text:
\section{Overview}

\IEEEPARstart{R}{igid}-body attitude control problems have received considerable research interest since the 1950s \cite{10.5[5]} and have been a long-standing staple in the nonlinear control along with a vast literature spanning many decades \cite{8.1[1]}, \cite{10.7[1]}, \cite{8.1[3]}, \cite{10.7}, \cite{18abd}, \cite{2010mengziyangauto}, \cite{du2016tac}, \cite{zddtcsI}, \cite{zddtcns}, \cite{jinxinauto}, \cite{jinxintcst}, with various applications in aerospace, underwater vehicles, marine engineering and robotics \cite{43abd}, \cite{renwei2008shu}, \cite{2018liutao-auto}, \cite{Dobook}.
For the investigation convenience, it is usually assumed that the states of the rigid body can be placed into an abstract manifold with appropriate dimensions, with the one-to-one corresponding relations of points between them \cite{2012tac[4]}.
Under this correspondence, the attitude dynamics of the rigid body give rise to a dynamical system on the boundaryless compact manifold $SO(3)$, i.e., the state space of attitude rotation matrices \cite{2011tac}, \cite{pedro2017auto}.
In fact, it is precisely because the topological property without contractibility of $SO(3)$, resulting from the compact property without boundary (see in \cite[Ex. 2.4.6]{8.1[9]}), that $SO(3)$ is not diffeomorphic to any Euclidean space. That is, for a differential equation on the attitude dynamics/kinematics with a locally Lipschitz right-hand side, the basin of attraction of its asymptotically stable equilibrium point is not contractible.

On the attitude configuration space $SO(3)$, the rigid-body attitude is usually parameterized for exploiting the redundancies
in the rotation-matrix descriptions. According to the attitude parametrization, attitude control strategies can be
categorized by choosing the Euler-angles parametrization (e.g., roll, yaw and pitch), exponential coordinates, the (modified) Rodrigues parametrization \cite{18abd} and the unit quaternion parametrization \cite{13-14}, etc.
However, different parameterizations lead to further topological difficulties. For example, considering that no globally nonsingular three-parameter representation exists, the above former three parametrizations would consequently cause the singularity issue \cite{12[14]}, \cite{10.7}.
This can seriously constrain admissible
attitudes and further complicates the path planning.
This obstacle on the allowable attitude motions has nothing to do with the
physically admissible attitude motions, while it is for purely
mathematical reasons, i.e., it is an inherent obstacle.
As a globally nonsingular parametrization (e.g., the covering map from
$\mathbb{S}^{3}$ to $SO(3)$ is everywhere a local diffeomorphism), the unit quaternion parametrization is often employed evolving on the three-dimensional unit sphere $\mathbb{S}^{3}$.
However, the well-known Rodrigues mapping from $\mathbb{S}^{3}$ to $SO(3)$ implies that two antipodal unit quaternions correspond to the same attitude in $SO(3)$. Corresponding to the above described topological constraints on $SO(3)$, the attitude control law designed on $\mathbb{S}^{3}$ is to stabilize two disconnected sets of quaternions consistent with the same physical attitude on $SO(3)$ \cite{2011tac}, \cite{2012tac}, \cite{2013tac}, \cite{13-14}. Without considering the double covering problem \cite{10.7}, the designed controller may induce the unwinding phenomenon where the rigid body will unnecessarily make a full rotation \cite{2012tac[4]}. To avoid the unwinding phenomenon, a dynamic mechanism is necessary to
resolve the ambiguity, i.e., which quaternion should be chosen for feedback, when an inconsistent quaternion-based feedback can be used simultaneously. More specifically, a hysteretic-based switching scheme is considered in terms of the aforementioned binary logic variable, to select which pole of $\mathbb{S}^{3}$ to regulate in a hysteretic fashion, see \cite{2011tac} for more detailed analysis. Later, this unwinding-free feedback strategy has been further applied into \cite{2012tac}, \cite{2013tac} and \cite{13-14}, which manages a tradeoff between the robustness to arbitrarily small measurement disturbances and a small amount of hysteresis-induced inefficiency caused by the unwinding phenomenon. Note that, in \cite{18abd}, \cite{20abd} and \cite{43abd} for the synchronization problem, the unwinding phenomenon has been solved by restricting the control gain, and the synchronization can be realized if and only if the imaginary part of a quaternion is zero. Coincidentally, in the later work \cite{2014duauto} on the synchronization issue, the authors have made that point emphatically in the reply to the comments on the quaternion ambiguity involved in \cite{2014duauto} made by Rezaee and Abdollahi \cite{com2014}, by assuming that the scale part of quaternion for each spacecraft is
always positive. Therefore, the quaternion ambiguity should be considered in the hybrid feedback strategy for the attitude investigation of rigid bodies, to exclude the unwinding phenomenon.

Compared with other global (non-singular) attitude representations (i.e., rotation matrix also called direction cosine matrix, abbr. DCM) and the axis-angle parameterization, the computational burden can be greatly reduced by making use of unit quaternions \cite[pg. 322]{shs1.1Ehsan2019[9]}. As the minimal globally nonsingular parametrization, unit quaternions are often used to parametrize rotation matrices on $SO(3)$; meanwhile, however, the unwinding phenomenon is introduced due to the double covering transformation from $SO(3)$ to state space $\mathbb{S}^{3}$ of unit quaternions \cite{2012tac[4]}, causing the rigid body to unnecessarily make a full rotation from the unstable saddle point to another stable equilibrium point
\cite{10.7}, \cite{chat2009tac}. Nevertheless, the topology constraint on $SO(3)$ never stops the researchers from using unit quaternions to design feedback control algorithms for attitude control \cite{du2016tac}, \cite{13-14}. To solve the topology constraint and realize the robust global attitude control, the design of the quaternion-based torque control law applies a binary logic variable \cite{2011tac, 2012tac}, where the hysteresis half-width arranges a tradeoff between the hysteresis-induced inefficiency for avoiding unwinding phenomenon and the robustness to measurement noise.
Hence, the salient feature on the attitude control problem leads to a captivating difficulty stemming from the well-known topological constraints of the rigid-body state space $SO(3)$ on rotation matrices: It is impossible to find any continuous feedback control law so that all equilibrium points of $SO(3)$ can be globally asymptotically stable, since there must be one other unstable equilibrium point in any smooth
vector field on $SO(3)$ if an attracting equilibrium point exists, as analyzed in \cite{2012tac}, \cite{2013tac}, \cite{36abd} and \cite{13-14}.

With continuous state-feedback control laws designed on the Lie group $SO(3)$ or the Lie group diffeomorphic to $SO(3)\times \mathbb{R}^{3}$ (like \cite{chat2009tac}), the best results that one can achieve are at most almost global stability (i.e., the attraction domain of the equilibrium is open and dense), where the desired attitude can be stabilized from any initial attitude, necessarily excluding a nowhere dense set of zero Lebesgue measure which is attracted to the antipode of the stabilized desired orientation. But yet, for the same reason, there exist no periodic or discontinuous feedback that can simultaneously stabilize a particular attitude robustly and globally: Any periodic or discontinuous feedback will create a rigid decision boundary, and it can be hijacked by any small measurement disturbance so that the real-time attitude attitude is far from the reference one, as verified in \cite{2011tac}. To compensate for the topological constraints on $SO(3)$ by simultaneously achieving the global and robust attitude control performance, many authors employ the hybrid feedback control strategy \cite{2011tac}, \cite{2012tac}, \cite{2013tac}, \cite{13-14}, \cite{8.1}, \cite{36abd}, \cite{15abd}, \cite{16abd}. More specifically, in \cite{2011tac}, the authors propose several hybrid control strategies based on the binary logic variable $h\in \{-1,1\}$ in terms of the \emph{sign} function, and verify the most appropriate one by considering the well-posed property of hybrid dynamic theory.
Later, the authors in \cite{2012tac}, \cite{2013tac} and \cite{13-14} employ the logic-variable based hybrid strategy to different investigations including the attitude stabilization/consensus/synchronizaiton, etc.
Differently, in \cite{8.1}, \cite{36abd}, \cite{15abd} and \cite{16abd}, the authors consider the inherent passivity properties of attitude dynamics to design an appropriate error function on $SO(3)$, and further establish an artificial potential functions. It is verified that, by coordinating a family of synergistic potential functions in a hybrid strategy with a hysteresis-based
switching mechanism by selecting an appropriate control law corresponding to the minimal potential function, the topological obstructions can be overcome. Based on the \cite{8.1}, the authors in \cite{36abd}, \cite{15abd} and \cite{16abd} propose the exp-synergistic potential functions on $SO(3)$, which are applied to velocity-free hybrid attitude stabilization.

\begin{table*}[tbp]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Attitude Representation & Globally represent  & Uniquely represent & Number of parameters \\ \hline

Euler angles (kinematically singular)&  No  & No & 3 \\ \hline

(Modified) Rodrigues parameters (geometrically singular)    & No & No& 3  \\ \hline

Quaternions& Yes & No& 4\\ \hline
Axis-angle & Yes & No & 4\\  \hline
Rotation matrix & Yes & Yes & 9\\  \hline
\end{tabular}
 \caption{Properties of attitude representations I.}\label{unique}
\end{table*}

This paper serves as a roadmap for the theoretical development of attitude control, including the following aspects:\\
\indent 1) A class of attitude representations, i.e., 9-parameter rotation matrix (globally and uniquely, no singularities), 3-parameter Euler angles (neither globally nor uniquely, kinematical singularity and Gimbal lock) and (modified) Rodrigues parameters (neither globally nor uniquely, geometrical singularity), 4-parameter axis-angle (globally but not uniquely) and unit quaternion (globally but not uniquely, unwinding phenomenon, reduced computation burden).\\
\indent 2) The relationships among those attitude representations, especially the connections among Gimbal lock, unwinding phenomenon and a nowhere dense set of zero Lebesgue measure, i.e., Gimbal lock (i.e., when the plane's head is straight up or straight down) includes a nowhere dense set of zero Lebesgue measure, not including the two antipodal equilibrium points. The unwinding phenomenon is caused when we ignore the two antipodal equilibrium points, so even when we remove the Gimbal lock problem using unit quaternions, then unwinding phenomenon could still occur.\\
\indent 3) Attitude control laws for the (almost) global control, including unit quaternion based control laws (continuous, discontinuous and hybrid cases) and DCM based control laws. Generally, the hybrid case focuses on the robust global attitude control by overcoming the topology constraints of boundaryless compact manifold $SO(3)$, such as a family of configuration error functions or synergistic potential functions, or binary logic variable based hybrid control. The continuous and discontinuous cases try to realize the global attitude control by adopting certain methods, for example modifying control laws over a set of zero Lebesgue measure or restricting the rotation angle for the global control. \\
\indent 4) The nominal robustness with respect to small perturbations (including measurement noises/disturbances) for avoiding chattering, and general robustness against bounded \emph{unstructured} uncertainties in both the translational dynamics and the rotational dynamics in terms of the presented control laws, where continuous and hybrid control laws are generally nominally robust, not including the purely discontinuous control law.\\
\indent 5) The technique tools used to prove the (almost) global attitude control and the robustness. The LaSalle's/hybrid invariance principle is better applied to aysmptotic stability for time-invariant or periodic equation, not including the time-varying (i.e, non-autonomous) systems. Other than LaSalle's/hybrid invariance principle, cross-term-added Lyapunov function or Barbalat's lemma are feasible for time-varying case. Comparison principle can be applied into non-autonomous/autonomous case, while Matrosov' theorem can be applied in all circumstance, i.e., the time-(in)variant systems, nonperiodic/time-dependent systems, etc. Non-autonomous systems could be dealt with Barbalat's lemma, Matrosov's theorem, comparison principle, etc.

