text:
\section{Introduction}
	\IEEEPARstart{T}{he} Furuta pendulum (also known as a rotary inverted pendulum) is an under-actuated system, which is formed by two bodies in series. The first one is rotating around a vertical axis followed by a pendulum rotating around a horizontal axis. It was first proposed by Furuta \cite{Furuta_1992} mainly to test different control laws \cite{1999}.  The Furuta pendulum is mainly used as a testbed of nonlinear control strategies and is also of educational value. Many of the works in the literature use the Furuta pendulum to illustrate a proposed control law \cite{1999,Furuta_1992,Gafvert,Wadi18}. However, these works did not elaborate on the dynamic model of the Furuta pendulum and they limited their work to a simplified model \cite{Jadlovsk__2013,Gar_a_Alar_on_2012}.

	Few papers dealt with the modeling of the Furuta pendulum. They highlighted the complexity of the dynamic model and the necessity to simplify it based on several assumptions. The authors of \cite{Furuta_1992} and \cite{Cazzolato_2011} proposed a linearized model based on a small angle assumption. Others neglected one or more of the cross coupling terms relating the two rotations \cite{Furuta_1992,Gafvert,Gar_a_Alar_on_2012, Antonio_Cruz_2015,Antonio_Cruz_2014}. The authors of \cite{Cazzolato_2011} showed, through simulations, that the aforementioned assumption is not acceptable and it could have an important effect on the dynamic response. The authors of \cite{Antonio_Cruz_2015} derived a significantly simplified system and carried a brief three-second test. All the experimental works on the Furuta pendulum are directed at testing various control strategies \cite{Aguilar_2008,Freidovich_2007}. The nonlinear dynamic behavior of the system, which includes Coriolis and centrifugal forces in the two rotating frames of motion, is often overlooked or linearized. However, the knowledge of the dynamic behavior of a system is highly desirable to design advanced nonlinear controllers for a given nonlinear system. In this paper, the developed Furuta pendulum model, which encompasses the aforementioned dynamics, is validated experimentally.

	The problem of identifying the different parameters of the Furuta pendulum is also treated in the literature \cite{Gar_a_Alar_on_2012, Schlee,Habib_2015}. Most of the authors dealing with this problem mentioned the high complexity of the model, especially the friction effects arising from the wide range of velocities the links go through. The parameters representing the dry and viscous frictions in the joints are one of the most difficult parameters to identify \cite{Wadi2017a,Antonio_Cruz_2015,Drewniak_2009, Makkar,Crisco_2007,Teixeira_2013, Fang_2012, Freidovich_2006}. Indeed, there is no universal model to represent damping, and the literature contains several models that attempt to capture the physics of this complex phenomenon \cite{Pennestr__2015,Keck_2017,Marques_2018}.

	When the Furuta pendulum is used in a passive mode, the gravity, which is the only external load, actuates only the second joint. In this case, the Furuta pendulum does not have a stable configuration, which makes any reproducible experimentation nearly impossible. Moreover, this will cause the passive Furuta pendulum to exhibit chaotic behavior such that the motion of both links become heavily dependent and sensitive to initial conditions. This fact could explain the reason of the nonexistence in the literature of dynamic models of a passively actuated Furuta pendulum. The closest work to passive excitation was in \cite{Chenglin_Hu_2009} where friction in a cart pendulum system was identified. Nevertheless, that work decoupled the pendulum from the cart and dealt with them separately in the identification phase. A significant portion of the literature work, attempting parameter estimation, chooses an active excitation of the setup, often alongside a controller that helps stabilize the system. However, a controller might mask system dynamics and thus hinder parameter identification results. Moreover, a dynamic model of the passive system could capture the physics of the system without the perturbation of the input torque. In this paper, to actuate both joints in the passive mode, we propose to tilt the Furuta pendulum. In this case, the Furuta pendulum has a stable equilibrium configuration for both links, which is defined by gravity. The oscillations of the pendulum around this equilibrium could then be studied analytically and experimentally. After tilting, the model reveals a contribution of the gravity in the two generalized forces applied to the two joints. The existence of the stable equilibrium configuration, around which the perturbed system oscillates, is also seen. The developed model includes the effect of the static and dynamic frictions, viscous friction, and the stribeck effect. We propose a novel Universal Adaptive Stabilizer-based model identification algorithm and apply it to estimate the parameters of the friction model in the joints of the Furuta pendulum. The proposed approach is tested in simulation and in experiment, and it is compared against grey box optimization-based parameter estimation. After identification, validation is performed by simulating the system model using the identified parameters and comparing that to the experimental time response of the system. The coefficient of determination, $ R^2 $, is used as a benchmark statistic, and the computation time the algorithm takes to run is also noted for comparison with the optimization-based approach.

	The rest of the paper is organized as follows: after the introduction, Section II introduces the model developed. The identification approach is presented in Section III. The comparison of the results, along with a discussion, are presented in Section IV. Some concluding remarks and future work are presented in Section V.

	\section{Materials and Methods}

	\begin{figure}
		\centering
		\includegraphics[width=8.25 cm]{Figs/11Hardware}
		\caption{The Furuta pendulum in its tilted position.}\label{Hardware}
	\end{figure}
	\begin{figure}
		\centering
		\includegraphics[width=8 cm]{Figs/Schematic}
		\caption{Schematic of the tilted setup}\label{Schematic}
	\end{figure}

		\begin{table*}
			\caption{Parameters of the Furuta Pendulum Model.}
			\label{table:parameters}
			\centering
			\begin{tabular}{lcll}
				\toprule
				{Parameter}&  Unit	& {Arm link}	& {Pendulum link}\\
				\midrule
				Mass &$ [Kg] $						& $ m_1=0.370 $						& $ m_2=0.128 $\\
				Inertia &$ [Kg \, m^2] $			& $ j_{1_z} \,= 3.09\times 10^{-3} $	& $ \begin{matrix}	j_{2_{x}}=5.25\times 10^{-3} \\ j_{2_{y}}=5.25\times 10^{-3} \\ j_{2_{z}}=2.91\times 10^{-6}		\end{matrix} $\\
				Pivot to $ CG $ distance &$ [m] $	& $ l_1\;=OG_1=0.0620 $			&	$ l_2=AG_2=0.0620 $\\
				Total length &$ [m] $				& $ L_1\;=OA=0.216 $			&	$ L_2=AB=0.316 $\\
				\bottomrule
			\end{tabular}
		\end{table*}
		\subsection{The Furuta Pendulum}
		Figure~\ref{Hardware} shows the Furuta pendulum in its tilted position. The angle $ \phi $ is taken around the fixed y-axis, which is pointing out of the page. The obtained tilted reference frame is $x_0-y_0-z_0$, where $z_0$ is now the axis of the first joint between the ground and the arm. The joint is disconnected from the existing motor and the pendulum is subjected only to its weight. Figure~\ref{Schematic} shows the schematic of the Furuta pendulum in its tilted position. The arm is rotating around the fixed axis $z_0$ by an angle $\theta_0$ and the pendulum is rotating with respect to the moving axis $x_1$ by an angle $\theta_1$. The different parameters of the two links are defined in Table~\ref{table:parameters}.

		\subsection{Dynamic Model of the Furuta Pendulum}
		Based on the schematic of the Furuta pendulum, shown in Figure~\ref{Schematic}, the dynamic model is elaborated. The Euler-Lagrange Formulation is used to derive the equations of the motion of the system.  Two generalized coordinates are required to fully describe the dynamics of the two-degrees-of-freedom system, and they are the angular displacements of the Arm and Pendulum links given by

		\begin{equation}\label{q}
			q=\begin{bmatrix}	\theta_0\\\theta_1 \end{bmatrix}
		\end{equation}

		The Euler-Lagrange equation, for the $ i^{th} $ generalized coordinate of the system can be written as follows:

		\begin{equation}\label{EL}
			\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} =Q_i\,
		\end{equation}

		where the kinetic energy, $ T $, is written as the sum of the linear and angular kinetic energies of the two links, which  is given by

		\begin{equation}\label{T}
			T = \sum_{i=1}^{2} \frac{1}{2} m_i {v}_{G_i}^{2} + \sum_{i=1}^{2} \frac{1}{2} m_{i}  {\omega}_{i-1}^{T}({J}_i {\omega}_{i-1}),
		\end{equation}

		where ${\omega}_{0}=\dot{\theta}_0 {z}_0$, ${\omega}_{1}=\dot{\theta}_0 {z}_0+\dot{\theta}_1 {x}_1$ and $ {v}_{G_i} $ is the velocity of $G_i$ ($ i=1 $ for the Arm and $  i=2 $ for the Pendulum).
		The potential energy of gravity is obtained following the scalar product between the global vertical axis and the center of gravity position vector in the local tilted frame of reference.

		\begin{equation}\label{V}
			V = - m_1 g ({z} \cdot {OG_1}) - m_2 g ({z} \cdot {OG_2})
		\end{equation}

		The generalized forces due to friction in the joints are written in \eqref{Qf} as the sum of friction torques acting on the links. This model, proposed in \cite{Brown_2016}, is chosen for numerous reasons. It is versatile in modeling different friction phenomenon, its parameters are physically meaningful and not added on an ad hoc basis, and it is continuously differentiable. The latter two properties are required by the proposed algorithm as upper and lower limits on the estimated parameters steady-state values need to be placed in the algorithm, and the time derivative of the friction term is a necessary part in the algorithm formulation.

		\begin{equation}\label{Qf}
			\begin{split}
				{f}_i(\dot{\theta}_i) =& F_{n_i} \mu_{d_i} tanh \left( 4\;\frac{\dot{\theta}_i}{\dot{\theta}_{t_i}}\right)
				+	F_{n_i}
				\cfrac{(\mu_{s_i} -\mu_{d_i})\; \cfrac{\dot{\theta}_i}{\dot{\theta}_{t_i}} }{ \cfrac{1}{4} \left(\cfrac{\dot{\theta}_i}{\dot{\theta}_{t_i}} \right)^2 + \cfrac{3}{4} }\\
				&+ \mu_{v_i} \dot{\theta}_i tanh \left( 4\;\frac{F_{n_i}}{F_{nt_i}}\right)
			\end{split}
		\end{equation}

		where $ F_{n_i} $ is the normal force in joint $ i $, which is a function of the system parameters as well as the state $ q $ and its derivatives; $ \mu_{d_i} $, $ \mu_{s_i} $, and $ \mu_{v_i} $ are the dynamic, static and viscous friction coefficients; $ \dot{\theta}_{t_i} $ is the transition angular velocity responsible for shaping the stribeck curve; and $ F_{nt_i} $ is the transition force that activates the viscous friction term.

