}_{x_{d}}(t)$. Similarly, filter block 2
takes ${g}_{x_{d}}(t)$ as input and provides ${g}_{v_{d}}(t)$ using equations
(21) and (22).

Figure 3: Filters illustration

Following the rigorous analysis presented in the subsequent sections, we can
use ${g}_{v_{d}}(t)$ as an estimate of $\ddot{x}_{d}(t)$, however, this
estimate can suffer from peaking phenomenon due to the low value of
$\gamma_{2}$ which might jeopardize the saturated thrust requirements to
design a smooth attitude tracking controller. To get a saturated thrust input,
a differential projection filter for a bounded estimate $g_{1}(t)$ of
$\ddot{x}_{d}(t)$ with design parameter $\gamma_{21}\in\mathbb{R}_{+}$ and
$g_{1}(0)=0$ is proposed as

$\begin{aligned}
&\dot{g}_{1}=Proj(g_{1},\phi)=\begin{cases}\phi-\frac{\nabla{f(g_{1})}\left(\nabla{f(g_{1})}\right)^{T}}{||\nabla{f(g_{1})}||^{2}}\phi
f(g_{1}),&\text{if }f(g_{1})>0\ \&\ \phi^{T}\nabla{f(g_{1})}>0\\\
\phi,&\text{otherwise}\end{cases},\end{aligned}$

(23)

where

$\phi=\frac{1}{\gamma_{21}}({g}_{v_{d}}-g_{1}),f(g_{1})=\frac{||g_{1}||^{2}-h_{2}^{2}}{\epsilon_{0}h_{2}^{2}},\nabla{f(g_{1})}=\frac{2g_{1}}{\epsilon_{0}h_{2}^{2}},$

where $h_{2}$ is the known upper bound of the $\ddot{x}_{d}$ and
$\epsilon_{0}>0$ is the tolerance.

###  3.3 Auxiliary Desired Attitude Construction

The desired attitude $R_{c}(t)$ is constructed exploiting the virtual control
input $f_{d}(t)$ given in (17) and the desired heading direction
$x_{B_{d}}(t)$ as

$\displaystyle R_{c}=[x_{B_{c}},y_{B_{c}},z_{B_{c}}],$ (24) $\displaystyle
z_{B_{c}}=\frac{f_{d}}{\|f_{d}\|}\quad y_{B_{c}}=-\frac{x_{B_{d}}\times
z_{B_{c}}}{\|x_{B_{d}}\times z_{B_{c}}\|}\quad x_{B_{c}}=y_{B_{c}}\times
z_{B_{c}}.$ (25)

Once we construct the desired attitude matrix a well defined†††We assume that
$x_{B_{d}}(t)\nparallel z_{B_{c}}(t)$. We note that one can circumvent this
singularity in differential flatness transformation using the Hopf Fibration
on SO(3) proposed in [25]. However this is out of scope of the current problem
formulation. $R_{c}(t)$ as given in (24), the desired angular velocity
$\Omega_{c}(t)$ and acceleration $\dot{\Omega}_{c}(t)$ corresponding to this
can be calculated as

$\displaystyle\hat{\Omega}_{c}=$ $\displaystyle R_{c}^{T}\dot{R}_{c},$ (26)
$\displaystyle\dot{\hat{\Omega}}_{c}=$ $\displaystyle
R_{c}^{T}\ddot{R}_{c}-\hat{\Omega}_{c}^{2},$ (27)

Note that the above calculations are essential to design an accurate attitude
tracking control law for the dynamics given in (9) and (10) 18, 19. However,
one requires information about the higher derivatives of the virtual input
$f_{d}(t)$ and heading direction $x_{B_{d}}(t)$ to calculate
$\dot{R}_{c},\ddot{R}_{c}$. Providing the information mentioned earlier is not
feasible as it involves higher derivatives of the desired trajectory and even
the multi-rotor UAV’s states such as rate of acceleration (jerk). This issue
is tackled by systematically defining an appropriate auxiliary filter dynamics
for the available signal $R_{c}(t)\in SO(3)$ to obtain a filtered signal
$R_{d}(t)\in SO(3)$ strictly belonging to the special orthogonal group and an
angular velocity $\Omega_{d}(t)$ associated to it as

$\displaystyle\dot{R}_{d}=$ $\displaystyle R_{d}\hat{\Omega}_{d},$ (28)
$\displaystyle\dot{\Omega}_{d}=$
$\displaystyle-\frac{1}{\gamma_{4}}e_{\Omega_{dc}}+e_{R}(R_{d},R_{c})+\frac{1}{\gamma_{3}}E(R_{d},R_{c})\Omega_{d},$
(29)

where $e_{R}(R_{d},R_{c})$ is the attitude error between $R_{c}$ and $R_{d}$
as described in Appendix A. $\gamma_{3},\gamma_{4}\in\mathbb{R}_{+}$ are the
constant design parameters. The term $E(R_{d},R_{c})$ is also described in
Appendix Geometric Tracking Control of a Multi-rotor UAV for Partially Known
Trajectories. The error
$e_{\Omega_{dc}}=\Omega_{d}+\frac{1}{\gamma_{3}}e_{R_{dc}}$ is carefully
designed based on the stability analysis in the subsequent section.

###  3.4 Effect of large attitude errors on $f$

Authors in 18 discussed the necessity of designing the thrust control input
$f$ to reduce its magnitude whenever there are large attitude errors. They do
so by multiplying the magnitude of the virtual control input $f_{d}$ with the
dot product of $z_{B_{c}}=R_{c}e_{3}$ and $z_{B}=Re_{3}$, i.e.,
$f=\|f_{d}\|e_{3}^{T}R_{c}^{T}Re_{3}$. The term $e_{3}^{T}R_{c}^{T}Re_{3}$
represents the cosine of the angle between $z_{B_{c}}$ and $z_{B}$, which can
make the control input $f$ negative whenever the angle is greater than
$90^{\circ}$, which is not feasible for the quadrotor to attain. To address
this, authors in 20 suggested using simply the magnitude of the virtual
control input, i.e., $f=\|f_{d}\|$ as the thrust control input. However, this
may cause instability in the overall system dynamics by generating
unnecessarily high thrust for large attitude errors. The careful analysis and
design of the thrust control input in equation (16) attenuates both of these
problems by multiplying the scaling quantity
$\sqrt{\frac{1+e_{3}^{T}R_{c}^{T}Re_{3}}{2}}$ to $\|f_{d}\|$ which is nothing
but cosine of half of the angle between $z_{B_{c}}$ and $z_{B}$. An
illustrative comparison of different thrust inputs with the respective virtual
control inputs in the case of 18, 20 and the proposed method w.r.t to the
angle between $z_{B_{c}}$ and $z_{B}$ is provided in Figure 4.

Figure 4: Thrust control input comparison. $\theta^{\prime}$ is the angle
between $z_{B_{c}}$ and $z_{B}$.

######  Remark 3.1.

Note that the proposed virtual control input (17) is always positive for a
tunable $k_{\alpha}$, which is a necessary condition for the existence of the
desired attitude $R_{c}(t)$. Moreover, the requirement of proportionally
reduced thrust magnitude for large attitude error is inherently captured in
the designed thrust input (16) in the whole configuration space (especially
angles greater than $90^{\circ}$).