The paper is organized as follows: Section \ref{section3.0} presents attitude representations for rigid body kinematics, including the connections among Gimbal lock, unwinding phenomenon and a nowhere dense set of zero Lebesgue measure. Section \ref{section4.0} presents attitude control laws, including unit quaternion based control laws (continuous, discontinuous and hybrid cases), DCM based control laws, analysis on (almost) global control, analysis on robustness and technique tools. Section \ref{section6} concludes the paper.

\section{Attitude Representation for Rigid Body Kinematics}\label{section3.0}

The first step for investigating attitude control problems of rigid-body systems is to choose a best description for rigid body orientation, which is a very fundamental and important topic. Indeed, choosing a good attitude coordinates could greatly facilitate investigations, i.e., simplifying mathematical calculations and analysis, avoiding geometrical/mathematical singularities or strictly nonlinear kinematic differential equations (KDEs). Generally, in order to choose a good attitude description, one need to consider the following four truths about rigid body attitude coordinates:\\
\indent 1) The minimum parameter used to describe the relative attitude orientation (i.e., angular displacement)
between two reference frames is three.\\
\indent 2) For any 3-parameter set of attitude coordinates, there exists at least one geometrical orientation so that the coordinates are singular, i.e., at least two coordinates are not unique.\\
\indent 3) Corresponding to any geometric singularity, the kinematic
differential equations (KDEs) are also singular.\\
\indent 4) A solution to avoid geometric singularities is to regularize the three-parameter representation: Once-redundant redundant four or more coordinates contain no any geometric/kinematicla singularity, which are universally determined.\\
According to the above truths, we analyze the following attitude representation for rigid-body kinematics.

\subsection{Rotation matrix}\label{section3.1}

In the three-dimensional space, the attitude of a rigid-body system denotes the relative rotation from the body frame $\mathcal {F}_{b}$ to the inertial reference frame $\mathcal {F}_{o}$, called rotation matrix. Rotation matrix is also called direction cosine matrix (DCM), since the axes of $\mathcal {F}_{b}$ and $\mathcal {F}_{o}$ are usually unit vectors, where only cosine terms are involved in the rotation matrix, see \cite[pg. 80-86]{2019Ehsan[27]} for more details. According to \cite[pg. 17]{abd2013shu}, the attitude of a rigid-body system can be described by a $3\times3$ orthogonal rotation matrix unitary determinant, i.e., an element of a Lie group:
\begin{equation*}\label{rm1}
\begin{split}
SO(3)=\{R\in \mathbb{R}^{3\times 3}: R^{T}R=I_{3\times 3}, \hbox{det} (R)=1\},\\
\end{split}
\end{equation*}
where $I$ denotes the $3\times3$ identity element and $\hbox{det} (R)$ is the determinant of matrix $R$.

Take a simple example to show how to use the rotation matrix \cite[pg. 17]{abd2013shu}. Let $R\in SO(3)$ be the rotation from $\mathcal {F}_{b}$ to $\mathcal {F}_{o}$, and given $X_{b}$ as the coordinates of a vector in $\mathcal {F}_{b}$, then the coordinates $X_{o}$ of this same vector in $\mathcal {F}_{o}$ can be denoted as
\begin{equation*}\label{rm2}
\begin{split}
X_{o}=RX_{b}.
\end{split}
\end{equation*}
In fact, the above property also holds for several frames using the composition of rotations, i.e., the noncommutative multiplication
of rotation matrices in terms of those frames.

The definition of rotation matrix shows that the rotation matrix is the most fundamental attitude representation, which can globally and uniquely denote any orientation as shown in Table \ref{unique}. However, the 9-parameter representation also exhibits high redundancy for describing a relative orientation. As shown in Table \ref{unique}, compared with the 3-parameter representation, there are six extra parameters causing redundancy due to the orthogonality condition $R^{T}R=I_{3\times 3}$. In practice, we usually apply the less redundant representation, rather than the 9-parameter DCM.

But at the cost of high redundancy, what we have to admit is that the DCM has a biggest asset, i.e., easily transforming vectors between any two reference frames. Furthermore, given any instant, the instantaneous DCM could be obtained via a rigorously linear differential equation as shown in \cite[pg. 85-86]{2019Ehsan[27]}. So linearity and universal applicability are major advantages of the kinematic differential equations for DCM, and there is no any geometric/kinematical singularities occurring in the kinematic differential equations using DCM.

\begin{figure}[!htb]
\centering
\includegraphics[width=3in,height=2in]{rpy.eps}
\caption{\cite[pg. 87]{2019Ehsan[27]} Yaw $\psi$, pitch $\textcolor[rgb]{0.00,0.00,0.00}{\theta}$, and roll $\phi$ Euler angles with respect to body-fixed axes $\{\vec{b}\}=\{b_{1}, b_{2},b_{3}\}$.}\label{rpy}
\end{figure}

\subsection{Euler angles: kinematically singular}\label{section3.2}

The Euler angles are defined to describe the orientation of a rigid body, including three angles with respect to a fixed
coordinate system. Both kinds of sequences symmetric Proper Euler angles (i.e., $z-y-z$, $x-z-x$, $y-x-y$, $z-x-z$, $x-y-x$, $y-z-y$, also called longitude of the ascending node denoted as $\tilde{\psi}$, inclination denoted as $\textcolor[rgb]{0.00,0.00,0.00}{\tilde{\theta}}$, and argument of the perihelion denoted as $\tilde{\phi}$) and asymmetric Tait-Bryan angles ($x-z-y$, $z-y-x$, $y-x-z$, $x-y-z$, $y-z-x$, $z-x-y$, also called yaw-pitch-roll angles) are called ``Euler angles''.

Take (yaw, pitch, roll):=($\psi$, $\textcolor[rgb]{0.00,0.00,0.00}{\theta}$, $\phi$) for example about the sequentially
displaced body-fixed axes $\{\vec{b}\}=\{b_{1}, b_{2},b_{3}\}$, as shown in Fig. \ref{rpy}. Successive multiplication (usually three elemental rotation matrices) of rotation matrices can derive the transformation mapping the vector ($\psi$, $\textcolor[rgb]{0.00,0.00,0.00}{\theta}$, $\phi$) into its corresponding rotation matrix as shown in \cite[pg.18]{abd2013shu}, which shows the connections between rotation matrix and Euler angles, i.e., $(\psi, \textcolor[rgb]{0.00,0.00,0.00}{\theta}, \phi)\longmapsto b_{3}(\psi)b_{2}(\textcolor[rgb]{0.00,0.00,0.00}{\theta})b_{1}(\phi)$, implying $R_{\vec{b}}(\psi, \textcolor[rgb]{0.00,0.00,0.00}{\theta}, \phi)=b_{3}(\psi)b_{2}(\textcolor[rgb]{0.00,0.00,0.00}{\theta})b_{1}(\phi)$, i.e., rotated about axis $b_{3}$ by the yaw
angle $\psi$, axis $b_{2}$ by the pitch angle $\textcolor[rgb]{0.00,0.00,0.00}{\theta}$ and axis $b_{1}$ by the roll angle $\phi$, respectively.

Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any
rotation matrix $R$ can be decomposed as a product of three elemental rotation matrices. Euler angle is a minimal 3-parameter representation of attitude as shown in Table \ref{unique}. The 3-parameter sets of (modified) Rodrigues parameters can be regarded as certain embedded subsets of Euclidean space $\mathbb{R}^{3}$, so that one can apply the analysis methods suited to $\mathbb{R}^{3}$. However, it is not globally defined for the transformation from their time rates of change to the angular velocity vector. In other words, the time derivatives of the Euler angles could not represent every possible angular velocity, causing mathematical singularity of the Euler angle kinematic differential equation (kinematical singularity, i.e., two angles are not uniquely defined). In fact, except rotation matrix, all parameterizations fail to uniquely represent the set of attitudes, and Euler angles are no exception. The angles are uniquely determined except for the singular case that two out of the three gimbals are identical, i.e. two axes have the same or opposite directions, usually called Gimbal lock, as shown in Fig. \ref{GB}. Also as analyzed in \cite[pg. 86-95]{2019Ehsan[27]}, the geometric singularity occurs when the \emph{pitch} is rotated $90^{\circ}$ up or down ($\pm 90^{\circ}$) in pitch-yaw-roll angles, or when an \emph{inclination} angle of $0^{\circ}$ or $180^{\circ}$ in the symmetric Proper Euler angles\footnotemark[1]. So Euler angle representation encounters a geometric singularity at certain specific values of the second Euler rotation angle (pitch angle and/or inclination angle) only, i.e., when the first and third angles are measured in the same plane, however, never encountering a singularity in the first or third angles. Actually, Gimbal lock would cause disastrous results if the rigid-body is in a steep ascent or dive, in a gimbal-based aerospace inertial navigation system.

\footnotetext [1]{The geometric singularity is when the plane's head is straight up or straight down as shown in Fig. \ref{GB}(b), corresponding to ``$\hbox{pitch}=\pm 90^{\circ}$'' or ``$\hbox{inclination}=0^{\circ}~or~180^{\circ}$'' using different sets of Euler angles, called Gimbal lock. More specifically, a singularity (Gimbal lock) occur only at certain specific values of the second rotation angle, never resulting from the first and third rotation angles (the two angles never lead to any singularity). That is, the singularity of attitude (Gimbal lock) may include many combinations of attitude angles, even if the second angle is fixed, which will be analyzed later.}

Therefore, in both kinds of sequences symmetric Proper Euler angles and asymmetric Tait-Bryan angles, a rigid-body is not allowed to rotate more than $90^{\circ}$ away from the singularity, so that they are limited to describe large (in particular arbitrary) rotations. Moreover, despite the kinematic differential equations of Euler angle could be linearized, however, the linearized one could only be valid for relatively small rotations. If not linearized, their fairly nonlinear kinematic differential equations will contain many computationally intensive trigonometric functions. Hence, Euler angles with the minimum number 3 are not always convenient as analyzed in a series of work
\cite[pg. 86-95]{2019Ehsan[27]} or \cite{meyer1966, 10.7[1]}, especially the symmetric Proper Euler angles, since the small departure rotations with vary small angles are always hovering around the singular attitude $0^{\circ}$ (very close to the singularity). Still and all, the 3-parameter Euler angle representation is a compact attitude description with a easily visualized coordinates.

\begin{figure*
\centering\subfigure[]{
\begin{minipage}[t]{0.33\textwidth}
\label{fig:subfig:c}
\includegraphics[width=2.2in,height=1.7in]{xyz.eps}
\end{minipage}}
\centering\subfigure[]{
\begin{minipage}[t]{0.33\textwidth}
\label{fig:subfig:c}
\includegraphics[width=2.2in,height=1.7in]{gimbal.eps}
\end{minipage}}
\caption{The rotor of the gyroscope$^{2}$. (a) The rotor of the gyroscope is kept in balance: Red gimbal ring rotates along $x$-axis (red line) to realize the pitch motion; Blue gimbal ring rotates along $y$-axis (green line) to realize the yaw motion; Green gimbal ring rotates along $z$-axis (blue line) to realize the roll motion. (b) The the rotor of the gyroscope is out of balance: Gimbal lock, i.e., two out of the three gimbals are in the same plane, so that one degree of freedom is lost.}\label{GB}
\end{figure*}

\subsection{Axis-angle}\label{section3.3}

Axis-angle representation is about a given normalized vector by a given rotation angle, which are mapped on a rotation matrix, so the axis-angle representation is a 4-parameter representation of attitude as shown in Table \ref{unique}.