		The final model is given in the following form:

		\begin{equation}\label{Model}
			{H(q)} {\ddot{q}} + {B(q,\dot{q})} + {G(q)}={Q(\dot{q})}
		\end{equation

		where the generalized inertia matrix is given by

		\begin{equation}\label{H}
			{H(q)}= \begin{bmatrix}	\begin{pmatrix}j_{1_z} + m_1l_1^2 \\+ (m_2 l_2^2 + j_{2_y})\; sin(\theta_1)^2 \\+j_{2_z}\; cos(\theta_1)^2\end{pmatrix} & -m_2 l_2 L_1\; cos(\theta_1)\\ \\
				-m_2 l_2 L_1\; cos(\theta_1) & j_{2_x} + m_2 l_2^2\end{bmatrix}
		\end{equation}

		the centrifugal and coriolis forces are given by

		\begin{equation}\label{B}
			{B(q,\dot{q})}=\begin{bmatrix} \begin{pmatrix}-(m_2l_2^2-j_{2_y}+j_{2_x})\;sin(2\theta_1)\;\dot{\theta}_0\; \dot{\theta}_1\\-m_2 l_2 L_1\; sin(\theta_1)\;\dot{\theta}_1^2\end{pmatrix} \\ \\
				(m_2l_2^2-j_{2_y}+j_{2_z})sin(\theta_1)\;cos(\theta_1)\;\dot{\theta}_0^2	\end{bmatrix}
		\end{equation

		the effect of gravity is given by

		\begin{equation}\label{G}
			{G(q)}=\begin{bmatrix} \begin{pmatrix} m_1 g l_1\; sin(\theta_0)\;sin(\phi) \\ \\ +m_2 g \begin{pmatrix} L_1\; sin(\theta_0)\\ -l_2 sin(\theta_1)\;cos(\theta_0) \end{pmatrix}\; sin(\phi) \end{pmatrix} \\ \\
				-m_2 g l_2 \begin{pmatrix}sin(\theta_0)\; cos(\theta_1)\; sin(\phi)\\ - sin(\theta_1)\; cos(\phi)\end{pmatrix}	\end{bmatrix}
		\end{equation}

		and the generalized forces due to friction are given by

		\begin{equation}\label{Q}
			{Q(\dot{q})}=\begin{bmatrix} {f}_0(\dot{\theta}_0) \\ {f}_1(\dot{\theta}_1)	\end{bmatrix}
		\end{equation}

		where the full term is presented in Equation~\eqref{Qf}.

		\subsection{Setup}
		The experimental setup is comprised of a Furuta Pendulum and a Dspace 1104 data acquisition system (Figure~\ref{Hardware}). The system is equipped with two Baumer rotary incremental encoders with a resolution of $ 40,000 $ counts/rev to record the angular positions of the two joints as a function of time. A sampling frequency of $ 10 $ \textit{kHz}  was used to capture the time response of the system. The acquisition time covers the full motion until the links stop moving.

		Now that the setup is explained, the preliminaries required for Universal Adaptive Stabilizer (UAS) based parameters estimation are provided in Section \ref{uinfo}. This is followed by description of the proposed parameters identification framework in Section \ref{Sec:paraID}, and the mathematical justification in Section \ref{mjusti}.

		\section{UAS-based High Gain Adaptive Parameter Identification Routine}
		\subsection{Universal Adaptive Stabilizer (UAS)}\label{uinfo}
		Nussbaum functions are switching functions used in high-gain adaptive control such as with a UAS. They are employed where the control direction is generally unknown. Hence, they are selected as no assumptions and no restrictions on the parameter signs and magnitudes are made. This has been successfully employed in \cite{Ali2017,Usman2019} in estimating the parameters of various other systems. Equation~\eqref{Ni1} shows some examples of Nussbaum functions.

		A Nussbaum function is a piecewise right continuous and locally Lipschitz function $N(.): [k',\infty) \to \mathbb{R}$ that satisfies

		\begin{equation}\label{Nuss}
			\begin{aligned}
				&\sup\limits_{k>k_0} \frac{1}{k-k_0} \int_{k_0}^{k} N(\tau) \,d\tau =+ \infty\\
				&\text{and}\\
				&\inf\limits_{k>k_0} \frac{1}{k-k_0} \int_{k_0}^{k} N(\tau) \,d\tau =- \infty\\
			\end{aligned}
		\end{equation}
		A Mittag-Leffler function was chosen to act as a Nussbaum function for the purpose of tuning the design parameters. It is of interest to note that a Mittag-Leffler function can act as a Nussbaum function under certain conditions that are documented in \cite{li_when_2009}. The Mittag-Leffler function depends on two positive real parameters, $\alpha$ and $\beta$. Authors in \cite{li_when_2009} determined the conditions under which the Mittag-Leffler function acts as a Nussbaum function, which are if $\alpha\in(2,3]$ and $\beta=1$. The Nussbaum function of Mittag-Leffler type is given in \eqref{Ni1} as $N_1(-\lambda z^\alpha)$.

		\begin{equation}\label{Ni1}		N_1(z)=\sum_{\gamma=0}^{\infty} \frac{z^\gamma}{\Gamma(\alpha\gamma+\beta)},\, \alpha\in(2,3],\, \beta=1\end{equation}
		\begin{equation}\label{Ni2}		N_2(z)=z\; cos(\sqrt{|z|})				\end{equation}
		\begin{equation}\label{Ni3}		N_3(z)=z^{2}\; cos(|z|)					\end{equation}
		\begin{equation}\label{Ni4}		N_4(z)=cos(\frac{\pi z}{2})\;e ^{z^{2}}	\end{equation}

		The above functions differ in how fast they vibrate. In other works \cite{Usman2019,Ali2017}, it was advantageous to choose a rapidly vibrating Nussbaum functions in applications involving DC motor and Li-ion Battery Dynamics. For that reason, the Mittag-Leffler Nussbaum function is used in this work.

		\begin{figure*}
			\centering
			\begin{tikzpicture}[auto, node distance=1.75cm,>=latex']
				\node [input] (input){};
				\node [place, below of=input, node distance=2cm] (p1) {};

				\node [block, right of=input, node distance=2cm] (UVP) {Plant};
				\node [block, below of=UVP, node distance=1cm] (UVM) {Model};
				\node [sum, right of=UVP, node distance=2.5cm] (sum2) {};
				\node [block, right of=sum2, node distance=2cm] (int) {$ \int_{0}^{t} ||e||_{2} \; dt$};
				\node [block, right of=int, node distance=3cm] (ML) {$H(\hat{q})N(k) e$};
				\node [block1, above of=ML, node distance=1.5cm] (ParaUP) {$ \int_{0}^{t} ||e||_{2} + \lambda_{u}(z_{u} - z_{Est}) + \lambda_{l}(z_{l} - z_{Est}) \; dt $};

				\node [output, right of=ML, node distance=2.5cm] (output) {};
				\node [place, right of=sum2, node distance=.5cm] (p2) {};
				\node [place, below of=p2, node distance=1cm] (p3) {};
				\node [place, below of=UVM, node distance=0.75cm] (p4) {};
				\node [place, right of=ParaUP, node distance=4cm] (p5) {};
				\node [place, above of=p5, node distance=1cm] (p6) {};
				\node [place, above of=ParaUP, node distance=.5cm] (p7) {};

				\draw [->] (UVM) -| node[pos=0.2] {${q}_{Est}$} (sum2);
				\draw [->] (UVP) -- node[pos=0.5] {${q}_{meas}$} (sum2);
				\draw [->] (sum2) -- node[pos=0.2] {$e$} (int);
				\draw [->] (int) -- node {$k$} (ML);
				\draw [->] (p2) --  (p3) -| node {} (ML);
				\draw [->] (ML) -- node {} (output) |- (p4) -- node[pos=0.25] {$u$} (UVM.south);
				\draw [->] (p2) |- node[pos=0.4] {$e$} (ParaUP);
				\draw [->] (ParaUP.east) -- (p5) |- (p1) |- node[pos=0.8] {$z_{Est}$} (UVM.west);
				\draw [->] (ParaUP.east) -| (p6) -| (p7) -- node[pos=0.1,xshift=-0.8 5cm,yshift=2mm] {$z_{Est}$} (ParaUP.north);
			\end{tikzpicture}
			\caption{Algorithm Flowchart}
			\label{AlgFlow}
		\end{figure*}