###  3.5 Attitude Tracking Controller

We now design the attitude control input $M$ 19 to track the auxiliary
reference signals $R_{d}(t),\Omega_{d}(t),\dot{\Omega}_{d}(t)$ as follows

$\displaystyle M=$ $\displaystyle-
k_{R}e_{R}(R,R_{d})-k_{\Omega}e_{\Omega}(R,\Omega,R_{d},\Omega_{d})+\Omega\times
J\Omega-J(\hat{\Omega}R^{T}R_{d}\Omega_{d}-R^{T}R_{d}\dot{\Omega}_{d}),$ (30)

where $\\{k_{R},k_{\Omega}\in\mathbb{R}_{+}\\}$ are constant control design
parameters. The attitude and angular velocity errors $e_{R}(R,R_{d})$ and
$e_{\Omega}(R,\Omega,R_{d},\Omega_{d})$ are defined in detail in Appendix
Geometric Tracking Control of a Multi-rotor UAV for Partially Known
Trajectories.

##  4 Theortical Analysis

To comment on the stability and error convergence of the complete system, we
first derive the comprehensive closed-loop error dynamics for individual
subsystems.

###  4.1 Position Tracking Error System

Using the relations given in (13) and (14) the time derivative of $e_{x}$ can
be given as

$\displaystyle\dot{e}_{x}=e_{\alpha}-\alpha_{x}\tanh(e_{x})-\tanh(e_{f}).$
(31)

The open-loop dynamics of the filter tracking error multiplied by mass
$me_{\alpha}(t)$ substituting the system dynamics (8) and auxiliary filter
dynamics (15) can be written as

$\displaystyle m\dot{e}_{\alpha}=$ $\displaystyle
mge_{3}-fRe_{3}-m\ddot{x}_{d}+\chi-mk_{\alpha}e_{\alpha}+m\tanh(e_{x})$ (32)

where $\chi=m\alpha_{{x}}Sech^{2}(e_{x})\dot{e}_{x}-m\alpha_{f}\tanh(e_{f})$.
To better understand the attitude dynamics coupling, the virtual control input
$f_{d}(t)$ is added and subtracted to the right-hand side of (32) as

$\displaystyle m\dot{e}_{\alpha}=$ $\displaystyle
mge_{3}+\Delta{f}+\chi-m\ddot{x}_{d}-mk_{\alpha}e_{\alpha}+m\tanh(e_{x})-f_{d},$
(33)

where $\Delta{f}=f_{d}-fRe_{3}$.

Using the expression for virtual control input from (17), the closed-loop
filter error dynamics can be written as

$\displaystyle m\dot{e}_{\alpha}=$
$\displaystyle\Delta{f}+\chi-m(\ddot{x}_{d}-g_{1})-mk_{\alpha}e_{\alpha}-m\tanh(e_{x})+mk_{\alpha}\tanh(e_{f}).$
(34)

To analyze the dynamical behavior of the proposed filter structures for
desired position reference derivatives $\frac{d^{n}x_{d}(t)}{dt^{n}}$, we
propose the following error system $e_{d}(t)$ as

$\displaystyle e_{d}=\begin{bmatrix}e_{g_{x}}\\\ e_{\ddot{x}_{d}}\\\
e_{g_{v}}\\\ e_{g_{1}}\\\
\end{bmatrix}=\begin{bmatrix}\dot{x}_{d}-{g}_{x_{d}}\\\
\ddot{x}_{d}-\dot{g}_{x_{d}}\\\ \dot{g}_{x_{d}}-{g}_{v_{d}}\\\
\ddot{x}_{d}-g_{1}\\\ \end{bmatrix}.$ (35)

Using the expressions from (18), (19), (21) and (23), the closed loop error
dynamics for error system (35) can be given as

$\displaystyle\begin{bmatrix}\dot{e}_{g_{x}}\\\ \dot{e}_{\ddot{x}_{d}}\\\
\dot{e}_{g_{v}}\\\
\dot{e}_{g_{1}}\end{bmatrix}=\begin{bmatrix}-\frac{1}{\gamma_{1}}e_{g_{x}}+\ddot{x}_{d}\\\
-\frac{1}{\gamma_{1}}e_{\ddot{x}_{d}}+\dddot{x}_{d}\\\
-\frac{1}{\gamma_{2}}e_{g_{v}}+\frac{1}{\gamma_{1}}e_{\ddot{x}_{d}}\\\
-\frac{1}{\gamma_{21}}e_{g_{1}}+\frac{1}{\gamma_{21}}e_{g_{v}}+\frac{1}{\gamma_{21}}{e}_{\ddot{x}_{d}}+\Phi_{g_{1}}+\dddot{x}_{d},\end{bmatrix}$
(36)

where

$\displaystyle\Phi_{g_{1}}=\frac{\nabla{f(g_{1})}(\nabla{f(g_{1})})^{T}}{||\nabla{f(g_{1})}||^{2}}\phi
f(g_{1})$ (37)

which appears only if $f(g_{1})>0\ \&\ \phi^{T}\nabla{f(g_{1})}>0$. Moreover,
${e}_{g_{1}}^{T}\Phi_{g_{1}}\leq 0$ (Please refer to the properties of
projection in 25).

###  4.2 Attitude Tracking Error System

The time derivatives of the attitude error system with its properties are
discussed in detail in Appendix Geometric Tracking Control of a Multi-rotor
UAV for Partially Known Trajectories, which will be helpful in the theoretical
claims provided in this section. For simplicity in the representation, we will
use the following notations for the various attitude and angular velocity
errors throughout the paper.

$\displaystyle{e}_{R}(R,R_{d})=e_{R_{d}},\quad{e}_{R}(R,R_{c})=e_{R_{c}},\quad{e}_{R}(R_{d},R_{c})=e_{R_{dc}},\quad
e_{\Omega}(R,\Omega,R_{d},\Omega_{d})=e_{\Omega_{d}}.$ (38)

Using the control input $M$, the closed loop error dynamics of angular error
$e_{\Omega_{d}}$ premultiplied with inertia matrix $J$ can be given as

$\displaystyle
J\dot{e}_{\Omega_{d}}=-k_{R}e_{R_{d}}-k_{\Omega}e_{\Omega_{d}}.$ (39)

###  4.3 Stability Analysis

The stability and convergence analysis is systematically presented in this
section. We first state the following lemmas, which will be utilized in the
subsequent analysis.

######  Lemma 4.1.

There exist positive constants $\alpha_{1},\alpha_{2}$ such that

$\displaystyle\|f_{d}\|\leq\alpha_{1},\quad\|f_{d}\|\geq\alpha_{2}.$ (40)

Further, $\alpha_{2}$ is strictly positive if

$\displaystyle k_{\alpha}<g-h_{2}-2.$ (41)

######  Proof 4.2.

Please refer to Appendix Geometric Tracking Control of a Multi-rotor UAV for
Partially Known Trajectories for the proof.

######  Lemma 4.3.

The quantity $\Delta{f}=f_{d}-fRe_{3}$ satisfies the following bound

$\displaystyle\|\Delta{f}\|\leq 2\alpha_{1}(\|e_{R_{dc}}\|+\|e_{R_{d}}\|).$
(42)

######  Proof 4.4.

Please refer to Appendix Geometric Tracking Control of a Multi-rotor UAV for
Partially Known Trajectories for the proof.

######  Lemma 4.5.

The desired angular velocity $\Omega_{c}$ satisfies the following bound

$\displaystyle\|\hat{\Omega}_{c}\|\leq$
$\displaystyle\rho_{1}(\|e_{\alpha}\|,\|\tanh(e_{x})\|,\|\tanh(e_{f})\|)+\rho_{01}.$
(43)

######  Proof 4.6.

Please refer to Appendix Geometric Tracking Control of a Multi-rotor UAV for
Partially Known Trajectories for the proof.