Define $\vec{k}\in \mathbb{R}^{3}$ as an arbitrary unit-length vector and $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}$ as the rotation angle about $\vec{k}$. So we derive the corresponding rotation matrix $\mathbf{R}(\textcolor[rgb]{0.00,0.00,0.00}{\Phi},\vec{k})\in SO(3)$ \cite[pg. 18]{abd2013shu},
\begin{equation}\label{Aa1}
\begin{split}
\mathbf{R}(\Phi, \vec{k})=I_{3}-\sin(\textcolor[rgb]{0.00,0.00,0.00}{\Phi})S(\vec{k})+(1-\cos(\textcolor[rgb]{0.00,0.00,0.00}{\Phi}))S^{2}(\vec{k}),
\end{split}
\end{equation}
where $I_{3}$ denotes the $3\times3$ identity element, and $S(\vec{k})$ is defined as for $\vec{k}=(k_{1}, k_{2}, k_{3})$
\begin{equation*}\label{Aa2}
\begin{split}
S(\vec{k})=\left[\begin{array}{ccc}
       0 & -k_{3}& k_{2}\\
       k_{3}& 0 & -k_{1}\\
       -k_{2}& k_{1}& 0
     \end{array}\right].
\end{split}
\end{equation*}
That is, $S(\cdot)$ denotes the skew symmetric matrix, so the cross product of any two vectors $X,Y \in \mathbb{R}^{3}$ can be expressed by the matrix multiplication, i.e., $X\times Y=S(X)Y$.

Therefore, it can be checked from (\ref{Aa1}) that the attitude matrix can be defined by axis-angle representation, i.e., a given set of parameters $\{\textcolor[rgb]{0.00,0.00,0.00}{\Phi}, \vec{k}\}$. For (\ref{Aa1}), it can be easily checked that $\mathbf{R}(\textcolor[rgb]{0.00,0.00,0.00}{\Phi},\vec{k})=\mathbf{R}(-\textcolor[rgb]{0.00,0.00,0.00}{\Phi},-\vec{k})$ since $S(-\vec{k})=-S(\vec{k})$. More specifically, one can get a unique rotation matrix from a given axis-angle parameters $\{\textcolor[rgb]{0.00,0.00,0.00}{\Phi}, \vec{k}\}$, but not vise versa, since there exist at least two antipodal sets of axis-angle parameters $(\textcolor[rgb]{0.00,0.00,0.00}{\Phi},\vec{k})$ and $(-\textcolor[rgb]{0.00,0.00,0.00}{\Phi},-\vec{k})$ for a given rotation $\mathbf{R}(\textcolor[rgb]{0.00,0.00,0.00}{\Phi},\vec{k})$. Indeed, for the case $\mathbf{R}(\textcolor[rgb]{0.00,0.00,0.00}{\Phi},\vec{k})=I_{3}$, one can find a series of axis-angle parameters $\{2n \pi, \vec{k}\}$ for any $n\in \mathbb{Z}$ and any unit-length vector $\vec{k}\in \mathbb{R}^{3}$.

As analyzed in \cite[pg. 102]{2019Ehsan[27]}, the kinematic differential equation using axis-angle contains a $\frac{0}{0}$-type mathematical singularity when $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}=0$ (i.e., zero rotation), so the axis-angle is not well suited in those cases with small motion feedback control. In addition, the axis-angle is not well suited for the large rotation either, since the axis-angle based mathematical expression contains not only polynomial fractions of degrees up to three but also trigonometric functions, which makes computations more complicated. As a result, axis-angle representation is less attractive to describe small/large arbitrary rotations, compared with other attitude representations.

\subsection{Quaternions (i.e., Euler symmetric parameters)}\label{section3.4}

As shown in Table \ref{unique}, unit quaternions and the axis-angle representation are both 4-parameter representation of attitude. In fact, unit quaternions are often regarded as an axis-angle representation, i.e., $q=(\cos \frac{\Phi}{2},\vec{k}\sin \frac{\Phi}{2})\in \mathbb{S}^{3}$ for given axis-angle parameter $\{\textcolor[rgb]{0.00,0.00,0.00}{\Phi}, \vec{k}\}$, where $\mathbb{S}^{n}:=\{x\in \mathbb{R}\times\mathbb{R}^{n}: x^{T}x =1\}$ is the $n$-dimensional unit sphere embedded in $\mathbb{R}^{n+1}$

According to the Rodrigues formula $\mathcal {R}(-q)=\mathcal {R}(q)=I+2\eta S(\epsilon)+2S^{2}(\epsilon)$
in terms of $\mathcal {R}: \mathbb{S}^{3}\rightarrow SO(3)$, where unit quaternion
$q=(\eta, \epsilon)\in\mathbb{S}^{3}$, every element of $SO(3)$ can be parameterized by two antipodal unit quaternions in $\mathbb{S}^{3}$.
For any two unit quaternions $q_{1}$ and $q_{2}$ with $q_{i}=(\eta_{i}, \epsilon_{i})\in \mathbb{S}^{3}$ for $i=\{1,2\}$, one has
\begin{equation*}\label{t1t}
\begin{split}
q_{1}\otimes q_{2}=\left[ \begin{array}{c}
                              \eta_{1}\eta_{2}- \epsilon^{T}_{1}\epsilon_{2} \\
                              \eta_{1}\epsilon_{2}+ \eta_{2}\epsilon_{1}+S(\epsilon_{1})\epsilon_{2}
                            \end{array}
\right].
\end{split}
\end{equation*}
This implies that each unit quaternion $q_{i}\in\mathbb{S}^{3}$ has an inverse $q^{-1}_{i}=(\eta_{i}, -\epsilon_{i})$.

As shown above, as the minimal globally nonsingular parametrization, unit quaternions are often used to parametrize rotation matrices on $SO(3)$. Meanwhile, however, given a certain attitude, there exist actually two sets of unit quaternions describing the same orientation in terms of $\mathcal {R}(-q)=\mathcal {R}(q)$, due to the \emph{double covering} transformation from $SO(3)$ to state space $\mathbb{S}^{3}$ of unit quaternions \cite{2012tac[4]}. As a result, this will cause the rigid body to unnecessarily make a full rotation from the unstable saddle point to another stable equilibrium point, called unwinding phenomenon \cite{10.7, chat2009tac}. As shown in Fig. \ref{shortest}, the stable one could specify the orientation by the shortest single axis rotation, while the other unstable saddle needs to specify by the longest.

As analyzed in \cite[pg. 103-111]{2019Ehsan[27]}, the kinematic differential equation of unit quaternions is rigorously linear if the angular velocities in all three directions depend only on time, and more generally considered bilinear if the angular velocities in all three directions are themselves coordinates. In fact, only the once-redundant unit quaternions (1 parameter is redundant compared with the 3-parameter case) retain a singularity-free attitude representation, whose kinematic differential equations are also linear analogous to the rotation matrix. Those properties show that unit quaternions are attractive attitude representation in some topics including attitude estimation problems using filters or observers where the KDEs are linearized. However, the kinematic differential equations for all 3-parameter attitude representations are nonlinear, containing geometrical and/or mathematical (kinematical) singularities.

\emph{The analysis on unwinding phenomenon and Gimbal lock:}\\
\indent--Given unit quaternion $q=(\eta, \epsilon)=(\cos \frac{\Phi}{2},\vec{k}\sin \frac{\Phi}{2})\in \mathbb{S}^{3}$. That is, $\eta=\cos \frac{\Phi}{2}$. Following \cite[(3.73)\&(3.91a)]{2019Ehsan[27]}, we get $\cos\Phi=\frac{1}{2}(C_{11}+C_{22}+C_{33}-1)$ where $C_{ii}$ denotes three diagonal elements of DCM.\\
\indent--Based on \cite[pg. 86-95]{2019Ehsan[27]}, Gimbal lock occurs when the \emph{pitch} is rotated $90^{\circ}$ up or down ($\pm 90^{\circ}$) in pitch-yaw-roll angles, or when an \emph{inclination} angle of $0^{\circ}$ or $180^{\circ}$ in the symmetric Proper Euler angles. Gimbal lock would cause disastrous results if the rigid-body is in a steep ascent or dive, in a gimbal-based aerospace inertial navigation system. Next we analyze the two cases when Gimbal lock occurs.

For the asymmetric pitch-yaw-roll angles, we define $(\theta_{1}, \theta_{2}, \theta_{3}):=(\psi, \textcolor[rgb]{0.00,0.00,0.00}{\theta}, \phi)$. Then we get the DCM in terms of the asymmetric pitch-yaw-roll angles, i.e.,
\begin{equation}\label{ypr-1}
\begin{split}
[C]=\left[\begin{array}{ccc}
       C_{2}C_{1} & C_{2}S_{1}& -S_{2}\\
       S_{3}S_{2}C_{1}-C_{3}S_{1}& S_{3}S_{2}S_{1}+C_{3}C_{1} & S_{3}C_{2}\\
       C_{3}S_{2}C_{1}+S_{3}S_{1}& C_{3}S_{2}S_{1}-S_{3}C_{1}& C_{3}C_{2}
     \end{array}\right]
\end{split}
\end{equation}
where element $C_{2}$ denotes $\cos (\theta_{2})$, and others are similarly defined.
When Gimbal lock occurs, i.e., $\theta_{2}=\pm 90^{\circ}$, we have $C_{2}=0$ and $S_{2}=\pm 1$, then the DCM (\ref{ypr-1}) becomes
\begin{equation}\label{ypr-2}
\begin{split}
[C]&=\left[\begin{array}{ccc}
       0 & 0& \mp1\\
       \pm S_{3}C_{1}-C_{3}S_{1}& \pm S_{3}S_{1}+C_{3}C_{1} & 0\\
       \pm C_{3}C_{1}+S_{3}S_{1}& \pm C_{3}S_{1}-S_{3}C_{1}& 0
     \end{array}\right]  \\
     &= \left[\begin{array}{ccc}
       0 & 0& \mp1\\
       - S(\theta_{1}\mp \theta_{3})& C(\theta_{1}\mp \theta_{3}) & 0\\
       C(\theta_{1}\mp \theta_{3})& - S(\theta_{3}\mp \theta_{1})& 0
     \end{array}\right].
\end{split}
\end{equation}
That is, $C_{13}=\mp1$, $C_{11}=C_{33}=0$, $C_{22}=C(\theta_{1}\mp \theta_{3})$.

For the symmetric Proper Euler angles, we define $(\theta_{1}, \theta_{2}, \theta_{3}):=(\tilde{\psi}, \textcolor[rgb]{0.00,0.00,0.00}{\tilde{\theta}}, \tilde{\phi})$. Then we get the DCM in terms of the symmetric Proper Euler angles, i.e.,
\begin{equation}\label{noypr-1}
\begin{split}
[C]=\left[\begin{array}{ccc}
       C_{3}C_{1}-S_{3}C_{2}S_{1}& C_{3}S_{1}+S_{3}C_{2}C_{1}-& S_{3}S_{2}\\
        -S_{3}C_{1}-C_{3}C_{2}S_{1}&-S_{3}S_{1}+C_{3}C_{2}C_{1} & C_{3}S_{2}\\
        S_{2}S_{1} & -S_{2}C_{1}& C_{2}
     \end{array}\right]
\end{split}
\end{equation}
where element $C_{2}$ denotes $\cos (\theta_{2})$, and others are similarly defined.
For the Gimbal lock, i.e., $\theta_{2}=0^{\circ}~or~180^{\circ}$, we have $S_{2}=0$ and $C_{2}=\pm 1$, then the DCM (\ref{noypr-1}) becomes
\begin{equation}\label{noypr-2}
\begin{split}
[C]&=\left[\begin{array}{ccc}
       C_{3}C_{1}\mp S_{3}S_{1}& C_{3}S_{1}\pm S_{3}C_{1}-& 0\\
        -S_{3}C_{1}\mp C_{3}S_{1}&-S_{3}S_{1}\pm C_{3}C_{1} &0\\
        0 & 0& \pm1
     \end{array}\right]  \\
     &= \left[\begin{array}{ccc}
       C(\theta_{1}\pm \theta_{3})&  S(\theta_{3}\pm \theta_{1})& 0\\
        - S(\theta_{3}\pm\theta_{1})& -C(\theta_{1}\mp \theta_{3}) & 0\\
       0 & 0& \mp1\\
     \end{array}\right].
\end{split}
\end{equation}
That is, $C_{11}=C(\theta_{1}\pm \theta_{3})$, $C_{22}=-C(\theta_{1}\mp \theta_{3})$, $C_{33}=\mp1$.