		\subsection{Parameter Identification}\label{Sec:paraID}
		The models in Equation~\eqref{Model} and Equation~\eqref{Qf} are rewritten here to setup the parameter identification procedure as the following:

		\begin{equation}\label{EstModel}
			{H(\hat{q})} {\ddot{\hat{q}}} + {B(\hat{q},\dot{\hat{q}})\dot{\hat{q}}} + {G(\hat{q})}
			=
			{\hat{Q}(\dot{\hat{q}})}	+  {u}\\
		\end{equation}
		here $ {\ddot{\hat{q}}, \dot{\hat{q}}, \hat{q}} $ are the estimated  acceleration, velocity and position, respectively, and ${u}$ is the UAS input given by Equation~\eqref{unk}. Also note that the parameters of $ H $, $ B $, and $ G $ in \eqref{EstModel} are known, and $ {\hat{Q}}=\begin{bmatrix}
			{\hat{f}_0(\dot{\hat{\theta}}_0)}&{\hat{f}_1(\dot{\hat{\theta}}_1)}
		\end{bmatrix}^T$ is the estimated friction vector whose entries are given in \eqref{EstQf1}, and \eqref{EstQf2}.

		\begin{equation}\label{EstQf1}
			\begin{split}
				\hat{f}_0(\dot{\hat{\theta}}_0) =& F_{n_0} \hat{z}_1 tanh \left( 4\;\frac{\dot{\hat{\theta}}_0}{\hat{z}_4}\right)
				+	F_{n_0}
				\cfrac{(\hat{z}_2 -\hat{z}_1)\; \cfrac{\dot{\hat{\theta}}_0}{\hat{z}_4} }{ \cfrac{1}{4} \left(\cfrac{\dot{\hat{\theta}}_0}{\hat{z}_4} \right)^2 + \cfrac{3}{4} }\\
				&+ \hat{z}_3 \dot{\hat{\theta}}_0 tanh \left( 4\;\frac{F_{n_0}}{\hat{z}_5}\right)
			\end{split}
		\end{equation}
		\begin{equation}\label{EstQf2}
			\begin{split}
				\hat{f}_1(\dot{\hat{\theta}}_1) =& F_{n_1} \hat{z}_6 tanh \left( 4\;\frac{\dot{\hat{\theta}}_1}{\hat{z}_9}\right)
				+	F_{n_1}
				\cfrac{(\hat{z}_7 -\hat{z}_6)\; \cfrac{\dot{\hat{\theta}}_1}{\hat{z}_9} }{ \cfrac{1}{4} \left(\cfrac{\dot{\hat{\theta}}_1}{\hat{z}_9} \right)^2 + \cfrac{3}{4} }\\
				&+ \hat{z}_8 \dot{\hat{\theta}}_1 tanh \left( 4\;\frac{F_{n_1}}{\hat{z}_{10}}\right)
			\end{split}
		\end{equation}

		In these equations $ \hat{z}_n $ represents the frictional model parameter estimates where $ z_1 = \mu_{d_0} $, $ z_2 = \mu_{s_0} $, $ z_3 = \mu_{v_0} $, $ z_4 = \dot{\theta}_{t_0} $, $ z_5 = F_{nt_0} $, $ z_6 = \mu_{d_1} $, $ z_7 = \mu_{s_1} $, $ z_8 = \mu_{v_1} $, $ z_9 = \dot{\theta}_{t_1} $, $ z_{10} = F_{nt_1} $, the following equations complete the high-gain universal adaptive observer used for parameters estimation in this work.

		\begin{equation}\label{e} e(t)=\dot{q}(t) -	\dot{\hat{q}}(t)  \end{equation}

		\begin{equation}\label{kd} \dot{k}(t)= \left\lVert e(t) \right\rVert^2_2,\; k(t_0)=k_0>0\end{equation}
		\begin{equation}\label{Nk} N(k(t))=N_1(-\lambda k(t)^\alpha) \end{equation}
		\begin{equation}\label{unk} u(t)=	H(\hat{q}) N(k(t))e(t) \end{equation}

		In Equation~\eqref{Nk}, $\lambda = 1$, and $\alpha =3$, thus resulting in a Nussbaum function of the Mittag-Leffler form. In Equation~\eqref{unk} $ H $ is the generalized inertia matrix from \eqref{H}.

		Equation~\eqref{adapt} is used in identifying the friction model parameters $\hat{z}_n$ in Equation~\eqref{EstQf1}, and Equation~\eqref{EstQf2}.

		\begin{equation}\label{adapt}
			\dot{\hat{z}}_n(t)=(\gamma+\lambda_{nu} (z_{nu}-\hat{z}_{n}(t))+\lambda_{nl} (z_{nl}-\hat{z}_{n}(t)) ) \left\lVert e(t) \right\rVert_2
		\end{equation}

		where $\hat{z}_n(t)\in \Bbb R$ represents each parameter that needs to be identified, and $n \in {1, 2, \dots, N}$ refers to the parameter number where $N=10$ is the number of unknown parameters that need to be identified. Also, $\gamma$ is a tuning parameter to control the rate at which the parameters adapt.

		The proposed parameter evolution equation requires steady state upper and lower bounds $z_{nu}$ and $z_{nl}$, as well as their confidence levels $\lambda_{nu}$ and $\lambda_{nl}$, respectively. The parameters $\lambda_{nu},\lambda_{nl}>0$ and the parameters $z_{nu},z_{nl},\lambda_{nu},\lambda_{nl} \in \Bbb R$. The upper and lower bounds $z_{nu}$ and $z_{nl}$ are constants which represent the limits within which $\hat{z}_n(t)$ is desired to settle, as $t \to \infty$. The constants $\lambda_{nu}$ and $\lambda_{nl}$ represent the users confidence in their choice of steady state upper and lower bounds $z_{nu}$ and $z_{nl}$, respectively. The flowchart of the proposed parameter identification routine is presented in Figure~\ref{AlgFlow}.

		\subsection{Mathematical Justification}\label{mjusti}

		This section presents the mathematical justification behind the proposed friction model parameters identification routine. First, Lemma~\ref{adaptiveproof} shows that the adaptive observer's error heading to zero leads to boundedness of the parameters. Second, Theorem~\ref{boundnessproof} proves that the adaptive observer's error is driven to zero by the designed UAS input. Finally, Theorem~\ref{convergeproof} establishes the convergence of the estimated parameters to the true parameters values.

		\begin{lemma}\label{adaptiveproof}
			Let $ \hat{z}_n $ be given as in Equation~\eqref{adapt}. Given that $ \lambda_{nu}, \lambda_{nl}>0 $, \textbf{if} the parameter adaptation proposed in Equation~\eqref{adapt} is used and
			$ e(t)  \to 0 $ as $ t\to \infty $, \textbf{then} $ \hat{z}_n $ is bounded $ \forall t>t_0 $, for $ n \in \{1,2, \dots, 10\} $.
		\end{lemma}
		\begin{proof}[Proof of Lemma~\ref{adaptiveproof}]
			Observe that \eqref{adapt} is a stable Linear Time Invariant (LTI) system driven by the input $ (\gamma+\lambda_{nu} z_{nu}+\lambda_{nl} z_{nl} ) \left\lVert e(t) \right\rVert_2  $ as shown in Equation~\eqref{adapt_expand}.

			\begin{equation}\label{adapt_expand}
				\dot{\hat{z}}_n(t) + (\lambda_{nu}+\lambda_{nl}) \hat{z}_{n}(t)=(\gamma+\lambda_{nu} z_{nu}+\lambda_{nl} z_{nl} ) \left\lVert e(t) \right\rVert_2
			\end{equation}

			The solution of the above equation can be obtained by the properties of the convolution integral to be

			\begin{equation}\label{adapt_solution}
				\begin{split}
					&\hat{z}_n(t)=\hat{z}_n(t_0)e^{-(\lambda_{nu}+\lambda_{nl})t} \\
					&+(\gamma+\lambda_{nu} z_{nu}+\lambda_{nl} z_{nl})\int_{t_0}^{t}  \left\lVert e(t-\tau) \right\rVert_2 e^{-(\lambda_{nu}+\lambda_{nl})\tau} d\tau
				\end{split}
			\end{equation}

			Because $ \hat{z}_n(t),\lambda_{nl},\lambda_{nu},z_{nl},z_{nu} $ are all bounded for $ n \in \{1,2, \dots, 10\} $, and because $ \left\lVert e(t-\tau) \right\rVert_2>0 $ for all $ t $, then Equation~\eqref{adapt_solution} yields $ \hat{z}_n(t) \to \hat{z}_{n_\infty} $ that is bounded. This can be seen from the first term on the right hand of \eqref{adapt_solution} growing smaller as $ t\to\infty $, and following the assumption that $ e\to0 $ as $ t\to\infty $, the second term under the integral is bounded.

		\end{proof}

		With the boundedness of the parameters established in Lemma~\ref{adaptiveproof}, the assumption that the estimation error grows to zero is now proven in Theorem~\ref{boundnessproof}.