######  Lemma 4.7.

The filter errors given in equation (36) can be bounded as

$\displaystyle\|e_{g_{x}}(t)\|$
$\displaystyle\leq\sqrt{\|e_{g_{x}}(0)\|^{2}exp\left(-\frac{1}{\gamma_{1}}t\right)+\gamma_{1}^{2}h_{2}^{2}}\leq
max(\|e_{g_{x}}(0)\|,\gamma_{1}h_{2})=\alpha_{3},$ (44)
$\displaystyle\|e_{\ddot{x}_{d}}(t)\|$
$\displaystyle\leq\sqrt{\|e_{\ddot{x}_{d}}(0)\|^{2}exp\left(-\frac{1}{\gamma_{1}}t\right)+\gamma_{1}^{2}h_{3}^{2}}\leq
max(\|e_{\ddot{x}_{d}}(0)\|,\gamma_{1}h_{3})=\alpha_{4},$ (45)
$\displaystyle\|e_{g_{v}}(t)\|$
$\displaystyle\leq\sqrt{\|e_{g_{v}}(0)\|^{2}exp\left(-\frac{1}{\gamma_{2}}t\right)+\frac{\gamma_{2}^{2}}{\gamma_{1}^{2}}\|e_{\ddot{x}_{d}}\|^{2}}\leq
max\bigg{(}\|e_{g_{v}}(0)\|,\frac{\gamma_{2}}{\gamma_{1}}max(e_{\ddot{x}_{d}}(0),\gamma_{1}h_{3})\bigg{)}=\alpha_{5},$
(46) $\displaystyle\|e_{g_{1}}(t)\|$ $\displaystyle\leq
max\left(\|e_{g_{1}}(0)\|,\sqrt{{8}(\alpha_{4}^{2}+\alpha_{5}^{2})+2\gamma_{21}^{2}h_{3}^{2}}\right)=\alpha_{6}.$
(47)

######  Proof 4.8.

Please refer to Appendix Geometric Tracking Control of a Multi-rotor UAV for
Partially Known Trajectories for the proof.

Next, we derive the stability and convergence properties of the attitude
subsystem of the multirotor tracking the auxiliary desired attitude.

######  Theorem 4.9.

(Adapted from 19) The closed loop error dynamics of
$e_{1}(t)=[e_{R_{d}}(t)^{T},e_{\Omega_{d}}(t)^{T}]^{T}$ is exponentially
stable with the control input $M$ defined in (30) for the smooth desired
attitude reference $\left(R_{d}(t),\ \Omega_{d}(t),\
\dot{\Omega}_{d}(t)\right)$. Moreover, an estimate of the region of attraction
is given by

$\displaystyle\Psi(R(0),R_{d}(0))<2,$ (48)
$\displaystyle\|e_{\Omega_{d}}(0)\|^{2}<\frac{2k_{R}}{\lambda_{max}(J)}(2-\Psi(R(0),R_{d}(0)).$
(49)

######  Proof 4.10.

We start by proving that given condition (49), the sublevel set
$\mathcal{L}_{2}=\\{R,R_{d}\in\mathrm{SO(3)}|\Psi(R,R_{d})<2\\}$ is positively
invariant set. Consider the following Lypunov candidate

$\displaystyle
V_{1}=\frac{1}{2}e_{\Omega_{d}}^{T}e_{\Omega_{d}}+k_{R}\Psi(R,R_{d})$ (50)

Taking the time derivative of $V_{1}(t)$ after substituting the dynamics (39)
and (82) yields

$\displaystyle\dot{V}_{1}=-k_{\Omega}\|e_{\Omega_{d}}\|^{2}.$ (51)

Equation (51) along with (48) and (49) implies that

$\displaystyle k_{R}\Psi(R(t),R_{d}(t))\leq V_{1}(t)\leq
V_{1}(0)<2k_{R}\quad\forall t>0.$ (52)

This proves that the sub-level set $\mathcal{L}_{2}$ is a positively invariant
set such that $\Psi(R(t),R_{d}(t))<2$ and hence the attitude errors are well
defined.

To prove the exponential stability, let’s consider another Lyapunov candidate

$\displaystyle
V_{2}=\frac{1}{2}e_{\Omega_{d}}^{T}e_{\Omega_{d}}+k_{R}\Psi(R,R_{d})+c_{1}e_{\Omega_{d}}^{T}e_{R_{d}},$
(53)

where $c_{1}$ is a positive constant. Using the properties of configuration
error function, for $e_{1}(t)=[e_{R_{d}}(t)^{T},e_{\Omega_{d}}(t)^{T}]^{T}$ we
get

$\displaystyle e_{1}^{T}W_{11}e_{1}\leq V_{1}\leq e_{1}^{T}W_{12}e_{1},$ (54)

where

$\displaystyle W_{11}=\begin{bmatrix}k_{R}&\frac{c_{2}}{2}\\\
\frac{c_{2}}{2}&\frac{\lambda_{min}(J)}{2}\end{bmatrix},\quad
W_{12}=\begin{bmatrix}2k_{R}&\frac{c_{2}}{2}\\\
\frac{c_{2}}{2}&\frac{\lambda_{max}(J)}{2}\end{bmatrix}.$ (55)

Taking time derivative of $V_{2}(t)$ and substituting the appropriate error
dynamics using (39), (82) and (83) will yield

$\displaystyle\dot{V}_{2}=$ $\displaystyle-
k_{\Omega}e_{\Omega_{d}}^{T}e_{\Omega_{d}}-c_{1}k_{R}e_{R_{d}}^{T}J^{-1}e_{R_{d}}+c_{1}k_{\Omega}e_{R_{d}}^{T}J^{-1}e_{\Omega_{d}}+c_{1}e_{\Omega_{d}}^{T}E^{T}(R,R_{d})e_{\Omega_{d}}.$
(56)

Further taking bounds using the properties in Appendix Geometric Tracking
Control of a Multi-rotor UAV for Partially Known Trajectories in equation (56)
will yield

$\displaystyle\dot{V}_{2}\leq$
$\displaystyle-\underbrace{\left[k_{\Omega}-\frac{c_{1}}{2}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}\right]}_{\bar{\lambda}_{1}}\|e_{\Omega_{d}}\|^{2}-\underbrace{\left[\frac{c_{1}k_{R}}{\lambda_{max}(J)}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}\right]}_{\bar{\lambda}_{2}}\|e_{R_{d}}\|^{2}.$
(57)

Furthermore, we can write

$\displaystyle\dot{V}_{2}\leq$ $\displaystyle-\bar{\beta}V_{2},$ (58)

where
$\bar{\beta}=\frac{min(\bar{\lambda}_{1},\bar{\lambda}_{2})}{\lambda_{max}(W_{12})}$.
We choose the parameter so that $\bar{\beta}$ is positive. Then, the relation
(58) invokes global uniform exponential stability (based on Theorem 4.10 in
26).

Now, we are ready to state our main results.