\footnotetext [2]{https://blog.csdn.net/AndrewFan/article/details/60981437.}

\begin{figure}[!htb]
\centering
\includegraphics[width=2.8in,height=1.2in]{unGL.eps}
\caption{Gimbal lock and unwinding phenomenon.}\label{unGL}
\end{figure}

Following \cite[(3.73)\&(3.91a)]{2019Ehsan[27]}, we get $\cos\Phi=\frac{1}{2}(C_{11}+C_{22}+C_{33}-1)$, so we do the following analysis using the asymmetric pitch-yaw-roll angles, since (\ref{noypr-2}) is more complicated than (\ref{ypr-2}). When Gimbal lock occurs in terms of (\ref{ypr-2}), we get $\cos\Phi=\frac{1}{2}(C_{11}+C_{22}+C_{33}-1)=\frac{1}{2}(0+C(\theta_{1}\mp \theta_{3})+0-1)=\frac{1}{2}(C(\theta_{1}\mp \theta_{3})-1)$. Then we do the following analysis:\\
\indent (a) if $C_{22}=1$, i.e., $\{1\}\subset C(\theta_{1}\mp \theta_{3})$ (like $\theta_{1}-\theta_{3}$=0), we obtain $\cos\Phi=\frac{1}{2}(1-1)=0$, so $\Phi=\frac{\pi}{2}$;\\
\indent (b) if $C_{22}=-1$, i.e., $\{-1\}\subset C(\theta_{1}\mp \theta_{3})$ (like $\theta_{1}+\theta_{3}=\pi$), we obtain $\cos\Phi=\frac{1}{2}(-1-1)=-1$, so $\Phi=\pi$, i.e., $\eta=\cos \frac{\Phi}{2}=0$.
In summary, Gimbal lock includes the case where $\eta=0$. However, Gimbal lock does not includes the cases where $\eta=\pm1$, as shown in the following analysis\footnotemark[3]:\\
\indent (c) when $\eta=\cos \frac{\Phi}{2}=\pm1$, then $\Phi=0~or~2\pi$, i.e., $\cos\Phi=1$, since $\cos\Phi=\frac{1}{2}(C_{11}+C_{22}+C_{33}-1)=\frac{1}{2}(0+C(\theta_{1}\mp \theta_{3})+0-1)=\frac{1}{2}(C(\theta_{1}\mp \theta_{3})-1)$, then it should hold $1=\frac{1}{2}(C(\theta_{1}\mp \theta_{3})-1)$, i.e., $C(\theta_{1}\mp \theta_{3})$ should be 3, which is impossible, resulting in a paradox.

As a result, Gimbal lock (i.e., when the plane's head is straight up or straight down as shown in Fig. \ref{GB}(b), corresponding to ``$\hbox{pitch}=\theta=\pm 90^{\circ}$'' or ``$\hbox{inclination}=\tilde{\theta}=0^{\circ}~or~180^{\circ}$'') includes the case where $\eta=0$, not including the cases where $\eta=\pm1$, as shown in Fig. \ref{unGL}. This is reasonable, since when the plane's head is straight up or straight down, we could only ensure the second rotation angle $\theta=\pm 90^{\circ}$ or $\tilde{\theta}=0^{\circ}~or~180^{\circ}$, but we could not ensure the first and third rotation angles, so that Gimbal lock may include many combinations of attitude angles.

\footnotetext [3]{Gimbal lock does not includes the cases where $\eta=\pm1$ when we do analysis using the symmetric Proper Euler angles, since Gimbal lock is the same singularity for all set of Euler angles, i.e., when the plane's head is straight up or straight down.}

As analyzed above, the essence of Gimbal lock is that we cannot control the attitude state where the pitch angle approaches to
$\pm 90^{\circ}$ (north/south pole) in the asymmetric Tait-Bryan angles. Fortunately, unite quaternions representation can overcome the difficulty by supporting spherical linear interpolation. However, because the covering is double ($q$ and $-q$ map to the same rotation), the rotation path may turn either the ``short way'' (less than $180^{\circ}$) or the ``long way'' (more than $180^{\circ}$), see Fig. \ref{shortest}. As a result, unwinding phenomenon arises, corresponding the cases where $\eta=\pm1$. To summarize, the unwinding phenomenon is caused when we ignore the two closed-loop equilibrium points $\mathcal {M}_{0}=\{q\in \mathbb{S}^{3}:\eta=\pm1\}$. That is, even when the Gimbal lock problem does not occur using unit quaternions, unwinding phenomenon follows. Therefore, in existing literature, they usually restrict the rotation angle to be in $[-\pi, \pi)$ and thus obtain the almost global attitude control for example in \cite{30abd}. Another method is applying the switching logic variable in order to derive the global attitude control, where both of the two equilibrium points $\eta=\pm1$ is globally asymptotically stable.

\begin{figure}[!htb]
\centering
\includegraphics[width=2.8in,height=2.2in]{shortest.eps}
\caption{Unit constraint sphere surface, where a point denotes a specific orientation and two antipodes ($q_{1}$ and $-q_{1}$) denote the same orientation. When $q_{2}$ goes to unstable saddle point $-q_{1}$ on $\mathbb{S}^{3}$, then it is close to the physical equilibrium on manifold $SO(3)$; however, it will leave the physical equilibrium on manifold $SO(3)$ and then come back the physical equilibrium, since it will leave the unstable saddle point $-q_{1}$ to the stable equilibrium point $q_{1}$ on $\mathbb{S}^{3}$.}\label{shortest}
\end{figure}

\subsection{(Modified) Rodrigues parameters}\label{section3.5}

Rodrigues parameters and modified Rodrigues parameters are also three-parameter representations derived from unit quaternions denoted as $q=(\eta, \epsilon^{T})\in \mathbb{S}^{3}$ where $\epsilon\in \mathbb{R}^{3}$. So Rodrigues parameters or modified Rodrigues parameters are also minimal. Similar to Eluer angles, the three-parameter sets of (modified) Rodrigues parameters can be regarded as certain embedded subsets of Euclidean space $\mathbb{R}^{3}$, so that one can apply the analysis methods suited for $\mathbb{R}^{3}$.

According to \cite[pg. 20]{abd2013shu}, we present the Rodrigues vector
\begin{equation}\label{MRP1}
\begin{split}
\rho:=\frac{1}{\eta} \epsilon=\vec{k} \tan(\textcolor[rgb]{0.00,0.00,0.00}{\Phi}/2)
\end{split}
\end{equation}
where $\vec{k}\in \mathbb{R}^{3}$ is an arbitrary unit-length vector and $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}$ is the rotation angle. We call the three elements of $\rho$ the Rodrigues parameters.

As a minimal 3-parameter representation of attitude, rotations $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}=\pm \pi$ do not make sense in (\ref{MRP1}). Hence, Rodrigues vectors could not represent rotation angles $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}=\pm \pi$, corresponding to $\eta=0$ in terms of the analysis on connections between unwinding phenomenon and Gimbal lock in Section \ref{section3.2}. It can be checked from (\ref{MRP1}) that the small angle behavior of Rodrigues parameters could be more linear than that of any Euler angle representation, since $\tan(\textcolor[rgb]{0.00,0.00,0.00}{\Phi}/2)\approx \textcolor[rgb]{0.00,0.00,0.00}{\Phi}/2$ for small angle $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}/2$. So linearizing Rodrigues parameters could nicely remove from the singularities $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}=\pm \pi$.

Next, we present a modified representation called modified Rodrigues parameters \cite[pg. 20]{abd2013shu}, i.e.,
\begin{equation}\label{MRP2}
\begin{split}
\bar{\rho}:=\frac{1}{1+\eta} \epsilon=\vec{k} \tan(\textcolor[rgb]{0.00,0.00,0.00}{\Phi}/4),
\end{split}
\end{equation}
where $\vec{k}\in \mathbb{R}^{3}$ is an arbitrary unit-length vector and $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}$ is the rotation angle. We call the three elements of $\rho$ the Rodrigues parameters.

As a minimal 3-parameter representation of attitude, rotations $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}=\pm 2\pi$ corresponding to $\eta=-1$ do not make sense in (\ref{MRP2}). Hence, modified Rodrigues vectors could not represent rotation angles $\textcolor[rgb]{0.00,0.00,0.00}{\Phi}=\pm 2\pi$ and/or $\eta=-1$. That is, the
singularity in (\ref{MRP2}) has moved to $\pm 2\pi$ from $\pm \pi$ in the Rodrigues vector (\ref{MRP1}). So, except the complete rotation from the current orientation to the original orientation, modified Rodrigues parameters could describe any rotation angle.

\subsection{Comparisons}\label{section3.6}

As analyzed in the former subsections, each attitude representation has its strengths and weaknesses
compared to the others:

--As shown in Table \ref{unique}, 3-parameter representation includes Euler angles and (modified) Rodrigues parameters, which could neither globally nor uniquely define the attitude rotations. According to the analysis in Section \ref{section3.2} for Euler angles, the geometric singularity occurs at certain specific values of the second Euler rotation angle (pitch angle at $\pm 90^{\circ}$ and/or inclination at angle $0^{\circ}$ or $180^{\circ}$), so the Euler angles representation is suitable for vehicles which do not need the pitch angles (i.e., no vertical maneuvers), including land vehicles and ships \cite{Dobook}. Also, as analyzed in Section \ref{section3.5}, (modified) Rodrigues parameters, which are derived from unit quaternion, could provide a continuous and unique attitude rotation if we limit rotation angles between $-\pi$ and $\pi$ for Rodrigues parameters and between $0$ and $2\pi$ for modified Rodrigues parameters (i.e., rotation angle is less than $2\pi$), respectively. If there is no constraint on rotation angles, then geometrical singularity occurs at ($\pm2\pi$) $\pm\pi$ for (modified) Rodrigues parameters.

--As shown in Table \ref{unique}, 4-parameter representation includes quaternion and axis-angle, which could globally not uniquely define the attitude rotations. According to the analysis in Section \ref{section3.4} with unit quaternions significant advantages, unit quaternions are often used to represent the attitude on $SO(3)$ since the computational burden can be greatly reduced compared with rotation matrix.
However, the once-redundant unit-quaternion representation needs 4 parameters, which is an over-parameterization of the boundaryless compact manifold $SO(3)$, so that the transformation $q\mapsto SO(3)$ is a double covering map (i.e., two-to-one map). Therefore, there exist two equilibrium solutions in the unit quaternion space for each equilibrium solution on $SO(3)$, one is stable equilibrium solution while the other is unstable saddle point, causing unwinding phenomenon. More specifically, when trajectories starts near a desired
equilibrium solution on manifold $SO(3)$, then it may diverge on $SO(3)$ if the corresponding equilibrium solution on the quaternion space is the unstable saddle point, then it comes back to the same equilibrium solution on manifold $SO(3)$ after traveling a large distance.
According to the analysis on axis-angle in Section \ref{section3.3}, we know that axis-angle representation is similar to unit quaternion, i.e., globally but not uniquely, with similar constraints of double covering map.