		%

		\begin{theorem}\label{boundnessproof}
			Let $ e(t)=\dot{q}(t) -\dot{\hat{q}}(t) $ as in Equation~~\eqref{e}, where the dynamics of $ q(t) $ are given by Equation~\eqref{Qf} --- Equation~\eqref{Q}, and the dynamics of $ \hat{q}(t) $ are given by Equation~\eqref{EstModel} --- \eqref{EstQf2}, $ u $ is as given in Equation~\eqref{unk}. \textbf{If} $H(q(t))$ and $H(\hat{q}(t))$ are invertible for all $t\geq t_0$ \textbf{Then}, $ e(t) \to 0 $ as $ t\to \infty $.
		\end{theorem}
		\begin{proof}[Proof of Theorem~\ref{boundnessproof}]
			Let the system dynamics in Equation~\eqref{Model} and system model in Equation~\eqref{EstModel} be split into the dynamics and the friction counterparts and written as follows
			\begin{equation}
				\label{reDynamicModel}
				\begin{split}
					{\ddot{{q}}}&={H({q})}^{-1} [{{Q}(\dot{{q}})}- {B({q},\dot{{q}})}\dot{{q}} - {G({q})}]	\\
					{\ddot{\hat{q}}}&={H(\hat{q})}^{-1} [{\hat{Q}(\dot{\hat{q}})}- {B(\hat{q},\dot{\hat{q}})}\dot{\hat{q}} - {G(\hat{q})}]+{H(\hat{q})}^{-1}{u}	\\
				\end{split}
			\end{equation}
			where it is understood that all the estimated quantities $ \hat{[\quad ]} $ vary with time, $ {u} $ is as defined in Equation~\eqref{EstModel}, and ${H({q})}^{-1}, {H(\hat{q})}^{-1}$ exist by assumption.

			The error dynamics $ \dot{e} ={\ddot{q}}-{\ddot{\hat{q}}} $ can then be expanded and written as

			\begin{equation}\label{eDynamics1}
				\begin{split}
					\dot{e}&={H}^{-1}{Q}-\hat{H}^{-1}\hat{{Q}} + \hat{H}^{-1}\hat{{B}}\dot{\hat{q}}-{H}^{-1}{B}\dot{q} \\
					&+ \hat{H}^{-1}\hat{{G}}-{H}^{-1}{G}-\hat{H}^{-1}{u}
				\end{split}
			\end{equation}
			where for ease of use we write  $H\equiv H(q), \hat H \equiv H(\hat q), G\equiv G(q), \hat G \equiv G(\hat q), B\equiv B({q},\dot{{q}}), \hat{B}=B(\hat{q},\dot{\hat{q}}), Q\equiv Q(q)$ and $\hat Q = \hat Q(\hat q)$.

			In Equation~\eqref{eDynamics1}, we add and subtract $ \Lambda e $, where $ \Lambda $ is a positive definite design matrix.

			\begin{equation}\label{eDynamics2}
				\begin{split}
					\dot{e}&={H}^{-1}{Q}-\hat{H}^{-1}\hat{{Q}} + \hat{H}^{-1}\hat{{B}}\dot{\hat{q}}-{H}^{-1}{B}\dot{q} \\
					&+ \hat{H}^{-1}\hat{{G}}-{H}^{-1}{G}-\hat{H}^{-1}{u} + \Lambda e - \Lambda e\\
					\dot{e}&={H}^{-1}{Q}-\hat{H}^{-1}\hat{{Q}} + \hat{H}^{-1}\hat{{B}}\dot{\hat{q}}-{H}^{-1}{B}\dot{q} \\
					&+ \hat{H}^{-1}\hat{{G}}-{H}^{-1}{G}-\hat{H}^{-1}{u} + \Lambda \dot{q} - \Lambda \dot{\hat{q}} - \Lambda e\\
					\dot{e}&={H}^{-1}{Q}-\hat{H}^{-1}\hat{{Q}} + \hat{H}^{-1}\hat{{G}}-{H}^{-1}{G} - \hat{H}^{-1}{u} - \Lambda e\\
					&- (\Lambda - \hat{H}^{-1}\hat{{B}})\dot{\hat{q}} + (\Lambda - {H}^{-1}{B})\dot{q}
				\end{split}
			\end{equation}

			The following notation simplifications are applied to Equation~\eqref{eDynamics2}.
			\begin{equation}\label{eDynamics3}
				\begin{split}
					\dot{e}&=\underbrace{{H}^{-1}{Q}-\hat{H}^{-1}\hat{{Q}}}_\mathcal{Q}
					+ \underbrace{\hat{H}^{-1}\hat{{G}}-{H}^{-1}{G}}_\mathcal{G}-\hat{H}^{-1}{u} - \Lambda e\\
					&- \underbrace{(\Lambda - \hat{H}^{-1}\hat{{B}})}_{\mathcal{B}_2}\dot{\hat{q}} + \underbrace{(\Lambda - {H}^{-1}{B})}_{\mathcal{B}_1}\dot{q} \\
					&=\mathcal{Q} + \mathcal{G} + \mathcal{B}_1\dot{q} - \mathcal{B}_2\dot{\hat{q}} -\hat{H}^{-1}{u} - \Lambda e
				\end{split}
			\end{equation}

			In Equation~\eqref{eDynamics3}, we add and subtract $ \mathcal{B}_2 \dot{q}$.

			\begin{equation}\label{eDynamics4}
				\begin{split}
					\dot{e}&=\mathcal{Q} + \mathcal{G} + \mathcal{B}_1\dot{q} - \mathcal{B}_2 \dot{\hat{q}} -\hat{H}^{-1}{u} - \Lambda e + \mathcal{B}_2 \dot{q} - \mathcal{B}_2 \dot{q}\\
					\dot{e}&=\mathcal{Q} + \mathcal{G} + \mathcal{B}_1\dot{q} - \mathcal{B}_2 \dot{q} + \mathcal{B}_2 (\dot{q} - \dot{\hat{q}})  -\hat{H}^{-1}{u} - \Lambda e\\
					\dot{e}&=\mathcal{Q} + \mathcal{G} + (\mathcal{B}_1-\mathcal{B}_2)\dot{q} + \mathcal{B}_2 e - \Lambda e - \hat{H}^{-1}{u} \\
					\dot{e}&=\mathcal{Q} + \mathcal{G} + (\mathcal{B}_1-\mathcal{B}_2)\dot{q} - (\Lambda - \mathcal{B}_2) e - \hat{H}^{-1}{u} \\
				\end{split}
			\end{equation}

			To prove the stability of the error dynamics above, pre-multiply \eqref{eDynamics4} by $ e^T $ to get
			\begin{equation}\label{eDynamics5}
				\begin{split}
					e^T\dot{e}&=e^T\mathcal{Q} + e^T\mathcal{G} + e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} - e^T(\Lambda - \mathcal{B}_2) e \\
					&- e^T\hat{H}^{-1}{u} \\
				\end{split}
			\end{equation}

			Equation~\eqref{eDynamics5} can be rearranged and rewritten as the following inequality taking into account that terms like $ e^T\mathcal{Q} $ can be used to form $ 2 e^T \mathcal{Q} \leq e^Te+\mathcal{Q}^T\mathcal{Q} $ as explained in Appendix~\ref{APP}. \textbf{Also, The substitution $ u = \hat{H} N(k) e $ is used}.
			\begin{equation}\label{eDynamics6}
				\begin{split}
					e^T\dot{e} + e^T(\Lambda - \mathcal{B}_2) e&\leq 3e^Te+\mathcal{Q}^T\mathcal{Q} + \mathcal{G}^T\mathcal{G} \\
					&  + \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q}  \\
					&- e^T\hat{H}^{-1}\hat{H} N(k) e \\
				\end{split}
			\end{equation}
			which is further simplified considering $ \dot{k}(t)= \left\lVert e(t) \right\rVert^2_2 =e^T e  $ and that $ N(k) $ is scalar.
			\begin{equation}\label{eDynamics7}
				\begin{split}
					e^T\dot{e} + e^T(\Lambda - \mathcal{B}_2) e&\leq 3\dot{k}+\mathcal{Q}^T\mathcal{Q} + \mathcal{G}^T\mathcal{G} - N(k) \dot{k}\\
					&  + \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q}  \\
				\end{split}
			\end{equation}