######  Theorem 4.11.

Consider the multi-rotor dynamics (7), (8), (9) and (10), with control inputs
(16) and (30). The closed loop dynamics of overall system errors
$e(t)=[e_{1}(t)^{T},e_{2}(t)^{T},e_{3}(t)^{T}]^{T}$ where
$e_{1}(t)=[e_{R_{d}}(t)^{T},e_{\Omega_{d}}(t)^{T}]^{T}$,
$e_{2}(t)=[e_{R_{dc}}^{T}(t),e_{\Omega_{dc}}^{T}(t)]^{T}$ and
$e_{3}(t)=[e_{\alpha}^{T}(t),\tanh(e_{x}(t))^{T},\tanh(e_{f}(t))^{T}]^{T}$, is
uniformly ultimately bounded, provided the following gain conditions are
satisfied

$\displaystyle
mk_{\alpha}-\frac{m\alpha_{{x}}(3\alpha_{{x}}+1)}{2}-\frac{m\alpha_{f}}{2}-\rho_{03}>0$
, (59) $\displaystyle m\alpha_{x}-\frac{m\alpha_{{x}}^{2}}{2}>0$ , (60)
$\displaystyle\frac{1}{2\gamma_{3}}-{2{\alpha_{1}}^{2}}>0$ , (61)
$\displaystyle\frac{m\alpha_{f}}{2}-\frac{m\alpha_{{x}}}{2}>0$ , (62)
$\displaystyle
k_{\Omega}-\frac{c_{1}}{2}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}>0$ , (63)
$\displaystyle\frac{c_{1}k_{R}}{\lambda_{max}(J)}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}-{2{\alpha_{1}}^{2}}>0$
, (64)

where the constant $\rho_{03}$ is defined subsequently.

######  Proof 4.12.

For the sake of simplicity, we proceed in the following steps to prove the
theorem claims. Step 1 Let’s first consider the Lyapunov candidate $V_{3}(t)$
for the translational error $e_{3}(t)$ of multi-rotor

$\displaystyle V_{3}=$
$\displaystyle\frac{1}{2}me_{\alpha}^{T}e_{\alpha}+m\ln\left(\cosh(e_{x_{1}})\right)+m\ln\left(\cosh(e_{x_{2}})\right)+m\ln\left(\cosh(e_{x_{3}})\right)+\frac{1}{2}m\tanh(e_{f})^{T}\tanh(e_{f}).$
(65)

Taking the time derivative of $V_{3}(t)$ and substituting equations (15), (31)
and (34) yields

$\displaystyle\dot{V}_{3}=$ $\displaystyle-
mk_{\alpha}e_{\alpha}^{T}e_{\alpha}+e_{\alpha}^{T}\Delta{f}+e_{\alpha}^{T}\chi-
me_{\alpha}^{T}e_{g_{1}}-me_{\alpha}^{T}\tanh(e_{x})+mk_{\alpha}e_{\alpha}^{T}\tanh(e_{f})$
$\displaystyle+m\tanh^{T}(e_{x})\left(e_{\alpha}-\alpha_{x}\tanh(e_{x})-\tanh(e_{f})\right)+m\tanh(e_{f})^{T}\left(-k_{\alpha}e_{\alpha}+\tanh(e_{x})-\alpha_{f}\tanh(e_{f})\right)$
(66)

Further solving (4.12) and taking bounds on $\dot{V}_{2}(t)$ yield

$\displaystyle\dot{V}_{3}\leq$ $\displaystyle-
mk_{\alpha}\|e_{\alpha}\|^{2}-m\alpha_{x}\|\tanh(e_{x})\|^{2}-m\alpha_{f}\|\tanh(e_{f})\|^{2}+\|e_{\alpha}\|\|\Delta{f}\|+\|e_{\alpha}\|\|\chi\|+m\|e_{\alpha}\|\|e_{g_{1}}\|.$
(67)

Solving equation (67) by substituting the dynamics of $\dot{e}_{x}(t)$ from
(31) in $\chi$, and using equations (42) and (47) produces

$\displaystyle\dot{V}_{3}\leq$ $\displaystyle-
mk_{\alpha}\|e_{\alpha}\|^{2}-m\alpha_{f}\|\tanh(e_{f})\|^{2}+\|e_{\alpha}\|\left(m\alpha_{{x}}\|Sech^{2}(e_{x})\|\|\tanh(e_{f})\|+m\alpha_{f}\|\tanh(e_{f}))\|\right)-m\alpha_{x}\|\tanh(e_{x})\|^{2}$
$\displaystyle+\|e_{\alpha}\|\|Sech^{2}(e_{x})\|\big{(}m\alpha_{{x}}\|e_{\alpha}\|+m\alpha_{{x}}^{2}\|\tanh(e_{x})\|\big{)}+m\|e_{\alpha}\|\alpha_{6}+2\alpha_{1}\|e_{\alpha}\|\big{(}\|e_{R_{dc}}\|+\|e_{R_{d}}\|\big{)},$
(68)

which modifies to,

$\displaystyle\dot{V}_{3}\leq$ $\displaystyle-
mk_{\alpha}\|e_{\alpha}\|^{2}-m\alpha_{x}\|\tanh(e_{x})\|^{2}-m\alpha_{f}\|\tanh(e_{f})\|^{2}+m\alpha_{{x}}\|e_{\alpha}\|^{2}+m\alpha_{{x}}^{2}\|e_{\alpha}\|\|\tanh(e_{x})\|$
$\displaystyle+m\alpha_{{x}}\|e_{\alpha}\|\|\tanh(e_{f})\|+m\alpha_{f}\|e_{\alpha}\|\|\tanh(e_{f})\|+m\|e_{\alpha}\|\alpha_{6}+2\alpha_{1}\|e_{\alpha}\|\big{(}\|e_{R_{dc}}\|+\|e_{R_{d}}\|\big{)},$
$\displaystyle\leq$ $\displaystyle-
mk_{\alpha}\|e_{\alpha}\|^{2}-m\alpha_{x}\|\tanh(e_{x})\|^{2}-m\alpha_{f}\|\tanh(e_{f})\|^{2}+m\alpha_{{x}}\|e_{\alpha}\|^{2}+\frac{m\alpha_{{x}}^{2}}{2}\|e_{\alpha}\|^{2}+\frac{m\alpha_{{x}}^{2}}{2}\|\tanh(e_{x})\|^{2}+\frac{m\alpha_{{x}}}{2}\|e_{\alpha}\|^{2}$
$\displaystyle+\frac{m\alpha_{{x}}}{2}\|\tanh(e_{f})\|^{2}+\frac{m\alpha_{f}}{2}\|e_{\alpha}\|^{2}+\frac{m\alpha_{f}}{2}\|\tanh(e_{f})\|^{2}+\frac{m}{2}\|e_{\alpha}\|^{2}+\frac{m}{2}\alpha_{6}^{2}+\|e_{\alpha}\|^{2}+{2{\alpha_{1}}^{2}}\|e_{R_{dc}}\|^{2}+{2{\alpha_{1}}^{2}}\|e_{R_{d}}\|^{2}.$
(69)

Further manipulation yield

$\displaystyle\dot{V}_{3}\leq$
$\displaystyle-\bigg{(}mk_{\alpha}-m\alpha_{{x}}-\frac{m\alpha_{{x}}^{2}}{2}-\frac{m\alpha_{{x}}}{2}-\frac{m\alpha_{f}}{2}-\frac{m}{2}-1\bigg{)}\|e_{\alpha}\|^{2}-\left(m\alpha_{x}-\frac{m\alpha_{{x}}^{2}}{2}\right)\|\tanh(e_{x})\|^{2}$
$\displaystyle-\left(m\alpha_{f}-\frac{m\alpha_{{x}}}{2}-\frac{m\alpha_{f}}{2}\right)\|\tanh(e_{f})\|^{2}+{2{\alpha_{1}}^{2}}\|e_{R_{dc}}\|^{2}+{2{\alpha_{1}}^{2}}\|e_{R_{d}}\|^{2}+\frac{m}{2}\alpha_{6}^{2}.$
(70)