--In contrast, rotation matrix could provide a global and unique representation for all attitudes without any singularity. However, the 9 elements of the DCM are trigonometric functions, bringing more cumbersome computations, as analyzed in Section \ref{section3.1}. Compared with works for using rotation matrix (directly using the Lie group $SO(3)$) to avoid singularities, there is a weakness using unit quaternions, i.e., non-uniqueness. However, as presented in Table \ref{unique}, attitude rotations necessitate dealing with nine elements of the rotation matrix, and each element includes several
trigonometric functions making it harder to deal with. Therefore, as a global (non-singular) parameterization equivalent to rotation matrix, unit quaternion is simpler to deal with, compared with rotation matrix. Refer to Fig. \ref{r6} for the connections among those attitude representations.

\begin{figure*
\centering
\includegraphics[width=6in,height=3.3in]{relationship.eps}
\caption{Comparisons among the six kinds of attitude representations in Table \ref{unique}, where ``DCM'' denotes direction cosine matrix and ``KDE'' denotes kinematic differential equation.}\label{r6}
\end{figure*}
\begin{figure
\centering
\includegraphics[width=3in,height=2.4in]{3style.eps}
\caption{Axis angle $\Phi$ and unit quaternion $q=(\eta, \epsilon)=(\cos \frac{\Phi}{2},\vec{k}\sin \frac{\Phi}{2})$: continuous control, discrete control and hybrid control. For $\eta=1$, we have $\Phi\in \{4n\pi\}=\{0, 4\pi, 8\pi, 12\pi, \ldots\}$ with $n=1,2,3,\ldots$; for $\eta=0$, we have $\Phi\in\{(1+4n)\pi, (3+4n)\pi\}=\{\pi, 3\pi, 5\pi, 7\pi, 9\pi, \ldots\}$; for $\eta=-1$, we have $\Phi\in \{2(2n+1)\pi\}=\{2\pi, 6\pi, 10\pi, \ldots\}$.}\label{3style}
\end{figure}

\section{Attitude Control }\label{section4.0}
This section analyzes unit quaternion based control laws and DCM based control laws for (almost) global control in Section \ref{section4.1}, \ref{section4.2} and \ref{section4.3} respectively, as well as robustness analysis in Section \ref{section4.4} and technique tools' analysis in Section \ref{section4.5}.

\subsection{Unit quaternion based control laws}\label{section4.1}

The non-uniqueness issue of unit quaternions: \\
\indent --Essential reason of non-uniqueness: Non-uniqueness is caused by the fact that the boundaryless
compact manifold $SO(3)$ is not diffeomorphic to any Euclidean space, so that the map $\mathcal {R}: \mathbb{S}^{3}\rightarrow SO(3)$ is double covering, i.e., $\mathcal {R}(-q)=\mathcal {R}(q)$ for any $q\in \mathbb{S}^{3}$. \\
\indent --Solution: Solving the non-uniqueness by considering the double covering issue and thus adopting the switching control law as in \cite{2011tac}, in order to avoid the unwinding phenomenon and realize the global attitude control.

Consider the case where $q=(\eta, \epsilon)=(\cos \frac{\Phi}{2},\vec{k}\sin \frac{\Phi}{2})\in \mathbb{S}^{3}$ for given axis-angle parameter $\{\Phi, \vec{k}\}$, with the two closed-loop equilibrium points $\mathcal {M}_{0}=\{q\in \mathbb{S}^{3}:\eta=\pm1\}$.

\emph{Unit-quaternion-based control for the kinematics: continuous control, discrete control and hybrid control.}

1) Continuous control: robustly but not globally controlled.\\
\indent --Consider the continuous controller $\omega=-\epsilon$, used to stabilize the equilibrium point $q=(1,0,0,0)$, then another equilibrium point $q=(-1,0,0,0)$ is unstable saddle point, i.e., the attitude is not globally stabilized. When $\Phi\rightarrow 2\pi$ clockwise as shown in Fig. \ref{3style}, it will suddenly rotate reversely from certain point $\textbf{a}$ nearby $2\pi$, tend to point $\textbf{b}$ nearby $\Phi=4\pi$ and finally be stabilized at stable equilibrium point $q=(1, 0,0,0)$ with $\Phi=4\pi$, i.e., unwinding phenomenon occurs. That is, for $\Phi\in\{2\pi, 6\pi, 10\pi, \ldots\}$, it will unnecessarily rotate $2\pi$ reversely and finally be stabilized at the stable equilibrium point $q=(1,0,0,0)$ , i.e., causing unwinding phenomenon as shown in Fig. \ref{3style}.

2) Discrete control: neither robustly nor globally controlled (almost globally).\\
\indent --Consider the discrete controller $\omega=-\eta\epsilon$, i.e., $\omega=-\epsilon$ is to stabilize the equilibrium point $q=(1,0,0,0)$ while $\omega=\epsilon$ is to stabilize the equilibrium point $q=(-1,0,0,0)$. When $\Phi\rightarrow 2\pi$ clockwise then it will cross certain point $\textbf{a}$ nearby $2\pi$ as shown in Fig. \ref{3style} and finally be stabilized using $\omega=-\epsilon$ at stable equilibrium point $q=(-1, 0,0,0)$ with $\Phi=2\pi$; When $\Phi\rightarrow 4\pi$ clockwise then it will cross certain point $\textbf{a}$ nearby $4\pi$ and finally be stabilized using $\omega=\epsilon$ at stable equilibrium point $q=(1, 0,0,0)$ with $\Phi=4\pi$. As a result, even if the state arrives around $\textbf{a}$, it will not rotate reversely, thus no unwinding phenomenon occurs. However, if the initial state starts from $\mathcal {M}=\{q\in \mathbb{S}^{3}:\eta=0\}$ (i.e., a nowhere dense set of zero Lebesgue measure) for $\Phi\in \{\pi, 3\pi, 5\pi, \ldots\}$ as shown in Fig. \ref{3style}, then it will take arbitrarily long time to the two closed-loop equilibrium points $\mathcal {M}_{0}=\{q\in \mathbb{S}^{3}:\eta=\pm1\}$, and even never achieve at $\mathcal {M}_{0}$. Consequently, the attitude is almost globally stabilized, not globally stabilized. For example, the controller $\omega=-\epsilon$ switches to $\omega=\epsilon$ at $\Phi=\pi$, i.e., the initial state for the equilibrium point $q=(-1, 0,0,0)$ is actually $\mathcal {M}=\{q\in \mathbb{S}^{3}:\eta=0\}$. Hence, every time when $h$ switches, then $\mathcal {M}=\{q\in \mathbb{S}^{3}:\eta=0\}$ is actually the initial state for the next stabilization round, i.e., every time $h$ switches, it will take arbitrarily long time to achieve the next stable state. Moreover, at $\Phi\in\{\pi, 3\pi, 5\pi, 7\pi, 9\pi, \ldots\}$ as shown in Fig. \ref{3style}, switching controller between $\omega=-\epsilon$ and $\omega=\epsilon$ is not robust, i.e., any noise may affect the measurement of $\eta$ causing chattering at the discontinuity.

3) Hybrid control: robustly and globally controlled.\\
\indent --Consider the discrete controller $\omega=-h\epsilon$ where $h$ is reset at $h\eta=-\delta$ for constant $0<\delta<1$, i.e., $\omega=-\epsilon$ is to stabilize the equilibrium point $q=(1,0,0,0)$ where $h=-1$ switches to $h=1$ at certain point $\textbf{d}_{1}$ for $\eta=(\geq)\delta$ as shown in Fig. \ref{3style}; while $\omega=\epsilon$ is to stabilize the equilibrium point $q=(-1,0,0,0)$ where $h=1$ switches to $h=-1$ at certain point $\textbf{d}_{2}$ for $\eta=(\leq)-\delta$ as shown in Fig. \ref{3style}. As a result, as analyzed in the discrete case, no unwinding phenomenon occurs. Similar to the discrete case where $h$ should switch at $\mathcal {M}=\{q\in \mathbb{S}^{3}:\eta=0\}$ (i.e., the North pole in Fig. \ref{3style}), but no robustness holds. So switching at $h\eta=-\delta$ (i.e., switching $h=-1$ to $h=1$ at $\eta\geq \delta$ and switching $h=1$ to $h=-1$ at $\eta\leq -\delta$) means that the switching point at original points $\mathcal {M}=\{q\in \mathbb{S}^{3}:\eta=0\}$ has been removed to $\textbf{d}$, thus robustness holds. Consequently, the attitude is globally stabilized.

\subsection{DCM based control laws}\label{section4.2}

To realize the robust global attitude control, \cite{2011tac} proposes the binary logic variable based quaternion hybrid control law (transforming the unstable saddle point to the stable equilibrium point), where the hysteresis half-width arranges a tradeoff between the hysteresis-induced inefficiency for avoiding unwinding phenomenon and the robustness to measurement noise. Differently, a class of synergistic potential functions based hybrid control could also overcome the topology constraints on $SO(3)$ (preventing all potential functions from arriving their unstable saddle points) and thus realize the robust global control using (globally and nonsingularly defined) DCM on $SO(3)$ \cite{TL2}, \cite{15abd}, \cite{8.1}, \cite{16abd}, \cite{8.2}.

In the synergistic potential functions based hybrid control, one needs to construct a family of synergistic potential functions usually based on the basic potential function--modified trace function--with the angular warping technique \cite{15abd}, \cite{16abd}, or a class of more intuitive and straightforward expelling configuration error functions \cite{TL2}.
Generally, all the hybrid attitude control laws consist in three term: a proportional term, a derivative term resulting form the angular velocity that provides damping, as well as a feedforward torque. Differently, the proportional term depends on the discretely changing gradient of potential functions in the potential functions based hybrid control \cite{8.1}, \cite{15abd}, while depends on the continuously changing state $\epsilon$ and binary logic variable $h$ \cite{2011tac}. For the proportional term depending on the discrete gradient of potential functions, one also needs to smooth the hybrid controller via applying a dynamic (continuous but nonsmooth) interpolation method between the switching potential-energy terms based on the backstepping method \cite{8.2} or based on the integral action directly on the proportional term \cite{15abd}, in order to produce a continuous torque command. The purpose is to response the potential caveat of the hybrid feedback, i.e., some unmodeled dynamics--flexible structures, sloshing fuel or overwhelm the available actuators--maybe excited by the discontinuities in the commanded torque \cite{8.1}, since the potential function is switched and those control-induced functions are to stabilize a unique desired point. There does not exist such problems in the binary logic variables $h\in\{-1,1\}$ based hybrid control, since there only exist two control paradigms, i.e., both of the two critical points $(1, 0,0,0)$ and $-(1,0,0,0)$ could be stabilized respectively by logic variables $h=1$ and $h=-1$.

In summary, comparisons between different hybrid control laws can be summarized as:
1) All of them deal with robust global attitude control, i.e., overcoming the topology constraints of boundaryless compact manifold $SO(3)$, based on configuration error functions (in \cite{TL2}) or synergistic potential functions (in \cite{15abd}), or binary logic variable based hybrid control (in \cite{2011tac}).
2) \cite{TL2} and \cite{15abd} apply the global and nonsingular rotation matrices without adopting any attitude parametrization. In \cite{TL2} and \cite{15abd}, the switching among synergistic functions implies the global attitude control, since all potential functions are prevented from arriving their unstable saddle points. So constructing a family of synergistic functions is challenging. \cite{2011tac} applies the unit quaternion parametrization, where the computational burden can be greatly reduced and the switching of a logic variable based control law implies the global attitude control (by transforming the unstable saddle point to a stable equilibrium point), but analyzing the robustness of the hybrid control law brings difficulties.