			Integrating both sides of the above system yields
			\begin{equation}\label{eDynamics8}
				\begin{split}
					&\int_{t_0}^{t} e^T\dot{e} \;d\tau + \int_{t_0}^{t} e^T(\Lambda - \mathcal{B}_2) e\;d\tau\leq
					3\int_{t_0}^{t}\dot{k}\;d\tau \\
					&+\int_{t_0}^{t}\mathcal{Q}^T\mathcal{Q}\;d\tau + \int_{t_0}^{t}\mathcal{G}^T\mathcal{G}\;d\tau - \int_{t_0}^{t}N(k) \dot{k}\;d\tau\\
					&  + \int_{t_0}^{t}\dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q}\;d\tau  \\
				\end{split}
			\end{equation}
			Applying a change in variable in the $ \int_{t_0}^{t}N(k) \dot{k}\;d\tau $ term, we get
			\begin{equation}\label{eDynamics9}
				\begin{split}
					&\frac{1}{2}e^Te + \int_{t_0}^{t} e^T(\Lambda - \mathcal{B}_2) e\;d\tau\leq
					3(k(t)-k(t_0)) \\
					&+\int_{t_0}^{t}\mathcal{Q}^T\mathcal{Q}\;d\tau + \int_{t_0}^{t}\mathcal{G}^T\mathcal{G}\;d\tau - \int_{k_0}^{k}N(k)\;dk\\
					&  + \int_{t_0}^{t}\dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q}\;d\tau  \\
				\end{split}
			\end{equation}
			Dividing both sides by $ k(t)-k(t_0) $ and rearranging yields
			\begin{equation}\label{eDynamics10}
				\begin{split}
					\frac{1}{2}\frac{e^Te}{k(t)-k(t_0)} &+ \frac{\int_{t_0}^{t} e^T(\Lambda - \mathcal{B}_2) e\;d\tau}{k(t)-k(t_0)}		\leq 	3	 \\
					&-\frac{1}{k(t)-k(t_0)}\int_{k_0}^{k}N(k)\;dk\\ &+\frac{\int_{t_0}^{t}\mathcal{Q}^T\mathcal{Q}\;d\tau}{k(t)-k(t_0)}
					+ \frac{\int_{t_0}^{t}\mathcal{G}^T\mathcal{G}\;d\tau}{k(t)-k(t_0)} \\
					&+\frac{\int_{t_0}^{t}\dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T (\mathcal{B}_1-\mathcal{B}_2)\dot{q}\;d\tau}{k(t)-k(t_0)}  \\
				\end{split}
			\end{equation}
			Dividing both sides by the terms on the R.H.S. except for $ \frac{1}{k(t)-k(t_0)} \int_{k_0}^{k}N(k)\;dk $ yields
			\begin{equation}\label{eDynamics11}
				\begin{split}
					&\begin{pmatrix}\frac{1}{2}\frac{e^Te}{k(t)-k(t_0)} +\\ \\ \frac{\int_{t_0}^{t} e^T(\Lambda - \mathcal{B}_2) e\;d\tau}{k(t)-k(t_0)}\end{pmatrix}	\begin{pmatrix}3+\frac{\int_{t_0}^{t}\mathcal{Q}^T\mathcal{Q}\;d\tau}{k(t)-k(t_0)}
						+ \frac{\int_{t_0}^{t}\mathcal{G}^T\mathcal{G}\;d\tau}{k(t)-k(t_0)} \\ \\
						+\frac{\int_{t_0}^{t}\dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T (\mathcal{B}_1-\mathcal{B}_2)\dot{q}\;d\tau}{k(t)-k(t_0)}] \end{pmatrix}^{-1}
					\leq 		 \\
					&1-\frac{\int_{k_0}^{k}N(k)\;dk}{k(t)-k(t_0)}
					\begin{pmatrix}3+\frac{\int_{t_0}^{t}\mathcal{Q}^T\mathcal{Q}\;d\tau}{k(t)-k(t_0)}
						+ \frac{\int_{t_0}^{t}\mathcal{G}^T\mathcal{G}\;d\tau}{k(t)-k(t_0)} \\ \\
						+\frac{\int_{t_0}^{t}\dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T (\mathcal{B}_1-\mathcal{B}_2)\dot{q}\;d\tau}{k(t)-k(t_0)}] \end{pmatrix}^{-1} \\
				\end{split}
			\end{equation}

			It can be noted from the above that with the exception of the $ \frac{1}{k(t)-k(t_0)} \int_{k_0}^{k}N(k)\;dk $ term on the R.H.S. of Equation~\eqref{eDynamics10}, all the terms are quadratic forms and positive. Now if the term $ k(t) $ is monotonically increasing, i.e. $ k(t) \to \infty $ as $ t \to \infty $, then the Nussbaum function term on the R.H.S. can, as a whole, take values approaching $ \pm \infty $ by the definition of a Nussbaum function as presented in Equation~\eqref{Nuss}. This violates the inequality above. Consequently, the assumption that $ k(t) \to \infty $ as $ t \to \infty $ is false and $ k(t) $ must, therefore, be bounded. However, $ \dot{k}(t) $ being a non-decreasing function by definition in \eqref{kd} and $ k(t) $ being bounded implies that $ k(t) \to k_{\infty} $ as $ t \to \infty $. This means that $ \dot{k}(t) \to 0 $ as $ t \to \infty $ i.e. $ e^{T}e \to 0 $, or $ e \to 0 $. This establishes the required result.
		\end{proof}

		With the estimation error going to zero established in Theorem~\ref{boundnessproof}, the convergence of the estimated parameters to the real parameters is now shown in Theorem~\ref{convergeproof}. It is also worth noting that verifying the assumptions that $H(q(t))$ and $H(\hat{q}(t))$ are invertible is simple and can be explicitly checked by numerical computation. Further, before proceeding to the next result, some notation is defined, to facilitate the a condensed representation of the next result and its proof. Thus motivated let $i\in\{0,1\}$, and $j,k\in\{1,2,\cdots,9,\ell\}$. Here we use the set $\{1,2,\cdots,9,\ell\}$ and let $\ell=10$ so that in the statement of Theorem \ref{convergeproof} the subscript value of 10 can be easily visualized. Now consider the following definitions.
		\begin{equation}\label{Paraproof21}	a_i(t)=F_{n_i} \mu_{d_i} tanh \left( 4\;\frac{\dot{\theta}_i(t)}{\dot{\theta}_{t_i}(t)}\right)\\\end{equation}
		\begin{equation}\label{Paraproof22}	\hat{a}_{ijk}(t)=F_{n_i} \hat{z}_j(t) tanh \left( 4\;\frac{\dot{{\theta}}_i(t)}{\hat{z}_k(t)}\right)\end{equation}
		\begin{equation}\label{Paraproof23}	b_i(t)=\mu_{v_i} \dot{\theta}_i(t) tanh \left( 4\;\frac{F_{n_i}}{F_{nt_i}}\right)\\\end{equation}
		\begin{equation}\label{Paraproof24}	\hat{b}_{ijk}(t)=\hat{z}_j(t) \dot{{\theta}}_i(t) tanh \left( 4\;\frac{F_{n_i}}{\hat{z}_k(t)}\right)\\\end{equation}
		\begin{equation}\label{Paraproof25}	c_i(t)=\cfrac{F_{n_i}  \mu_{s_i}\; \cfrac{\dot{\theta}_i(t)}{\dot{\theta}_{t_i}(t)} }{ \cfrac{1}{4} \left(\cfrac{\dot{\theta}_i(t)}{\dot{\theta}_{t_i}(t)} \right)^2 + \cfrac{3}{4} }\\\end{equation}
		\begin{equation}\label{Paraproof26}	\hat{c}_{ijk}(t)=\cfrac{F_{n_i} \hat{z}_j(t)\; \cfrac{\dot{{\theta}}_i(t)}{\hat{z}_k(t)} }{ \cfrac{1}{4} \left(\cfrac{\dot{{\theta}}_i(t)}{\hat{z}_k(t)} \right)^2 + \cfrac{3}{4} } \\\end{equation}
		\begin{equation}\label{Paraproof28}	d_i(t)=	\cfrac{F_{n_i}  \mu_{d_i}\; \cfrac{\dot{\theta}_i(t)}{\dot{\theta}_{t_i}(t)} }{ \cfrac{1}{4} \left(\cfrac{\dot{\theta}_i(t)}{\dot{\theta}_{t_i}(t)} \right)^2 + \cfrac{3}{4} }\\\end{equation}
		\begin{equation}\label{Paraproof27}	\hat{d}_{ijk}(t)=\cfrac{F_{n_i}  \hat{z}_j(t)\; \cfrac{\dot{{\theta}}_i(t)}{\hat{z}_k(t)} }{ \cfrac{1}{4} \left(\cfrac{\dot{{\theta}}_i(t)}{\hat{z}_k(t)} \right)^2 + \cfrac{3}{4} }\end{equation}
		\begin{theorem}\label{convergeproof}
			Considering the definitions in \eqref{Paraproof21}-\eqref{Paraproof27} let $$A(t)=\begin{bmatrix}
				{a_0-\hat{a}_{014}}&{b_0-\hat{b}_{035}}&{c_0-\hat{c}_{024}}&{d_0-\hat{d}_{014}}\\
				{a_1-\hat{a}_{169}}&{b_1-\hat{b}_{18\ell}}&{c_1-\hat{c}_{179}}&{d_1-\hat{d}_{169}}
			\end{bmatrix},$$ and $x=\begin{bmatrix}
				{1}&{1}&{1}&{1}
			\end{bmatrix}^T$. For the parameter identification problem described in Section~\ref{Sec:paraID}, suppose the conditions required for Theorem 1 to hold are satisfied. Suppose there exists a time instant $t^*>t_0$ such that for $t\to\infty, t>t^*$, $x$ does not belong to the nullspace of $A(t), A(t)\neq 0$,and all the following functions: $tanh \left( 4\;\frac{\dot{\theta}_i(t)}{\dot{\theta}_{t_i}(t)}\right)$, $tanh \left( 4\;\frac{\dot{{\theta}}_i(t)}{\hat{z}_k(t)}\right)$, $tanh \left( 4\;\frac{F_{n_i}}{F_{nt_i}}\right)$, $tanh \left( 4\;\frac{F_{n_i}}{\hat{z}_k(t)}\right)$ tend to 1. \textbf{Then} the parameter adaptation law in Equation~\eqref{adapt} leads to $ \hat{z}_n \to z_{n_{true}} $ as $ t\to \infty $ for $ n \in \{1,2, \dots, 10\} $.
		\end{theorem}

		\begin{proof}[Proof of Theorem~\ref{convergeproof}]
			By the assumptions Theorem~\ref{boundnessproof} is satisfied, this gives $ e\to 0 $ as $ t\to \infty $. As a result $ k(t)\to k_\infty $ (a constant), which further leads $ {u}\to 0 $ following the definition of $u$ in \eqref{Nk}-\eqref{unk}. As $ e\to 0 $, therefore by definition of $e$ in \eqref{e} we get that ${\dot{\hat{q}}} \to {\dot{q}}$, and $ {\ddot{\hat{q}}} \to {\ddot{q}} $.