Step 2 Now, let’s consider another Lyapunov candidate $V_{4}(t)$ for auxiliary
attitude errors $e_{2}(t)$ as

$\displaystyle
V_{4}(t)=\Psi(R_{d},R_{c})+\frac{1}{2}e_{\Omega_{dc}}^{T}e_{\Omega_{dc}},$
(71)

Taking the time derivative and using equations (28), (29), (82), (83) and (84)
yields

$\displaystyle\dot{V}_{4}(t)=$
$\displaystyle-\frac{1}{\gamma_{3}}e_{R_{dc}}^{T}e_{R_{dc}}--\frac{1}{\gamma_{4}}e_{{\Omega}_{dc}}^{T}e_{{\Omega}_{dc}}-e_{R_{dc}}^{T}R_{d}^{T}R_{c}\Omega_{c}+\frac{1}{\gamma_{3}}e_{{\Omega}_{dc}}^{T}E(R_{d},R_{c})R_{d}^{T}R_{c}\Omega_{c}$
(72)

Taking bounds on $\dot{V}_{4}(t)$ yields

$\displaystyle\dot{V}_{4}(t)\leq$
$\displaystyle-\frac{1}{2\gamma_{3}}\|e_{R_{dc}}\|^{2}-\frac{1}{2\gamma_{4}}\|e_{{\Omega}_{dc}}\|^{2}+\left(\frac{\gamma_{3}}{2}+\frac{\gamma_{4}}{8\gamma_{3}^{2}}\right)\|\Omega_{c}\|^{2}$
(73)

Step 3 To prove the stability of dynamics of error $e(t)$, let’s consider the
Lyapunov candidate as

$\displaystyle V=$ $\displaystyle V_{2}(t)+V_{3}(t)+V_{4}(t)$ (74)

Taking the derivative of $V(t)$ using (73), (4.12) and (57) yields

$\displaystyle\dot{V}\leq$
$\displaystyle-\left(mk_{\alpha}-m\alpha_{{x}}-\frac{m\alpha_{{x}}^{2}}{2}-\frac{m\alpha_{{x}}}{2}-\frac{m\alpha_{f}}{2}-\frac{m}{2}-1\right)\|e_{\alpha}\|^{2}-\left(m\alpha_{x}-\frac{m\alpha_{{x}}^{2}}{2}\right)\|\tanh(e_{x})\|^{2}$
$\displaystyle-\left(m\alpha_{f}-\frac{m\alpha_{{x}}}{2}-\frac{m\alpha_{f}}{2}\right)\|\tanh(e_{f})\|^{2}+\frac{m}{2}\alpha_{6}^{2}-\left(\frac{1}{2\gamma_{3}}-{2{\alpha_{1}}^{2}}\right)\|e_{R_{dc}}\|^{2}+\left(\frac{\gamma_{3}}{2}+\frac{\gamma_{4}}{8\gamma_{3}^{2}}\right)\|\Omega_{c}\|^{2}$
$\displaystyle-\left(k_{\Omega}-\frac{c_{1}}{2}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}\right)\|e_{\Omega_{d}}\|^{2}-\frac{1}{2\gamma_{4}}\|e_{{\Omega}_{dc}}\|^{2}-\left(\frac{c_{1}k_{R}}{\lambda_{max}(J)}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}-{2{\alpha_{1}}^{2}}\right)\|e_{R_{d}}\|^{2}$
(75)

From lemma (4.5) and using the properties of hyperbolic functions, we can
state that for some $\|e_{\alpha}\|<\bar{e}_{\alpha}$, the function
$\rho_{1}()$ is locally Lipschitz. Thus,

$\displaystyle\|\hat{\Omega}_{c}\|^{2}\leq
L_{1}^{2}\|e_{\alpha}\|^{2}+\rho_{02}^{2},$ (76)

where $L_{1}$, a function of $\bar{e}_{\alpha}$ is the Lipschitz constant and
$\rho_{02}$ is a positive constant. Substituting the (76) in (4.12) yields

$\displaystyle\dot{V}\leq$
$\displaystyle-\underbrace{\left[mk_{\alpha}-\frac{m\alpha_{{x}}(3\alpha_{{x}}+1)}{2}-\frac{m\alpha_{f}}{2}-\rho_{03}\right]}_{\lambda_{1}}\|e_{\alpha}\|^{2}-\underbrace{\left[m\alpha_{x}-\frac{m\alpha_{{x}}^{2}}{2}\right]}_{\lambda_{2}}\|\tanh(e_{x})\|^{2}-\underbrace{\left[\frac{1}{2\gamma_{3}}-{2{\alpha_{1}}^{2}}\right]}_{\lambda_{4}}\|e_{R_{dc}}\|^{2}-\underbrace{\left[\frac{1}{2\gamma_{4}}\right]}_{\lambda_{5}}\|e_{{\Omega}_{dc}}\|^{2}$
$\displaystyle-\underbrace{\left[\frac{m\alpha_{f}}{2}-\frac{m\alpha_{{x}}}{2}\right]}_{\lambda_{3}}\|\tanh(e_{f})\|^{2}-\underbrace{\left[k_{\Omega}-\frac{c_{1}}{2}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}\right]}_{\lambda_{6}}\|e_{\Omega_{d}}\|^{2}-\underbrace{\left[\frac{c_{1}k_{R}}{\lambda_{max}(J)}-\frac{c_{1}k_{\Omega}}{2\lambda_{min}(J)}-{2{\alpha_{1}}^{2}}\right]}_{\lambda_{7}}\|e_{R_{d}}\|^{2}$
$\displaystyle+\underbrace{\left[\left(\frac{\gamma_{3}}{2}+\frac{\gamma_{4}}{8\gamma_{3}^{2}}\right)\rho_{02}^{2}+\frac{m}{2}\alpha_{6}^{2}\right]}_{D},$
(77)

where
$\rho_{03}=\frac{m}{2}+1+\frac{4\gamma_{3}^{2}+\gamma_{4}}{8\gamma_{3}^{2}}L_{1}^{2}$.
Equation (4.12) implies

$\displaystyle\dot{V}\leq$
$\displaystyle-\underbrace{min(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda_{7})}_{\lambda_{V}}\|e\|^{2}+D$
(78)

The relation (78) invokes uniformly ultimately bounded (UUB) stability since
for $\|e(t)\|^{2}>\frac{D}{\lambda_{V}}$, $\dot{V}(t)$ is negative-definite.
However, note that (78) is derived based on (76), which demands
$\|e_{\alpha}\|\leq\bar{e}_{\alpha}$. Hence, $\bar{e}_{\alpha}$ should be
chosen such that
$\bar{e}_{\alpha}\geq\sqrt{2\frac{max(V(0),\frac{D}{\lambda_{V}})}{m}}$, and
accordingly the gain condition related to $\lambda_{1}$ has to be satisfied.
This would ensure $\|e_{\alpha}(t)\|\leq\bar{e}_{\alpha}$, $\forall t\geq 0$,
and hence, the UUB result would follow.