\subsection{Analysis on (almost) global control}\label{section4.3}

\begin{table*}[tbp]
\centering
\begin{tabular}{|c|c|c|}
\hline
Attitude Representation & almost global control\footnotemark[4] & global control \\ \hline

Euler angles  &  ~~  & \cite{10.7[1]} via synthesizing a class of control laws \\ \hline

~ & ~ & \cite{renwei2008shu} restricting the rotation angle to be in $[0, \pi]$ \\
(Modified) Rodrigues parameters  & ~ &  \cite{10.2} via modifying control laws over a set of zero Lebesgue measure\\ \hline

~& ~ & \cite{2011tac, zddtcsI, zddtcns} with binary logic variables\\
Quaternions&  \cite{30abd}, \cite{10.6}   & \cite{31abd} with smooth control (rotation angle of the attitude error is different from $\frac{\pi}{2}$) \\ \hline
Axis-angle & ~~&~ \\  \hline
~ & ~~& \cite{TL9} by switching controllers \\
~ & \cite{TL5, TL10}& \cite{TL1, TL2} with multiple configuration error functions \\
Rotation matrix & \cite{TL1, TL2} with smooth control &  \cite{8.1, 8.2} and \cite{15abd} with a family of synergistic potential functions   \\
~ & ~& \cite{TL11} via globalizing the local stable manifold by the backward flow map \\ \hline
\end{tabular}
 \caption{Properties of attitude representations II.}\label{almost}
\end{table*}

As shown in Table \ref{unique}, only DCM could globally and uniquely represent the attitude. However, this does not mean that the attitude system can achieve global control under rotation matrix (especially when the controller is continuous) such as \cite{TL1, TL2} with smooth control besides \cite{TL5, TL10}, since rotation matrix belongs to Lie group $SO(3)$ whose internal inherent topology (not contractible, boundaryless compact manifold, not diffeomorphic to any Euclidean space) completely excludes the existence of any continuous feedback control in order to achieve global attitude stabilization on $SO(3)$. As well, unit quaternion based continuous control law could not realize the global attitude control, either, such as \cite{30abd}. Hence, the internal inherent topology of $SO(3)$ means that any continuous control using any attitude representation (or directly DCM) is impossible to realize the global attitude stabilization on $SO(3)$.

As a result, no matter which parameter is used, i.e., unit quaternions, DCM or (modified) Rodrigues parameters, designing discontinuous control laws is usually required in order to realize the global control. Moreover, considering that purely discontinuous control laws are usually not robust to any measurements thus causing chattering as analyzed in the coming Section \ref{section4.4}, researchers usually design switching control laws (also called hybrid control laws), such as \cite{2011tac} with binary logic variables using unit quaternions, \cite{TL1, TL2} with multiple
configuration error functions using DCM, \cite{8.1, 8.2} and \cite{15abd} with a family of synergistic potential functions using DCM, \cite{TL9} by switching controllers using DCM where the attraction region for each control mode covers the configuration space almost globally.

However, it is not always the case. Besides the hybrid control laws, the following cases also bring the global control as shown in Table \ref{almost}: the case in \cite{TL11} via globalizing the local stable manifold by the backward
flow map using DCM; the case in \cite{mengziy2019auto} where only one equilibrium is considered using unit quaternion; the case in \cite{31abd} with smooth control using unit quaternions provided that the rotation angle of the attitude error is different from $\pi/2$; the case in \cite{10.5} by restricting control gains for global asymptotic stability proved by the Lyapunov method, where the designed nonlinear control law using unit quaternions is model-independent (moments of inertia is not involved) and gain matrices are symmetric positive definite. If the control gains are not restricted in \cite{10.5}, i.e., for any choice of control gains, then the desired equilibrium could only be locally asymptotically stabilized, instead of the global result. However, there is no systematic synthesis procedure choosing those control gains in such case \cite{10.5}, which could only be done by trial and error. In addition, for the modified Rodrigues parameters representation, modifying these control laws over a set of zero Lebesgue measure is feasible to realize the global asymptotic stabilization control over the whole manifold $SO(3)$, for example using an open-loop strategy applied over an arbitrarily small finite interval \cite{10.2}. As well, the modified Rodrigues parameters representation could realize the global attitude control by restricting the rotation angle to be in $[0, \pi]$ \cite{renwei2008shu}.

\footnotetext [4]{\cite{TL2} Consider an equilibrium of a dynamic system located
at the origin. The equilibrium is almost globally asymptotically stable, if it is asymptotically
stable and almost all trajectories converge to it, i.e., the set of
the initial states that do not asymptotically converge to the origin
has zero Lebesgue measure.}

\subsection{Analysis on robustness}\label{section4.4}

Generally, there exist two kinds of robustness, i.e., one is the nominal robustness with respect to small perturbations (including measurement noises/disturbances) for avoiding chattering, and the other is the general robustness against bounded \emph{unstructured} uncertainties in both the translational dynamics and the rotational dynamics.

For hybrid attitude control to achieve \emph{global} attitude control, a hysteresis-based switching algorithm is often designed where the nominal robustness holds for avoiding chattering like \cite{2011tac}, \cite{TL1}, \cite{TL2}, \cite{TL9}. Generally, for hybrid control systems, nominal robustness holds if the hybrid assumption conditions (i.e., Ass. 6.5 in \cite{2012teel}) are satisfied, like the analysis in \cite{2011tac}. The nominal robustness holds in \cite{10.5} to model errors and parametric uncertainties, since the designed nonlinear control law in \cite{10.5} is model-independent (moments of inertia is not involved) and gain matrices are symmetric positive definite. Despite one of the designed nonlinear control laws \cite{10.7} is also model-independent (i.e., nominal robustness holds), however, the control gains are scalars, not including matrices gains.

As well, the general robustness against \emph{unstructured} uncertainties implies that the tracking/stabilization errors are
uniformly ultimately bounded, for example, \cite{TL1, TL2} and \cite{TL5} apply an integral control term to achieve robust global
exponential/asymptotic stability in the presence of disturbances. In many works, the authors only consider one kind of robustness, for example \cite{2011tac}, \cite{8.1} and \cite{15abd} for nominal robustness, \cite{mengziy2019auto} for general robustness, while in \cite{13-14} and in most works done by \emph{Taeyoung Lee}, both kinds of robustness are simultaneously considered for the global attitude control, like \cite{TL1}, \cite{TL2}, \cite{TL3}, \cite{TL4}, \cite{TL9}.

Different from other works using Euler angles with kinematical singularities like \cite{10.7[1]} or quaternions with the ambiguities (causing the unwinding phenomenon) \cite{2011tac}, \cite{13-14}, the works of \emph{Taeyoung Lee} directly use the special Euclidean group $SO(3)$, which are intrinsic and coordinate-free, thereby completely avoiding any singularities and ambiguities, as well as those works using a family of ``synergistic'' potential functions \cite{8.1} and \cite{15abd} for realizing global attitude control. For the case where the special Euclidean group is used, classical trace functions \cite{TL10} or modified trace functions \cite{8.1} are usually designed as Lyapunov functions to derive the global attitude stability.

\subsection{Analysis on technique tools}\label{section4.5}

\footnotetext [5]{A nonlinear system is said to be autonomous if it does not
depend explicitly on time \cite{TL4barbalat}. }

Not all theoretical methods are available to all situations. For example, the LaSalle's/hybrid invariance principle, which is better applied to aysmptotic stability of certain set points for time-invariant (i.e., autonomous\footnotemark[5]) \cite[Pg. 1370]{10.7[20I]} or periodic equation \cite{10.1}, \cite{10.2}, \cite{2011tac}, \cite{8.1}, not including the time-varying (i.e, non-autonomous) systems \cite{10.4}. Other than LaSalle's/hybrid invariance principle, we may use cross-term-added Lyapunov function or Barbalat's lemma for the time-varying case \cite{TL4}, \cite{TL5}. Comparison principle can be applied into non-autonomous/autonomous cases \cite{2015tangauto},
\cite[Chapter IX]{10.7[20I]}. Matrosov' theorem can be applied in all circumstances \cite{5.2shs[38]}, i.e., time-(in)variant systems, nonperiodic/time-dependent systems, etc. In summary, one may use other methods for non-autonomous systems, for example Barbalat's lemma \cite[Pg. 105]{TL4barbalat}, Matrosov's theorem, comparison principle, etc.

Despite \cite{10.5} and \cite{10.7} both design model-independent quaternion control laws for attitude control systems. However, the former uses the LaSalle's theorem to derive the global asymptotic stability when there is no cross-term in the designed Lyapunov function; while the latter applies Barbalat's theorem to derive the global asymptotic stabilization control where a cross term is added in the designed Lyapunov function. Even if the designed Lyapunov function does not contain any cross terms, Barbalat's lemma is also feasible to derive the stability control for the unit-quaternion based autonomous attitude control systems \cite{30abd}, \cite{31abd}. In fact, there is no need to use LaSalle's invariance principle when the Lyapunov function contains a cross term like \cite{TL2, TL9}. The work in \cite{10.7[20I]} about robotic manipulators further explores why the LaSalle's invariance principle and Barbalat's lemma are not required, when cross-term is added in the designed Lyapunov function. In fact, in non-autonomous systems, there is no need to use the LaSalle's/hybrid invariance principle if one could find a Lyapunov function with strictly negative derivative. However, for the passive/dissipative systems (i.e., attitude systems), one have to apply the LaSalle's/hybrid invariance principle when the designed Lyapunov function is semi-negative, unless adding a cross term in the Lyapunov function \cite{10.7}.

\section{Conclusion and Discussion}\label{section6}
The attitude control problem is presented in this paper. A class of attitude representations are clarified and compared according to the parameter set and geometric/kinematicla singularities, including 9-parameter rotation matrix, 3-parameter Euler angles and (modified) Rodrigues parameters, 4-parameter axis-angle and unit quaternions. According to attitude representations, almost global and global attitude control can be achieved based on different control laws where the nominal and general robustness are analyzed, including continuous control laws (i.e., modifying control laws over a set of zero Lebesgue measure or restricting the rotation angle for the global control), discontinuous control laws without nominal robustness and hybrid control laws (i.e., switching controllers or multiple configuration error functions for the global control), etc. According to different attitude representations, technical tools are presented for deriving the (almost) global stabilization, i.e., the LaSalle's/hybrid invariance principle for aysmptotic stability for time-invariant or periodic equation, cross-term-added Lyapunov function or Barbalat's lemma for time-varying case, comparison principle for non-autonomous/autonomous case, Matrosov' theorem for all circumstance (the time-(in)variant systems, nonperiodic/time-dependent systems).