			Considering this, the model of the system in Equation~\eqref{Model} and the estimator dynamics in Equation~\eqref{EstModel} can be written as
			\begin{equation}\label{Paraproof}
				\begin{split}
					{H(q)}{\ddot{{q}}}+ {B({q},\dot{{q}})}\dot{q} + {G({q})}&={{Q}(\dot{{q}})}	\\
					{H(\hat{q})} {\ddot{\hat{q}}} + {B(\hat{q},\dot{\hat{q}})\dot{\hat{q}}} + {G(\hat{q})}	&=	{\hat{Q}(\dot{\hat{q}})}	+  {u}.\\
				\end{split}
			\end{equation}
			Because $ u \to 0 $, $ {\dot{\hat{q}}} \to {\dot{q}} $, $ {\ddot{\hat{q}}} \to {\ddot{q}} $, subtracting the lower equation in \eqref{Paraproof} from the upper one, and substituting $ \hat{q} = q $ provides
			\begin{equation}\label{Paraproof0}
				{Q}(\dot{{q}}) - \hat{Q}(\dot{{q}})	= \begin{bmatrix}	{f}_0(\dot{{\theta}}_0) \\ {f}_1(\dot{{\theta}}_1) \end{bmatrix} -
				\begin{bmatrix}	\hat{f}_0(\dot{{\theta}}_0) \\ \hat{f}_1(\dot{{\theta}}_1) \end{bmatrix} =0.
			\end{equation}
			Now using the definitions of the friction force model in \eqref{Qf}, and the concerned estimates as defined in \eqref{EstQf1}, \eqref{EstQf2} one can rewrite \eqref{Paraproof0} as \eqref{Paraproof1} using the notation in \eqref{Paraproof21}-\eqref{Paraproof27}.

			\begin{equation}\label{Paraproof1}
				\underbrace{\begin{bmatrix}
						{a_0-\hat{a}_{014}}&{b_0-\hat{b}_{035}}&{c_0-\hat{c}_{024}}&{d_0-\hat{d}_{014}}\\
						{a_1-\hat{a}_{169}}&{b_1-\hat{b}_{18\ell}}&{c_1-\hat{c}_{179}}&{d_1-\hat{d}_{169}}
				\end{bmatrix}}_{A(t)}	\underbrace{\begin{bmatrix} 1 \\1\\1 \\ 1 \end{bmatrix}}_{x}=0
			\end{equation}
			We have arrived at \eqref{Paraproof1} because as $t\to\infty$ we have $e\to 0$. Recall that, by the assumptions of the theorem, there exists a time instant $t^*>t_0$ such that for $t\to\infty, t>t^*$, $x$ does not belong to the nullspace of $A(t), A(t)\neq 0$.  So for such $t>t^*$ from \eqref{Paraproof1} we can write ${\hat{a}_{014}=a_0}$, ${\hat{b}_{035}=b_0}$, ${\hat{c}_{024}=c_0}$, ${\hat{d}_{014}=d_0}$,
			${\hat{a}_{169}=a_1}$, ${\hat{b}_{18\ell}=b_1}$, ${\hat{c}_{179}=c_1}$, ${d_1=\hat{d}_{169}}$.

			Now considering ${\hat{a}_{014}=a_0}$, ${\hat{b}_{035}=b_0}$, ${\hat{c}_{024}=c_0}$, ${\hat{d}_{014}=d_0}$ along with their definitions in \eqref{Paraproof21}-\eqref{Paraproof27}, the following are obtained.

			\begin{equation}\label{Param1}
				F_{n_0} \hat{z}_1 tanh \left( 4\;\frac{\dot{{\theta}}_0}{\hat{z}_4}\right)
				=
				F_{n_0} \mu_{d_0} tanh \left( 4\;\frac{\dot{\theta}_0}{\dot{\theta}_{t_0}}\right)
			\end{equation}
			\begin{equation}\label{Param2}
				\hat{z}_3 \dot{{\theta}}_0 tanh \left( 4\;\frac{F_{n_0}}{\hat{z}_5}\right)
				=
				\mu_{v_0} \dot{\theta}_0 tanh \left( 4\;\frac{F_{n_0}}{F_{nt_0}}\right)
			\end{equation}
			\begin{equation}\label{Param3}
				\cfrac{F_{n_0} \hat{z}_2\; \cfrac{\dot{{\theta}}_0}{\hat{z}_4} }{ \cfrac{1}{4} \left(\cfrac{\dot{{\theta}}_0}{\hat{z}_4} \right)^2 + \cfrac{3}{4} }
				=
				\cfrac{F_{n_0}  \mu_{s_0}\; \cfrac{\dot{\theta}_0}{\dot{\theta}_{t_0}} }{ \cfrac{1}{4} \left(\cfrac{\dot{\theta}_0}{\dot{\theta}_{t_0}} \right)^2 + \cfrac{3}{4} }\
			\end{equation}

			As per the assumptions of the theorem, all the $tanh(\cdot)$ terms tend to unity. Considering the equations \eqref{Param1} and Equation~\eqref{Param2} with this gives $\hat{z}_1 	=	\mu_{d_0} $, $\hat{z}_3 	=	\mu_{v_0}$.

			Now considering \eqref{Param1} again with $\hat{z}_1 	=	\mu_{d_0}$ and explcitly writing the $tanh(\cdot)$ terms gives
			\begin{equation}\label{tanz4}
				tanh \left( 4\;\frac{\dot{{\theta}}_0}{\hat{z}_4}\right) = tanh \left( 4\;\frac{\dot{\theta}_0}{\dot{\theta}_{t_0}}\right).
			\end{equation}
			Now from the definition of the hyperbolic tangent function, and from its graph, it is known that its inverse can be found, i.e. so let $p=tanh \left( 4\;\frac{\dot{{\theta}}_0}{\hat{z}_4}\right) = tanh \left( 4\;\frac{\dot{\theta}_0}{\dot{\theta}_{t_0}}\right)$. Then, $tanh^{-1}(p)=  4\;\frac{\dot{{\theta}}_0}{\hat{z}_4} =   4\;\frac{\dot{\theta}_0}{\dot{\theta}_{t_0}}$, leading to $ \hat{z}_4 = \dot{\theta}_{t_0} $. Now using this exact same process with \eqref{Param2}, considering $\hat{z}_3 	=	\mu_{v_0}$ and inverting the $\tanh(\cdot)$ terms provides $  \hat{z}_5 = F_{nt_0} $. Further using the fact that $\hat{z}_4 = \dot{\theta}_{t_0} $ and considering \eqref{Param3} gives us $\hat{z}_2	=	\mu_{s_0}$.

			The exact same procedure as followed from after \eqref{Paraproof1} to the line above, can be used to explicitly analyze ${\hat{a}_{169}=a_1}$, ${\hat{b}_{18\ell}=b_1}$, ${\hat{c}_{179}=c_1}$, ${d_1=\hat{d}_{169}}$. Using this process provides the  following to $ z_6 = \mu_{d_1} $, $ z_7 = \mu_{s_1} $, $ z_8 = \mu_{v_1} $, $ z_9 = \dot{\theta}_{t_1} $, $ z_{10} = F_{nt_1} $. This completes the proof.
		\end{proof}
		It is worth mentioning that the assumptions of the above theorem are not necessarily restrictive. This is because during experimentation it has been routinely observed that all the $tanh(\cdot)$ terms frequently approach unity. So it is easy to pick time intervals at which $e\approx 0$, and $tanh(\cdot)\approx 1$ so that the the values of $\hat{z}_i(t),i\in\{1,2,\cdots,10\}$ in such an interval of time can be used as candidates for parameters estimates. Also, once such estimated parameters values are available, they can be fed into \eqref{Paraproof21}-\eqref{Paraproof27} to reconstruct matrix $A$ mentioned in Theorem \ref{convergeproof}. This will easily allow ascertaining if the nullspace related assumptions in Theorem \ref{convergeproof} are met. If they are not met then estimates from another interval in time where $e\approx 0$, and $tanh(\cdot)\approx 1$, can be checked for satisfaction of the nullspace assumption.

		\begin{table*}
			\centering
			\caption{Friction model parameter identification problem setup.}
			\label{table:ID0}
			\begin{tabular}{lccccc}
				\toprule
				{Parameter}&{$\mu_{d_0}$}	& {$\mu_{s_0}$}	& {$\mu_{v_0}$}  & {$\dot{\theta}_{t_0}$} &  {$ F_{nt_0} $} \\
				\midrule
				$ Z_{nl} $&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$\\
				\midrule
				$ Z_{nu} $ &$0.0750$&$0.0750$&$0.0100$&$0.0100$&$0.1$\\
				\midrule
				$ \lambda_l $	&$50$&$50$&$50$&$50$&$50$\\
				\midrule
				$ \lambda_u $	&$1$&$1$&$1$&$1$&$1$\\
				\midrule[1pt]
				{Parameter}&{$\mu_{d_1}$}	& {$\mu_{s_1}$}	& {$\mu_{v_1}$}  & {$\dot{\theta}_{t_1}$} &  {$ F_{nt_1} $}\\
				\midrule
				$ Z_{nl} $&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$&$2.22\times 10^{-16}$&$2.22\times 10{-16}$\\
				\midrule
				$ Z_{nu} $ &$0.150$&$0.151$&$0.0100$&$0.0100$&$0.100$\\
				\midrule
				$ \lambda_l $	&$50$&$50$&$50$&$50$&$50$\\
				\midrule
				$ \lambda_u $	&$1$&$1$&$1$&$1$&$1$\\
				\bottomrule
			\end{tabular}
		\end{table*}

		\section{Results and Discussion}

		In this section, the proposed parameter identification algorithm is tested in simulation, then experimental validation is performed. In all experiments, the links start with a zero initial angular velocity and a specified initial angular position. The motion is monitored until the system returns to the equilibrium position.