######  Remark 4.13.

It’s worth noting that $D$ can be made arbitrarily small by choosing the
filter gains $\gamma_{1},\gamma_{2},\gamma_{21},\gamma_{3},\gamma_{4}$
appropriately. This relaxes the unnecessary burden on the controller gains for
such formulations. Moreover, if $k_{\alpha}$ is chosen such that the condition
(41) is also satisfied then the desired attitude $R_{c}(t)$ always exists as
$\|f_{d}(t)\|\geq\alpha_{2}>0$ is guaranteed.

##  5 Conclusion

In this paper, we introduced a trajectory-tracking controller for multi-rotor
unmanned aerial vehicles (UAVs) with limited knowledge of the desired
trajectory, restricted to position and heading information. Using the
auxiliary filter dynamics, the unavailability of the higher derivative of the
desired trajectory is tackled to design both attitude and translational motion
controllers hierarchically. It is theoretically shown that the designed thrust
is saturated and always positive, avoiding singularities in the desired
attitude construction. The designed controller has proven to be ultimately
bounded in the extended error space considering attitude and position error
dynamics through rigorous theoretical analysis.

Future work would focus on experimental validation and real-world
implementations to further validate the controller’s robustness and
adaptability. Additionally, exploring potential enhancements to accommodate a
desired trajectory with unbounded derivatives (i.e. relaxing Assumption 2.3)
and external disturbances could further broaden the scope of applications for
multi-rotor UAVs.

*Acknowledgments This work is supported by TCS Research, India, under the TCS Research Scholar Program FP-140.

*Conflict of interest The authors declare no potential conflict of interests.

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Preliminaries on error formulation on $\mathrm{SO(3)}$ To design the attitude
tracking controller on a nonlinear space $\mathrm{SO(3)}$, we start with
constructing a positive definite configuration error function $\Psi$. We then
choose the configuration error vector and a velocity error vector in the
tangent space $T_{R}\mathrm{SO(3)}$ utilizing the derivatives of $\Psi$.
Considering the carefully crafted error system, we design the controller
following the Lyapunov direct method. Owing to the superior tracking
performance for large initial attitude errors, we consider the following
configuration error function 19
$\Psi:\mathrm{SO(3)}\times\mathrm{SO(3)}\rightarrow\mathbb{R}$ for a given
pair of desired attitude and angular velocity
$R_{1}\in\mathrm{SO(3)},\Omega_{1}\in\mathbb{R}^{3}$, and the current attitude
and angular velocity $R_{2}\in\mathrm{SO(3)},\Omega_{2}\in\mathbb{R}^{3}$

$\displaystyle\Psi(R_{2},R_{1})=2-\sqrt{1+tr[R_{1}^{T}R_{2}]}.$ (79)

Based on the configuration error function, we define an attitude error vector
$e_{R}(R_{1},R_{2}):\mathrm{SO(3)}\times\mathrm{SO(3)}\rightarrow\mathbb{R}^{3}$
and an angular velocity error vector
$e_{\Omega}:SO(3)\times\mathbb{R}^{3}\times\mathrm{SO(3)}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$
as

$\displaystyle
e_{R}(R_{2},R_{1})=\frac{1}{2\sqrt{1+tr[R_{1}^{T}R_{2}]}}(R_{1}^{T}R_{2}-R_{2}^{T}R_{1})^{\vee},$
(80) $\displaystyle
e_{\Omega}(R_{2},\Omega_{2},R_{1},\Omega_{1})=\Omega_{2}-R_{2}^{T}R_{1}\Omega_{1}.$
(81)

Throughout this article, we will use the following well-established properties
19 of these error variables in a sub-level set
$\mathcal{L}_{2}=\\{R_{1},R_{2}\in\mathrm{SO(3)}|\Psi(R_{2},R_{1})<2\\}$ of
$\Psi$

  * •

$\Psi$ is positive definite about $R_{2}=R_{1}$ and has only one critical
point $R_{2}=R_{1}$.

  * •

$\|e_{R}\|^{2}\leq\Psi\leq 2\|e_{R}\|^{2}$.

  * •

$\|e_{R}\|=sin\frac{\|\theta\|}{2}$, where $exp(\hat{\theta})=R_{1}^{T}R_{2}$
i.e. $\|e_{R}\|<1$.

The time derivative of the configuration function, attitude error and angular
velocity
$\Psi(R_{2},R_{1}),e_{R}(R_{2},R_{1}),e_{\Omega}(R_{2},\Omega_{2},R_{1},\Omega_{1})$
can be given as

$\displaystyle\frac{d}{dt}{\Psi}(R_{2},R_{1})=e_{R}(R_{2},R_{1})^{T}e_{\Omega}(R_{2},\Omega_{2},R_{1},\Omega_{1}),$
(82)
$\displaystyle\frac{d}{dt}{e}_{R}(R_{2},R_{1})=E(R_{2},R_{1})e_{\Omega}(R_{2},\Omega_{2},R_{1},\Omega_{1}),$
(83)
$\displaystyle\frac{d}{dt}J{e}_{\Omega}(R_{2},\Omega_{2},R_{1},\Omega_{1})=J\dot{\Omega}_{2}+J(\hat{\Omega}_{2}R_{2}^{T}R_{1}\Omega_{1}-R_{2}^{T}R_{1}\dot{\Omega}_{1}),$
(84)

where $E(R_{2},R_{1})\in\mathbb{R}^{3\times 3}$ is given as

$\displaystyle E(R_{2},R_{1})=$
$\displaystyle\frac{1}{2\sqrt{1+tr[R_{1}^{T}R_{2}]}}\left(tr[R_{2}^{T}R_{1}]I-R_{2}^{T}R_{1}+2e_{R}e_{R}^{T}\right).$
(85)

Using the fact that $\|E(R_{2},R_{1})\|=\frac{1}{2}$ from 19, we can state
that

$\displaystyle\|\frac{d}{dt}{e}_{R}(R_{2},R_{1})\|\leq\frac{1}{2}\|e_{\Omega}(R_{2},\Omega_{2},R_{1},\Omega_{1})\|.$
(86)

Proof of Lemmas

Lemma 4.1

######  Proof .1.

From equation (17) we can write

$\displaystyle\|f_{d}\|\leq
mg+m\|g_{1}\|+2m\|\tanh(e_{x})\|+mk_{\alpha}\|\tanh(e_{f})\|.$ (87)

From the projection based differential estimate given in equation (23), we
have $\|g_{1}\|\leq h_{2}$ and the fact that $\|\tanh(v)\|\leq 1\ \forall
v\in(-\infty,\infty)$, we can write

$\displaystyle\|f_{d}\|\leq
mg+mh_{2}+2\sqrt{3}m+m\sqrt{3}k_{\alpha}=\alpha_{1}.$ (88)

Also, from the third row of equation (21) we get

$\displaystyle
f_{d_{z}}=mg-m{g}_{1_{z}}+2m\tanh(e_{z_{1}})-mk_{\alpha}\tanh(e_{f_{z}})$ (89)

Further calculating for the lower bound of the $f_{d_{z}}(t)$, we have

$\displaystyle\|f_{d_{z}}\|\geq mg-mh_{2}-2m-mk_{\alpha}=\alpha_{2}.$ (90)

$\alpha_{2}$ is strictly positive for the given choice of $k_{\alpha}$.