redpajama_set_name: RedPajamaArXiv


text:
Hans Urs von Balthasar: A Noble Spirit
By J. Peter Pham|2016-12-29T10:44:59-06:00December 17th, 2016|Categories: Books, Christian Humanism, Christianity, Featured, Hans Urs von Balthasar|
Shortly before his death, Hans Urs von Balthasar addressed the question posed by many of those disconcerted by the large number of his books: Where must one start to understand him?…
Tragedy Under Grace: Reinhold Schneider on the Experience of the West, by Hans Urs von Balthasar, (San Francisco: Ignatius Press, 1997)
Hans Urs von Balthasar died twenty-eight years ago, just three days before he was to be inducted into the College of Cardinals, the vindication of his nearly sixty years of prodigious—and often controversial—scholarship. Perhaps Providence decided that the Sacred College needed no more distinguished members. Perhaps Balthasar needed no other honors.
His achievements as a systematic theologian, patristics scholar, spiritual writer, apologist, and translator of the highest rank place him in unique class among twentieth-century scholars. While his voluminous theological works, including some eighty books and hundreds of articles, have, in the years following his death, been taken up in earnest by numerous theologians, his profound excursions into other fields, especially letters, have largely been ignored. And where they have been noted, these texts have, more often than not, not been fully appreciated.
Yet, as T.S. Eliot once noted, "In my beginning is my end." And readers of Balthasar the theologian need to recall that his 1929 doctoral dissertation, which had as a subject the idea of the end of the world in modern German literature from Gotthold Lessing to Ernst Bloch, was in Language and Culture. Judging from his citations, throughout his career Balthasar continued to regard playwrights, poets, and novelists as theological sources as important as the Fathers or the Schoolmen. He was well-read in the literature of half a dozen languages—numbering among his friends men of letters like Albert Beguin, Georges Bernanos, and Reinhold Schneider—and has been called the most cultivated man of his age. Shortly before his death, ever the meticulous scholar, Balthasar addressed the question posed by many of those disconcerted by the large number of his books: Where must one start to understand him? While he advised starting with the trilogy of the Aesthetics (7 volumes), the Theodramatics (5 volumes), and the Theologic (3 volumes), he was quick to warn of the "too many betrayals" which would result if consideration was not given to his works on spirituality and the literary-philosophical works, among which he singled out his writings on Reinhold Schneider whom he lauded for his "genuine prophetic quality." It is therefore of great fortune to readers in the English world that Ignatius Press has made available for the first time Balthasar's Tragedy Under Grace: Reinhold Schneider on the Experience of the West, which the author prepared just before his death in 1988.
As he does frequently in his writings, Balthasar uses the occasion to discuss another author to expound on some of his own reflections, in this case the tragic figure of the twentieth-century German man of letters Reinhold Schneider, whom he single-handedly rescues from obscurity. In his book, following the outlines of Schneider's literary and dramatic career, Balthasar weaves three circles, each consisting of three forms. The first contains the origins, whose "characteristic is to be the most tenaciously retained and most passionately superseded, for the first book of a poet and thinker always remains his destiny." Schneider's (and Balthasar's) path began with Camoes's Portugal to which the Spain of Philip II bespeaks the antithesis. On the basis of both of these is the question raised of the relationship between spiritual and worldly power, between the Prussia of Fichte and that of Frederick II. The second circle opens, spiraling above the first: England's antithetical form is laid upon Portugal's formlessness, permitting the discernment of guilt as a moving force in history. The answer to this is the mystical form of Russia. It is only in the third circle, then, that the form of the supernatural power that sets its mark on history is explicitly brought out. In this circle, the Church is initially displayed as the place of saints, that is, according to Balthasar, of absolute persons who express most fully what human nature is (Joan of Arc is the representative here) and is then described in the crossroads of her earthly existence between power and renunciation, conquest of the world and decline. The place for this is Rome. Finally, Balthasar's eye is turned to that form of Christian existence that was the goal of Schneider's categories from the outset: The holy man of a supernatural form, the religious knight at the border fortress of Marienburg looking toward the East, "clearing the earth in the humility of his Christian mission so that it can bear culture." And this, concludes Balthasar, is "no longer utopia and contradiction, a sin the first of the circles…but the summons of the hour to some individuals who dare confront it without a safety net."
While to go into detail concerning this argument is, to cite an image which first occurs in the legends surrounding Saint Augustine, something like trying to put the Mediterranean into a pail, it would perhaps suffice to outline just one of Balthasar's lyrical arguments in Tragedy Under Grace. He notes that "the decisive word that must be uttered here is tradition," stating that "it is obvious that research cannot abolish faith." Thus, Christian tradition means something quite different from traditionalism. In fact, it "means its exact opposite." He argues that "It is clear that what is commanded in history is something impossible, but it must be done; grace makes the impossible possible. Out of this relationship to the world, the knightly spirit forms itself for a resistance that will demand more than everything." Thus, tradition is the handing on of the gospel and of the fundamental attitude of Christ and of those who follow him vis-a-vis the "earthly power that overwhelms them."
One can also interpret this deepest meaning of tradition, as Schneider understood it and bequeathed it, in another way. Subject to so many stimuli, the Christian tradition became the "catalyzing unification" that provided a criterion for evaluation and discrimination. Herein, according to Balthasar, lies the nobility—"chivalry" is the word preferred by Schneider—that shines throughout the work and life of his friend Schneider, and perhaps it is the reason why his works find almost no readers today. It was Schneider's (and Balthasar's) belief that "There is no republic without an aristocracy." This is not meant absolutely as a social caste so much as a spirit. Both Schneider and his commentator Balthasar concerned themselves only with noble souls since "it is not possible for the masses to become a people." It is typical, Tragedy Under Grace concludes, for the modern period to do nothing for nobility and that "its concept of people remains revolutionary-proletarian: a mass of equal persons who allow themselves to be ruled. In reality, a people is an articulated organism with a natural summit. Where the instinct for the nobility has been extinguished, we no longer find the instinct for order and for the great laws of life."
It is perhaps the finest tribute to Hans Urs von Balthasar that, thanks to his solitary efforts, the literary and philosophical genius of his friend Reinhold Schneider continues to shine forth and that, despite the travails of modern life, the quest for a nobility of spirit continues to embolden the hearts of men.
No Monuments Are Safe
Today no monuments are safe.
Stone basalt marble
impervious once to all but Time
for such legends of our saints our martyrs our heroes
what haven now?
For we have trapped and harnessed
the lightning-bolts.
What if, some innocent day
they tear loose again?
Oh then
stumbling among our shattered icons
we may find
some shards
some charred fragments of pottery
an ancient seal
or a carved stele bearing symbols
we can no longer read.
A montage of messages from a wiser world.
We have heard of Dr. Faustus
who it was reported
and was dragged screaming
down to Hell.
And of Oxymandias
That great King of Kings
three thousand years ago
whose trunkless legs
stand near his shattered still-arrogant face
half-buried beneath the eternal desert sands.
—Louise Dauner
Books on the topic of this essay may be found in The Imaginative Conservative Bookstore. Republished with gracious permission from Modern Age (Summer 1999).
About the Author: J. Peter Pham
J. Peter Pham is Director of the Africa Center at the Atlantic Council in Washington, DC. He previously held an academic appointment as tenured Associate Professor of Justice Studies, Political Science, and Africana Studies at James Madison University in Harrisonburg, Virginia.
The Magi and the Obstinacy of Belief

redpajama_set_name: RedPajamaCommonCrawl


text:
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The Differences between Guardianship and Custody
By: M. Scott Gordon
Sometimes, even a fit, loving parent may have to take a break from caring for their child or children. There may be an illness in the family, or financial hardship. When this happens, there is a way for the children to be taken care of without that parent losing legal custody through guardianship. Through guardianship, a child can be legally placed with another person.
Custody in Chicago
The concept of custody in Chicago and IL most often comes up in regard to divorce. When a couple decides to obtain a divorce, one of the things required is a parenting agreement. Unlike in many other states, however, Illinois courts do not presume that joint custody is in the best interests of the child. No rebuttable presumption exists in regard to custody determinations at the beginning; the court takes into account every piece of evidence as to which parent's custody would be in the best interests of the child. Some of the factors include:
The circumstances of each parent (who can provide a safe, clean home);
The level of emotional attachment between each parent and the child;
The degree to which the child is ensconced in their current school and community (for example, how many friends the child has at their current school, and if they would have to move if custody changed);
The child's preferences, though this is usually not dispositive; and
Many other factors.
A custody agreement will be handed down by a court during a divorce proceeding.
Legal Guardianship
Unlike custody determinations, guardianship is awarded when a parent is unable to take care of their child. It is very common for the guardian to be a relative or close family friend, but this is not required; in fact, it is actually quite easy to be appointed a legal guardian in most cases. If you are over the age of 18, not under guardianship yourself, a U.S. resident, and have no felony convictions, you are eligible to be approved as a guardian.
If you are unable to care for your child, you can have someone you trust file a Petition For Guardianship. If the Court approves the request, then that individual will now be the child's guardian.
What happens if you want to cancel or revoke the guardianship? You can not automatically revoke a guardianship. Rather, you must Petition the Court and request that your child be returned to you and that guardianship be revoked. Generally, a Court will want you to demonstrate that the circumstances which originally caused the guardianship to be awarded (such as inability to care for the child) have been rectified and no longer exist.
Contact An Experienced Guardianship Attorney
It can be advantageous to have plans in place to provide for your children if you become unable to look after them. However, setting up a guardianship to your satisfaction must be done with care. The dedicated Chicago guardianship attorneys at Gordon & Perlut, LLC understand how sensitive these matters are, and will work hard to ensure the best possible outcome for you and your family. Contact us today to schedule an appointment at our Chicago or Skokie offices. We serve the entire Chicagoland area.
Gordon & Perlut, LLC is a Divorce Lawyer and Family Law Firm located in Chicago, IL. Our Chicago office has convenient parking with discounted parking rates with Chicago office visit. Additionally, we offer a suburban office located in Skokie, IL. We Serve clients in Cook, Lake and DuPage County including, Chicago, Evanston, Des Plaines, Mount Prospect, Park Ridge, Elk Grove Village, Glenview, Niles, Prospect Heights, Glencoe, Winnetka, Morton Grove, Northbrook, Wheeling, Arlington Heights, Schiller Park, Palatine, Riverside, Rolling Meadows and Wilmette.
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Natura offers very high quality herbal products - mostly oral care and digestive products - that incorporate certified organic herbs, wild harvested herbs, organic essential oils, and wild-harvested resins without the use of alcohol in any of our herbal extracts. A great deal of thought goes into the efficacy, delicate balance, and harmonious blending of the herbal products. Whenever possible, we use organic herbs and organic essential oils ingredients. It is very rare that an effective herbal extract formulation (or single herb extract) is also pleasant to swallow; however, all Natura formulations have that unique distinction. They are the practical result of years of experimentation and study by Ethnobotanical Research Scientist Dr. Mary Rodio, the Director of Arizona-based Natura Health Services, Inc. Savings are especially apparent in that you don't have to buy various botanicals to provide yourself with an effective formula.


To provide Córdoba with a laboratory that covers the need of experimenting and developing their skills for the graduates of Theatre, Music and Dance.
To put in touch the artists of the different branches of the Performing Arts.
To give the opportunity to artists of different Performing Arts´ fields like Dance, Music, flamenco… to experience working as actors .
To investigate the mechanisms that make possible the fusion between Contemporary Theatre, Flamenco and Poetry.
To reinforce the pedagogical value through the creative process.
To offer a platform for professional actors to retrain their skills investigating the dynamics of Theatre Masters.
To generate performances of high quality original work by the professionals that have participated in the trainings and researches in the Teatro Laboratorio.
To introduce our works in the professional world; Theatre programming, events, festivals and tours in international, national, regional, provincial and local levels.
To offer actors of the rest of the world , who speak English or Spanish as a first or second language, workshops of Contemporary Theatre with Flamenco and Poetry in Córdoba and in their cities.
To rescue the themes, poetry and rituals of our region in the Teatro Laboratorio´s creations.
To facilitate in Córdoba the specialization in Film and TV Acting by offering intensive courses and shortfilms projects.