		\subsection{Simulation}
		Here, the performance of the proposed algorithm is evaluated in simulation. The dynamic equations of motion for the Furuta pendulum were simulated from some initial conditions, and the angular position and velocity states are fed into the algorithm. The initial conditions were $ \theta_0(0) = 0^\circ $, $ \theta_1(0) = 120^\circ $, $ \dot{\theta}_0(0) = 0^\circ/sec $, and $ \dot{\theta}_1(0) = 0^\circ/sec $. The Runge-Kutta method of order $ 4 $ with a sampling time of $ 0.001 s $ was used to propagate the equations in time and solve the ODEs. Normally distributed Gaussian random white noise characterized by $ v \sim \mathcal{N}(0,\,\sigma=0.1)\, $ was injected into the simulated states to serve as measurements noise. Results achieved despite this, show the resilience of the proposed approach to noise. The initial conditions fed into the algorithm were also injected with similar noise to account for the difficulty associated with acquiring highly accurate and precise initial conditions. Additionally, this serves to simulate the ramifications of initial conditions mismatch in chaotic systems such as the Furuta pendulum. The adaptation parameters used to test the algorithm are detailed in Table~\ref{table:ID0}. The upper/lower parameters limits in Table~\ref{table:ID0} were used to form a normal distribution centered around the mean of each respective parameter bounds. The initial friction model parameters were sampled from this distribution to start the algorithm. The actual parameters and the initial guesses used to simulate the response are presented in Table~\ref{table:IDRSim}.

		\begin{figure*}
			\centering
			\includegraphics[width=16 cm]{Figs/Sim_BM_UAS}
			\caption{Performance of the UAS observer during the simulation.\label{Sim_UAS}}
		\end{figure*}
		\begin{figure*}
			\centering
			\includegraphics[width=16 cm]{Figs/Sim_BM_Para}
			\caption{Parameter trajectories during the simulation.\label{Sim_Para}}
		\end{figure*}

		\begin{figure*}
			\centering
			\includegraphics[width=14 cm]{Figs/Sim_BM_Val}
			\caption{Validation of the identified parameters during the simulation.\label{Sim_Val}}
		\end{figure*}

		\begin{figure*}
			\centering
			\includegraphics[width=14 cm]{Figs/Sim_BM_Val_Frq}
			\caption{Frequency portrait of the identified parameters from the simulation.\label{Sim_Val_Frq}}
		\end{figure*}
		\begin{table*}
			\centering
			\caption{Friction model parameter simulation identification results with the UAS method.\label{table:IDRSim}}
			\begin{tabular}{lccccc}
				\toprule
				{Parameter}&{$\mu_{d_0}$}	& {$\mu_{s_0}$}	& {$\mu_{v_0}$}  & {$\dot{\theta}_{t_0}$} &  {$ F_{nt_0} $} \\
				\midrule
				{Initial Guess}&$5.135\times 10^{-3}$&$5.875\times 10^{-3}$&$2.531\times 10^{-3}$&$4.853\times 10^{-2}$&$1.029\times 10^{-1}$\\
				\midrule
				{Estimate}&$4.793\times 10^{-4}$&$5.740\times 10^{-4}$&$2.424\times 10^{-4}$&$4.742\times 10^{-3}$&$9.479\times 10^{-3}$\\
				\midrule
				{Actual Value}&$5\times 10^{-4}$&$6\times 10^{-4}$&$2.5\times 10^{-4}$&$5\times 10^{-3}$&$10\times 10^{-3}$\\
				\midrule
				{Parameter}&{$\mu_{d_1}$}	& {$\mu_{s_1}$}	& {$\mu_{v_1}$}  & {$\dot{\theta}_{t_1}$} &  {$ F_{nt_1} $} \\
				\midrule
				{Initial Guess}&$5.705\times 10^{-3}$&$6.683\times 10^{-3}$&$2.399\times 10^{-3}$&$4.820\times 10^{-2}$&$9.720\times 10^{-2}$\\
				\midrule
				{Estimate}&$5.740\times 10^{-4}$&$6.688\times 10^{-4}$&$2.424\times 10^{-4}$&$4.742\times 10^{-3}$&$9.479\times 10^{-3}$\\
				\midrule
				{Actual Value}&$6\times 10^{-4}$&$7\times 10^{-4}$&$2.5\times 10^{-4}$&$5\times 10^{-3}$&$10\times 10^{-3}$\\
				\bottomrule
			\end{tabular}
		\end{table*}
		Figure~\ref{Sim_UAS} shows the UAS tracking results where it is noticed that the estimated states quickly approach and converge to the true simulated response of the system. Further, as the estimated states approach the simulated states, the norm of the error is observed to subside in magnitude and approaches zero, and the UAS gain is observed to also grow smaller as a consequence of the reduction in estimation error. Figure~\ref{Sim_Para} shows the time trajectories of the parameters of the friction model as the UAS converges to the correct states of the real system. It is seen that the parameters do indeed stabilize around some final values. The values considered to be the parameter estimates were taken just after the error norm reduced below some user defined threshold, which here is recommended to be below $ \left\lVert e(t) \right\rVert_2=0.01 $. The identified values are presented in Table~\ref{table:IDRSim}. Figure~\ref{Sim_Val} showcases a validation test of the chosen parameter estimates. The experimental response is plotted against the response realized from both the UAS estimated parameters as well as the initial parameters. The UAS-provided parameter estimates generate a response very similar to the simulated response, which establishes that the proposed approach was successful at identifying high quality parameter estimates in simulation.

		Further analysis is presented in frequency domain in Figure~\ref{Sim_Val_Frq} where both the phase and spectrum of the time responses of the angular positions of the arm and pendulum links are presented. It is evident that the algorithm is successful at moving from the initial estimates to ones that are more representative of the system. The portrait is quite complex considering the nonlinearity of the system, and it helps explain the compression and expansion effect present in the time responses presented as a wide array of harmonics is present in the response. The spectrum plots show good matching between estimated and actual parameters in this test.

		\subsection{Experiment}
		Here, the performance of the proposed algorithm is validated experimentally. The initial conditions were approximately $ \theta_0(0) = 0.2^\circ $, $ \theta_1(0) = 126^\circ $, $ \dot{\theta}_0(0) = 0^\circ/sec $, and $ \dot{\theta}_1(0) = 0^\circ/sec $. As with the test in simulation, the initial friction model parameter estimates were randomly generated also considering the upper/lower limits from Table~\ref{table:ID0}.

		The Runge-Kutta method of order $ 4 $ with a sampling time of $ 0.0001 s $ was also used here to propagate the equations in time and solve the ODEs. The adaptation parameters used to test the algorithm are detailed in Table~\ref{table:ID0}.

		Figure~\ref{Exp_UAS} shows the UAS tracking results where it is noticed that the estimated states quickly approach and converge to the true response of the system. Throughout that process, the norm of the error is observed to subside in magnitude and approach zero, and the UAS gain is observed to follow a similar trend to the estimation error norm. Unlike the simulated case in the previous section, the UAS gain does not completely reduce to zero and intermittent spikes are observed in the response. Figure~\ref{Exp_Para} shows the time trajectories of the parameters of the friction model as the UAS tracks the states of the real system. It is seen that the parameters settle and oscillate within some final range of values, which makes sense following the behavior of the UAS gain. The values considered to be the parameter estimates were taken just after the error norm reduced below some user defined threshold, which here is recommended to be below $ \left\lVert e(t) \right\rVert_2=0.05 $. Due to the extra uncertainty present in the experimental data, it is recommended to average the parameters estimates during adaptation between the error subsiding and the system motion seizing. This provided tighter bounds on the parameters estimates in the experimental test. The identified values are presented in Table~\ref{table:IDRExp}. Figure~\ref{Exp_Val} showcases a validation test of the chosen parameter estimates. The experimental response is plotted against the response realized from both the UAS estimated parameters as well as the initial parameter guesses. It is clear that the UAS-provided parameter estimates generate a response that is qualitatively similar to the experiment. However, unlike the simulation test case, an exact match is not realized between the experimental response and the estimated one. We reason this to be due to the limitations of friction models available in the literature in that they only approximate and not fully characterize the damping phenomenon. Nevertheless, the proposed approach proved to be computationally highly efficient at identifying quality parameter estimates using experimental data.