Lemma 4.3

######  Proof .2.

Substituting the expression of $f(t)$ in the expression of $\Delta{f}$, yields

$\displaystyle\Delta{f}=\|f_{d}\|R_{c}e_{3}-\|f_{d}\|\sqrt{\frac{1+e_{3}^{T}R_{c}^{T}Re_{3}}{2}}Re_{3},$
(91)

The upper bound of the $\Delta{f}$ can be calculated as

$\displaystyle\|\Delta{f}\|\leq\bigg{\|}\|f_{d}\|R_{c}e_{3}-\|f_{d}\|\sqrt{\frac{1+e_{3}^{T}R_{c}^{T}Re_{3}}{2}}Re_{3}\bigg{\|}.$
(92)

Substituting the bounds of $\|f_{d}\|$ from lemma 4.1 to equation (92)

$\displaystyle\|\Delta{f}\|\leq\alpha_{1}\bigg{\|}R_{c}e_{3}-\sqrt{\frac{1+e_{3}^{T}R_{c}^{T}Re_{3}}{2}}Re_{3}\bigg{\|}.$
(93)

The quantity $e_{3}^{T}R_{c}^{T}Re_{3}$ represents the cosine of the angle
between vectors $R_{c}e_{3}$ and $Re_{3}$. Let the angle be
$\theta^{{}^{\prime}}$, then from equation (93), we have

$\displaystyle\|\Delta{f}\|\leq\alpha_{1}\sqrt{1-2cos(\frac{\theta^{{}^{\prime}}}{2})cos(\theta^{{}^{\prime}})+cos^{2}(\frac{\theta^{{}^{\prime}}}{2})}.$
(94)

Using Rodrigues’ formula, let’s consider the following

$\displaystyle R_{c}^{T}R=exp(\hat{\theta}_{1}),\quad
R_{c}^{T}R_{d}=exp(\hat{\theta}_{2})\quad R_{d}^{T}R=exp(\hat{\theta}_{3}),$
(95)

In the sublevel set $\mathcal{L}_{2}$ where
$\theta^{{}^{\prime}}\leq\|\theta_{1}\|$ the following holds

$\displaystyle\sqrt{1-2cos(\frac{\theta^{{}^{\prime}}}{2})cos(\theta^{{}^{\prime}})+cos^{2}(\frac{\theta^{{}^{\prime}}}{2})}\leq
2sin(\frac{\|\theta_{1}\|}{2})=2\|e_{R_{c}}\|.$ (96)

Also, from (95) we have

$\displaystyle R_{c}^{T}R=(R_{c}^{T}R_{d})(R_{d}^{T}R),\quad
exp(\hat{\theta}_{1})=exp(\hat{\theta}_{2})exp(\hat{\theta}_{3}),\quad
exp(\hat{\theta}_{1})=exp(\hat{\theta}_{2}+\hat{\theta}_{3}).$ (97)

From (97), the following relation holds

$\displaystyle\|\theta_{1}\|\leq\|\theta_{2}\|+\|\theta_{3}\|.$ (98)

Since the attitude error is equal to $sin\frac{\|\theta_{i}\|}{2}$ and in the
sub-level set $\mathcal{L}_{2}$ i.e. $0\leq\|\theta_{i}\|<\pi,\forall
i=1,2,3$, its monotonically increasing. Hence,

$\displaystyle\|e_{R_{c}}\|\leq\|e_{R_{dc}}\|+\|e_{R_{d}}\|.$ (99)

From equation (99) and (96), the bounds of $\|f_{d}\|$ can be given as

$\displaystyle\|\Delta{f}\|\leq 2\alpha_{1}(\|e_{R_{dc}}\|+\|e_{R_{d}}\|),$
(100)

Lemma 4.5

######  Proof .3.

From (26) we have

$\displaystyle\|\hat{\Omega}_{c}\|\leq\|\dot{R}_{c}\|.$ (101)

The time derivative of $R_{c}$ can be written as

$\displaystyle\dot{R}_{c}=$
$\displaystyle[\dot{x}_{B_{c}},\dot{y}_{B_{c}},\dot{z}_{B_{c}}],$ (102)

where

$\displaystyle\dot{z}_{B_{c}}=$
$\displaystyle\frac{\dot{f}_{d}}{\|f_{d}\|}-\frac{f_{d}(f_{d}^{T}\dot{f_{d}})}{\|f_{d}\|^{3}},$
$\displaystyle\dot{y}_{B_{c}}=$
$\displaystyle\frac{\dot{A}}{\|A\|}-\frac{(A^{T}\dot{A})}{\|A\|^{3}}A,\quad
A=x_{B_{d}}\times z_{B_{c}}$ $\displaystyle\dot{x}_{B_{c}}=$
$\displaystyle\dot{y}_{B_{c}}\times{z}_{B_{c}}+{y}_{B_{c}}\times\dot{z}_{B_{c}}.$
(103)

We individually calculate the upper bounds of column vectors of $\dot{R}_{c}$.
The bounds on $\dot{z}_{B_{c}}$ can be given as

$\displaystyle\|\dot{z}_{B_{c}}\|\leq 2\frac{\|\dot{f_{d}}\|}{\|f_{d}\|}$
(104)

Similarly, the bounds on $\dot{y}_{B_{c}}$ can be given as

$\displaystyle\|\dot{y}_{B_{c}}\|\leq 2\frac{\|\dot{A}\|}{\|A\|}$ (105)

Since $x_{B_{d}}\nparallel z_{B_{c}}$, we have

$\displaystyle\|\dot{y}_{B_{c}}\|\leq\frac{2}{\delta_{A}}(\|\dot{x}_{B_{d}}\|+\|\dot{z}_{B_{c}}\|),$
(106)

where ${\delta_{A}}\geq\|x_{B_{d}}\times z_{B_{c}}\|$. Using, equation (.3)
and properties of $\mathrm{SO(3)}$ we can obtain the upper bound of
$\dot{x}_{B_{c}}$ as

$\displaystyle\|\dot{x}_{B_{c}}\|\leq\|\dot{z}_{B_{c}}\|+\|\dot{y}_{B_{c}}\|$
(107)

Now, the upper bound of $\|\hat{\Omega}_{c}\|$ can be written as

$\displaystyle\|\hat{\Omega}_{c}\|$
$\displaystyle\leq\|\dot{z}_{B_{c}}\|+\|\dot{y}_{B_{c}}\|+\|\dot{x}_{B_{c}}\|$
$\displaystyle\leq 2\|\dot{z}_{B_{c}}\|+2\|\dot{y}_{B_{c}}\|$
$\displaystyle\leq(4+\frac{4}{\delta_{A}})\frac{\|\dot{f_{d}}\|}{\|f_{d}\|}+\frac{4}{\delta_{A}}h_{5}.$
(108)

Let’s evaluate the bound on the time derivatives of $f_{d}$. The time
derivatives of $f_{d}$ can be given as

$\displaystyle\dot{f}_{d}=$
$\displaystyle-m\dot{g}_{1}+2mSech^{2}(e_{x})\dot{e_{x}}-mk_{\alpha}Sech^{2}(e_{f})\dot{e}_{f}.$
(109)

Substituting the expressions from (15), (23) and (31) in the (109) will yield

$\displaystyle\dot{f}_{d}=$
$\displaystyle-m\phi+m\Phi_{g_{1}}+2mSech^{2}(e_{x})\left(e_{\alpha}-\alpha_{x}\tanh(e_{x})-\tanh(e_{f})\right)+mk_{\alpha}^{2}e_{\alpha}-mk_{\alpha}\tanh(e_{x})+mk_{\alpha}\alpha_{f}\tanh(e_{f}).$
(110)