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Adobe Enterprise Manager (AEM) Assets lets you edit the metadata of multiple assets simultaneously so you can quickly propagate common metadata changes to assets in bulk. You can also edit the metadata for multiple collections in bulk.
To customize the metadata properties page, including adding, modifing, deleting metadata properties, use the Schema editor.
In the Assets console, navigate to the location of the assets you want to edit.
Select the assets for which you want to edit common properties.
Click the View Properties icon to open the properties page for the selected assets.
When you select multiple assets, the lowest common parent form is selected for the assets. In other words, the properties page only displays metadata fields that are common across the properties pages of all the individual assets.
Modify the metadata properties for selected assets. You can choose to modify both basic and advanced metadata.
To view the metadata editor for a specific asset, unselect the remaining assets in the list. The metadata editor fields are populated with the metadata for the particular asset.
In the properties page, you can remove assets from the asset list by unselecting them. The asset list has all the assets selected by default. The metadata for assets that you remove from the list is not updated.
At the top of assets list, select the checkbox adjacent to Title to toggle between selecting the assets and clearing the list.
Click Save from the toolbar to save the changes.
To append the new metadata with the existing metadata in fields that contain multiple values, select Apend mode. If you do not select this option, the new metadata replaces the existing metadata in the fields. Click Submit.
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From the Collections console, select the collections you want to edit.
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To view the metadata properties for a specific collection, unselect the remaining collections in the collections list. The metadata editor fields are populated with the metadata for the particular collection.
In the collection properties page, you can remove collections from the list of collections by unselecting them. The collections list has all the collections selected by default. The metadata for collections that you remove is not updated.
At the top of the list, select the checkbox adjacent to Title to toggle between selecting the collections and clearing the list.


text:
Whispers : Oldest and youngest players
Yankees closer Mariano Rivera (42) is introduced before Friday's game in Detroit.(Rick Osentoski/USA Today Sports photo)
Phil Rogers, Chicago Tribune reporter
The Yankees' Mariano Rivera, 43, was the oldest player on an opening-day roster, and the Nationals' Bryce Harper, 20, was the youngest. … It took the stripped-down Marlins 20 innings to score a run off the Nationals' starting pitchers, getting blanked by Stephen Strasburg and Gio Gonzalez before breaking through on Jordan Zimmermann. … The Dodgers are paying Ted Lilly $12 million and don't know what to do with him. He's stuck on the disabled list while they try to find a team that wants him. … The Indians are very encouraged by the progress of Justin Masterson and Ubaldo Jimenez, who were a combined 20-32 last season. … With Alex Rodriguez sidelined, Adam Dunn moved into the fourth spot on the all-time strikeout list on opening day. He entered the weekend 270 behind No. 3 Sammy Sosa, 512 behind Jim Thome and 561 behind all-time leader Reggie Jackson (2,597). … Australian Grant Balfour and Cuban Yoenis Cespedes are the only foreign-born players on the Athletics' roster. The Brewers lead the majors with 14. … Cespedes was the A's 14th different opening-day left fielder in 14 years. … Josh Rutledge was the Rockies' 13th different opening-day second baseman in 13 years. … After going 20-for-32 in stolen-base attempts last season, Andrew McCutchen worked hard on his baserunning in spring training. His 4-for-4 against the Cubs in the first series of the year should have advance scouts on point. … Jose Fernandez, the 20-year-old the Marlins will start Sunday against the Mets, is Cuban born but was in a Tampa high school two years ago. He could be great. … Felix Hernandez gave John Jaso a Rolex watch for catching his perfect game last season. Because Jaso since has been traded to the A's, Hernandez sent it over from the Mariners' clubhouse to the A's clubhouse on opening day and then smiled when Jaso got the A's first hit after Hernandez had retired 10 straight. … The Mariners have catcher Mike Zunino, their first pick in last year's draft, on a fast track. He's in Triple A, handling pitching prospects Danny Hultzen and James Paxton. … Uniform historian Paul Lukas says the Red Sox's Jackie Bradley Jr. has become the first big leaguer to wear the "Jr.'' on his jersey.
'I feel like I'm built to do shows in an environment like this': Laurence Holmes zigs when others zag — and it makes for must-listen sports radio
WSCR bosses talk risk-reward of rehiring Dan McNeil, diversity, politics and more
Column: Drivers, don't start your engines. NASCAR, put your money where your mouth is and cancel today's race after the noose incident involving Bubba Wallace.


A book launch for the new book 'Kazan, baran i dastarkhan' by Stalik Khankishiyev – about Azerbaijani cuisine - was held in Moscow with support from BAKU Magazine.
Leyla Aliyeva - The Head of the Russia Representation of Heydar Aliyev Foundation, Editor-in-Chief of 'Baku' magazine - as well Arzu Aliyeva, Feodor Bondarchuk, Yulia and Mikhail Gusman, Leonid Portnoy, and Ambassadors accredited to Russia were present in the ceremony.
Devoted by charcoal and grill-master Khankishiyev to the rich cuisine of Azerbaijan, the book is all about dishes cooked over an open fire. Seasoned with views from the author's Azerbaijani and foreigner friends about Azerbaijan's delicious food, the book stresses to make a table beautiful and unforgettable every meal should be covered with a special taste and fantasy. "Everyone has talented hands" the author says, "but not everyone puts them to use".
Every dish in the rich cuisine of Azerbaijan has a specific and unique taste, says Khankishiyev - including kebabs cooked over hot coals. The author says this is just the first volume of his book - the second one, which will also be dedicated mainly to Azerbaijani cuisine, will be coming out soon.
There was unanimous praise for Leyla Aliyeva's dextrous organisation of the event.
Another attendee was well-known Mossovet Theater actor Alexander Lenkov. "On a dull cloudy day BAKU Magazine roused our spirits by organizing this wonderful, sunny evening. It's really noticeable that ll events concerning the amazing nation of Azerbaijan are arranged on such a magnificent level.
Much-followed popular singer Diana Gurskaya said Mrs Leyla Aliyeva "is a really great person who achieves really projects, and supports worthwhile initiatives. We would do with more people like her these days!".
The book launch for "Kazan, baran i dastarkhan" went off with a bang, not least due to the support from BAKU Magazine – and made a really big splash for Azebaijan's delicious cuisine on tables around the world.


After a week of rain I cherished every moment the sun was out! Ian and I went on evening bike rides, strolled around the farmers market and checked out a new winery. We are continually exploring our surroundings – even after a year of living here! This weekend we head back to Milwaukee for Ian's high school reunion and to catch up with old friends.


This document describes AI's concerns about a number of countries and issues relevant to the agenda of the forthcoming session of the UN Commission on Human Rights. Specific country situations addressed are: Afghanistan, Bougainville, Cuba, East Timor, El Salvador, Equatorial Guinea, Guatemala, Haiti, Iran, Iraq, Israeli-occupied Territories, Myanmar, South Africa, Sudan, Togo, Former Yugoslavia, Zaire, Albania, Cambodia, Georgia, Romania, Somalia, Sri Lanka, Tadzhikistan, Algeria, China (incl. Tibet), Peru and Turkey. Specific issues discussed are: human rights and mass exoduses, women, street children, protection of human rights defenders and promotion of human rights instruments. A bibliography of relevant AI documents is appended.


The Forest Echoes Cabins offers three individual cabins in a quiet, friendly atmosphere 1/2 km from Cultus Lake on private forested acreage surrounded by Provincial Parkland.
Owner/operators Ellie, Gary and family invite you to enjoy a unique stay at their clean and friendly accommodations. The quiet, secluded, natural setting is perfect for couples and families. The cabins are only minutes from the lake (1/2 km), from one of the main Provincial Park trailheads (right off our road), and from the Cultus Lake Waterslides (2 km) - giving visitors easy access to all the area has to offer.
To read more about the cabins and view photos of their interiors see "The Cabins" page.
For rates and booking information see the Rates page.
To see more photos of Forest Echoes, the Lake, and surrounding area see one of the Photo Gallery pages.
If you wish to be informed about upcoming specials or interesting local events taking place please sign up on the mailing list page.
If you need more information or want to inquire about availability please visit the contact page.


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John Stamos Quotes
John Stamos quotes on life, family, entertainment, humor, positivity, love, and more.
26 Interesting Quotes By John Stamos
Born In: Cypress, California, U.S.
John Stamos is an American actor, singer, musician, and producer. He is best known for playing 'Jesse Katsopolis' in the 90s sitcom 'Full House.' His portrayal of 'Blackie Parrish' on the TV series 'General Hospital' earned him a 'Daytime Emmy Award' nomination in 1983. He has starred in many TV shows including 'You Again' (1986), 'Thieves' (2001), 'Jake in Progress' (2005-06), 'ER' (2005-09), and 'Grandfathered' (2015-16). He reprised the role of 'Jesse' in 'Fuller House' in 2016. He also has jewelry line 'St. Amos Jewelry,' the proceeds of which goes to the 'Childhelp' foundation. Known for his charm and positivity, John often shares his outlook and ideas about life. We have curated a few of his quotes on life, family, entertainment, humor, positivity, and love.
When I was 16 or 17, I remember kissing one of my first girlfriends, Kim Anderson, under a stairwell at Disneyland. I'll never forget that feeling.
This is an old family secret, and I just found this out recently, and it almost broke my heart. My mother said to me, 'I had never told you this, but God, you were an ugly baby'.
Me God Family Heart
Mary-Kate and Ashley, who played Michelle, were great. I miss them, I love them, and I need to borrow some money from them.
Love Need
In the last few years, losing my father, going through a divorce and not getting some jobs I really wanted, is making me a much more interesting person, I think. This all really does feel like a rebirth, a new chapter.
Me Think
I'm happier in my career now than I've ever been.
I'm finally looking older and inviting my wrinkles.
I went to Broadway and I've been doing some fun guest spots with 'Entourage' and 'Glee' and I'm ready to have my own show.
I started Pilates. I'm the only guy in there. They plot before I get there: 'How can we make John look ridiculous?' Because every exercise involved my legs up, like I'm in the stirrups or something.
I mean you pull the curtain away, and you see I'm just as insecure and neurotic and scared and vulnerable as anybody, you know.
I have never been so calculating as to sing some Barry White song to get a girl. But I do think it's very romantic to cook dinner and sit around the piano at night and sing together.
I have a trainer that I box with. Luckily, on ER, they'll tell me if I have a shirtless scene coming up and I'll have a few weeks to power it out.
I grew up doing sitcoms and theater and even playing with the Beach Boys, where you're programmed to perform, your body gets into a rhythm and you know it has to perform.
I got cast on ER, I knew I'd be playing a great character and I knew the show was great.
I don't want to replace anyone, especially Charlie Sheen.
For so many years, fans and friends have been wanting me to succeed and be back on TV every week, which hasn't happened since 'Full House.' I feel like I came through for them.
Every couple years, you know, these great things drop in my lap. It's been fantastic.
I once dated someone who worked at McDonald's. She came up and asked if I wanted a Big Mac.
I like to sing with people, whether they're good or not.
I do know I want to have children. That's the only thing missing in my life. I certainly thought I'd have some by now.
I like when guests come over early and we chop veggies and talk and play music.
Like Music
I'm not the guy who bursts into the room. I'm the most insecure person you'll meet if you get to know me.
You Me
I've had a fairy tale life. I had a perfect family, a beautiful childhood, an incredible upbringing. I lived a lot of life but a lot of good life.
Life Family
I've been been on the cover of TV Guide, on every single talk and entertainment show except Letterman. It's interesting being older and dealing with this kind of success. I'm more appreciative of it now, and I don't take it for granted.
I also find it interesting that a lot of people in their 30s are not married and don't have kids. There are a lot of people in this age bracket that are out there dating and trying to find love. And I never thought that at my age I would be.
People look at me and go, 'You must have it made. You have girls. You have a great life.' It's not true. I mean you pull the curtain away, and you see I'm just as insecure and neurotic and scared and vulnerable as anybody, you know.
Joining 'ER,' I felt like that kid who got the golden ticket in 'Charlie and the Chocolate Factory.' I've been offered chocolate bars all these years, but there had been no golden ticket. Just the stomachache that was called 'Jake in Progress.'
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