The quantity $\Phi_{g_{1}}$ can be bounded as

$\displaystyle\|\Phi_{g_{1}}\|\leq$
$\displaystyle\frac{\|\nabla{f(g_{1})}\|^{2}}{\|\nabla{f(g_{1})}\|^{2}}\|\phi\|\|f(g_{1})\|,$
$\displaystyle\leq$ $\displaystyle\|\phi\|\|f(g_{1})\|,$ (111)

and $\Phi_{g_{1}}$ appears when $f(g_{1})>0$ (i.e.
$h_{2}<\|g_{1}\|<h_{2}+\epsilon_{0}$), then

$\displaystyle\|\Phi_{g_{1}}\|\leq\|\phi\|\frac{\epsilon_{0}+2h_{2}}{h_{2}^{2}},$
(112)

where $\phi$ can be written in the form of filter errors as

$\displaystyle\phi=\frac{1}{\gamma_{21}}(e_{g_{1}}-e_{g_{v}}-{e}_{\ddot{x}_{d}}).$
(113)

Then from equation (36) and (110), we can write

$\displaystyle\|\dot{f}_{d}\|\leq$
$\displaystyle\frac{m}{\gamma_{21}}\left(1+\frac{\epsilon_{0}+2h_{2}}{h_{2}^{2}}\right)(\|e_{g_{1}}\|+\|e_{g_{v}}\|+\|{e}_{\ddot{x}_{d}}\|)+2m\left(\|e_{\alpha}\|+\alpha_{x}\|\tanh(e_{x})\|+\|\tanh(e_{f})\|\right)$
$\displaystyle+mk_{\alpha}^{2}\|e_{\alpha}\|+mk_{\alpha}\|\tanh(e_{x})\|+mk_{\alpha}\alpha_{f}\|\tanh(e_{f})\|.$
(114)

Using lemma 4.1, 4.7 and relation (.3), the equation (.3) can be written as

$\displaystyle\|\hat{\Omega}_{c}\|\leq$
$\displaystyle\rho_{1}(\|e_{\alpha}\|,\|\tanh(e_{x})\|,\|\tanh(e_{f})\|)+\rho_{01},$
(115)

where $\rho_{1}()$ is class $\kappa$-function and $\rho_{0_{1}}$ is a positive
constant.

Lemma 4.7

######  Proof .4.

For the error $e_{g_{x}}$, lets consider the following Lyapunov candidate

$\displaystyle V_{g_{x}}=\frac{1}{2}e_{g_{x}}^{T}e_{g_{x}}.$ (116)

Taking the time derivative of $V_{g_{x}}(t)$ and using (37) will yield

$\displaystyle\dot{V}_{g_{x}}=-\frac{1}{\gamma_{1}}e_{g_{x}}^{T}e_{g_{x}}+e_{g_{x}}^{T}\ddot{x}_{d}.$
(117)

This implies that

$\displaystyle\|e_{g_{x}}\|=\sqrt{2{V}_{g_{x}}(t)}\leq\sqrt{\|e_{g_{x}}(0)\|^{2}exp\left(-\frac{1}{\gamma_{1}}t\right)+\gamma_{1}^{2}h_{2}^{2}}.$
(118)

Similarly, for error $e_{\ddot{x}_{d}}(t)$, consider the following Lyapunov
candidate

$\displaystyle
V_{\ddot{x}_{d}}=\frac{1}{2}e_{\ddot{x}_{d}}^{T}e_{\ddot{x}_{d}}.$ (119)

Taking the time derivative of $V_{\ddot{x}_{d}}(t)$ and using (37) will yield

$\displaystyle\dot{V}_{\ddot{x}_{d}}=-\frac{1}{\gamma_{1}}e_{\ddot{x}_{d}}^{T}e_{\ddot{x}_{d}}+e_{\ddot{x}_{d}}^{T}\dddot{x}_{d}.$
(120)

This implies that

$\displaystyle\|e_{\ddot{x}_{d}}\|=\sqrt{2{V}_{\ddot{x}_{d}}(t)}\leq\sqrt{\|e_{\ddot{x}_{d}}(0)\|^{2}exp\left(-\frac{1}{\gamma_{1}}t\right)+\gamma_{1}^{2}h_{3}^{2}}.$
(121)

For error $e_{g_{v}}$, consider the following Lyapunov candidate

$\displaystyle V_{g_{v}}=\frac{1}{2}e_{g_{v}}^{T}e_{g_{v}}.$ (122)

Taking the time derivative of $V_{g_{v}}(t)$ and using (37) will yield

$\displaystyle\dot{V}_{g_{v}}=-\frac{1}{\gamma_{2}}e_{g_{v}}^{T}e_{g_{v}}+\frac{1}{\gamma_{1}}e_{g_{v}}^{T}e_{\ddot{x}_{d}}.$
(123)

This implies that

$\displaystyle\|e_{g_{v}}\|=\sqrt{2{V}_{g_{v}}(t)}\leq\sqrt{\|e_{g_{v}}(0)\|^{2}exp\left(-\frac{1}{\gamma_{2}}t\right)+\frac{\gamma_{2}^{2}}{\gamma_{1}^{2}}\|e_{\ddot{x}_{d}}\|^{2}}.$
(124)

Finally, we will analyze the dynamical behavior of the error $e_{g_{1}}(t)$ by
choosing the following Lyapunov candidate

$\displaystyle V_{g_{1}}=\frac{1}{2}e_{g_{1}}^{T}e_{g_{1}}.$ (125)

Taking the time derivative of $V_{g_{1}}(t)$ and using (37) will yield

$\displaystyle\dot{V}_{g_{1}}=$
$\displaystyle-\frac{1}{\gamma_{21}}e_{g_{1}}^{T}e_{g_{1}}+\frac{1}{\gamma_{21}}e_{g_{1}}^{T}e_{g_{v}}+\frac{1}{\gamma_{21}}e_{g_{1}}^{T}{e}_{\ddot{x}_{d}}+e_{g_{1}}^{T}\Phi_{g_{1}}+e_{g_{1}}^{T}\dddot{x}_{d}.$
(126)

Using the fact that $e_{g_{1}}^{T}\Phi_{g_{1}}\leq 0$, the bounds on
$\dot{V}_{g_{1}}(t)$ can be given as

$\displaystyle\dot{V}_{g_{1}}\leq-\frac{1}{2\gamma_{21}}\|e_{g_{1}}\|^{2}+\frac{2}{\gamma_{21}}\|e_{g_{v}}\|^{2}+\frac{2}{\gamma_{21}}\|{e}_{\ddot{x}_{d}}\|^{2}+\gamma_{21}h_{3}^{2}.$
(127)

Solving (127) for ${V}_{g_{1}}(t)$ using equations (45), (46)

$\displaystyle{V}_{g_{1}}(t)\leq{V}_{g_{1}}(0)exp(-\frac{1}{\gamma_{21}}t)+{4}\alpha_{4}^{2}+{4}\alpha_{5}^{2}+\gamma_{21}^{2}h_{3}^{2}.$
(128)

This implies that

$\displaystyle\|e_{g_{1}}(t)\|=\sqrt{2{V}_{g_{1}}(t)}\leq\sqrt{\|e_{g_{1}}(0)\|^{2}exp(-\frac{1}{\gamma_{21}}t)+\alpha_{7}},$
(129)

where
$\alpha_{7}={8}(\alpha_{4}^{2}+\alpha_{5}^{2})+2\gamma_{21}^{2}h_{3}^{2}$.