		The same analysis is performed  in the frequency domain. In Figure~\ref{Exp_Val_Frq}, the phase and the spectrum of the time responses of the angular positions of the arm and pendulum links are presented. Similar to the experimental results, the algorithm is successful at moving from the initial estimates to the ones that are more representative of the system. However, the spectrum plots do not show as good matching between the estimated and the actual parameters in this test as it is the case with simulation. This is due to the difficulty concerning the description of the actual friction present in the system. Still, very good qualitative parameter estimates were realized using the proposed approach. In the next section, we explore alternative methods to address this problem.
		\begin{table*}
			\centering
			\caption{Friction model parameter experimental identification results with the UAS method.\label{table:IDRExp}}
			\begin{tabular}{lccccc}
				\toprule
				{Parameter}&{$\mu_{d_0}$}	& {$\mu_{s_0}$}	& {$\mu_{v_0}$}  & {$\dot{\theta}_{t_0}$} &  {$ F_{nt_0} $} \\
				\midrule
				{Initial Guess}&$1.090\times 10^{-2}$&$8.239\times 10^{-3}$&$4.971\times 10^{-3}$&$2.726\times 10^{-3}$&$1.297\times 10^{-2}$\\
				\midrule
				{Estimate}&$1.575\times 10^{-3}$&$1.575\times 10^{-3}$&$3.006\times 10^{-4}$&$3.006\times 10^{-4}$&$2.065\times 10^{-3}$\\
				\midrule[1pt]
				{Parameter}&{$\mu_{d_1}$}	& {$\mu_{s_1}$}	& {$\mu_{v_1}$}  & {$\dot{\theta}_{t_1}$} &  {$ F_{nt_1} $} \\
				\midrule
				{Initial Guess}&$1.957\times 10^{-2}$&$1.796\times 10^{-2}$&$1.607\times 10^{-2}$&$1.342\times 10^{-2}$&$9.213\times 10^{-3}$\\
				\midrule
				{Estimate}&$3.046\times 10^{-3}$&$3.065\times 10^{-3}$&$3.006\times 10^{-4}$&$3.006\times 10^{-4}$&$2.065\times 10^{-3}$\\
				\bottomrule
			\end{tabular}
		\end{table*}
		\begin{figure*}
			\centering
			\includegraphics[width=14 cm]{Figs/Exp_BM_UAS}
			\caption{Performance of the UAS observer during the experiment.\label{Exp_UAS}}
		\end{figure*}
		\begin{figure*}
			\centering
			\includegraphics[width=16 cm]{Figs/Exp_BM_Para}
			\caption{Parameter trajectories during the experiment.\label{Exp_Para}}
		\end{figure*}
		\begin{figure*}
			\centering
			\includegraphics[width=14 cm]{Figs/Exp_BM_Val}
			\caption{Validation of the identified parameters during the experiment.\label{Exp_Val}}
		\end{figure*}
		\begin{figure*}
			\centering
			\includegraphics[width=14 cm]{Figs/Exp_BM_Val_Frq}
			\caption{Frequency portrait of the identified parameters from the experiment.\label{Exp_Val_Frq}}
		\end{figure*}

		\subsection{Comparison with conventional methods}
		Here, we compare the proposed approach to a conventional optimization-based one. Specifically, a nonlinear grey-box identification in the MATLAB environment is selected to exploit the known structure of the dynamic model and generate estimates for its parameters. The results for all the cases are shown in Figure~\ref{bar}, where the quality of fit is measured through the coefficient of determination $ R^2 $ metric and is presented in Equation~\ref{NRMSE}.

		\begin{equation}\label{NRMSE}
			{fit=100 \times \left(1- \frac{\sum ( \hat{x}-x_{ref})^2 }{\sum(x_{ref} - \bar{x}_{ref} )^2} \right)}
		\end{equation}
		where $ x_{ref} $ is the reference signal, and $ \hat{x} $ is the estimate of $ x_{ref} $.

		Aditionally, the computation time is computed using
		\begin{equation}\label{T_metric}
			\begin{split}
				&\text{Normalized Computation Time} =\\& \frac{\text{Computation Time}}{\text{Experiment Time} \times \text{Sampling Frequency}}
			\end{split}
		\end{equation}

		Here, the same simulation parameters and initial conditions presented in the previous sections are used. Due to the high nonlinearity of the problem, optimization alone is reportedly not always a viable solution to address this problem. Testing shows that in simulation, optimization alone is not capable of reaching acceptable parameter estimates with a goodness of fit of only $ 13\% $ and $ 3\% $ for the Arm Joint and the Pendulum Joint responses, respectively. The proposed algorithm, in contrast, provided quick estimates that provided much better goodness of fit metrics of $ 98\% $ for both responses.  We conclude that the proposed approach is more robust to noisy data than the optimization approach. In the experimental test, the proposed method, while it is able to quickly generate quality estimates, it provides estimates, which can be improved on with a goodness of fit of $ 77\% $ and $ 91\% $ for the responses of the Arm Joint and the Pendulum Joint, respectively. The optimization approach, however, exhibited a low performance in terms of goodness of fit with only $ 3\% $ and $ 24\% $ for the responses of the Arm Joint and the Pendulum Joint, respectively. That result was also achieved at a considerably higher computational cost with optimization taking approximately $ 500$ and $ 1300 $ times more computation time than the UAS approach for the simulation and the experiment, respectively. Table~\ref{table:Comp} summarizes these results.

		\begin{figure}
			\centering
			\includegraphics[width=3in]{Figs/Metrics}
			\caption{Performance Comparison.\label{bar}}
		\end{figure}

		\begin{table*}
			\centering
			\caption{Computation time and quality of estimates.\\ Computation time is presented along with experiment runtime.}\label{table:Comp}
			\begin{tabular}{llcccc}
				\toprule
				&& \multicolumn{2}{c}{Proposed}	& \multicolumn{2}{c}{Optimization}\\
				\cmidrule(r){3-4} \cmidrule(l){5-6}
				&& Arm Joint&Pendulum Joint& Arm Joint&Pendulum Joint \\
				\midrule
				\multirow{2}{*}{Sim.}&{Test Duration}\quad\quad\,\;[s]			& \multicolumn{2}{c}{$35$}  & \multicolumn{2}{c}{$35$}  	\\
				&{Computation Time}\;[s]			& \multicolumn{2}{c}{$ 3.43$}     & \multicolumn{2}{c}{$ 4504.6 $}	\\
				&		{Goodness of fit}\quad\;\,\,[\%]				& $ 98.97 $		& $ 99.53 $& $12.77$ & $2.80$	\\
				\midrule
				\multirow{2}{*}{Exp.}&{Test Duration}\quad\quad\,\;[s]			& \multicolumn{2}{c}{$20.28$} & \multicolumn{2}{c}{$20.28$}   	\\
				&{Computation Time}\;[s]			& \multicolumn{2}{c}{$ 19.49$ } & \multicolumn{2}{c}{$ 10373.7$ }\\
				&     {Goodness of fit}\quad\;\,\,[\%]					& $77.84$&$91.99$						& $ 3.40 $			& $ 24.11 $	\\
				\bottomrule
			\end{tabular}
		\end{table*}

		\section{Conclusion}
		This paper presented a dynamic analysis of a passive tilted Furuta pendulum. The pendulum was tilted to ensure the existence of a stable equilibrium configuration with the weight of each link as the only external force applied to the system. A complete analytical model was derived where full inertia of the pendulum is taken into account, and a comprehensive friction model was selected form the literature. The problem of parameter estimation of the friction in the setup was addressed. A UAS-based high gain observer was designed to estimate the friction model parameters, and it was tested in simulation and experimentally. The obtained results showed that the developed algorithm is effective at identifying parameters in this class of systems. Nonetheless, it was also concluded that the friction phenomenon is not fully described by the 5-parameter per joint friction model. Simulation results, even with a high level of injected and propagated dynamic noise, were close to the dynamic behavior of the system. Experimental results were qualitatively good when compared to conventional optimization based parameter estimation strategies. A frequency analysis was performed and it showed that the system is highly nonlinear with multiple harmonics detected. Differences between the simulation and the experiments are mainly attributed to the difficulty associated with characterizing friction in the joints. Future work includes the implementation of nonlinear control strategy using the developed model to showcase the performance a more accurate model of the system would permit.

		\appendices

		\section{Proof Addendum}\label{APP}
		\begin{remark}\label{rmk}
			Here we show how the inequalities in the mathematical justification are formed.
		\end{remark}
		Take the difference of quadratic forms below where the terms are the same as those in Equation~\eqref{eDynamics6}
		\begin{equation} (e-(\mathcal{B}_1-\mathcal{B}_2)\dot{q})^{T}(e-(\mathcal{B}_1-\mathcal{B}_2)\dot{q})\ge0 \end{equation}
		Expanding the above difference gives
		\begin{equation}
			\begin{split}
				e^Te - e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} &- ( (\mathcal{B}_1-\mathcal{B}_2)\dot{q} )^Te \\
				&+ ( (\mathcal{B}_1-\mathcal{B}_2)\dot{q} )^T( (\mathcal{B}_1-\mathcal{B}_2)\dot{q} ) \ge 0
			\end{split}
		\end{equation}
		Simplifying
		\begin{equation}
			\begin{split}
				e^Te - e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} &- \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^Te \\
				&+ \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} \ge 0
			\end{split}
		\end{equation}
		Rearranging
		\begin{equation}
			\begin{split}
				e^Te + \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} \ge
				2e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q}
			\end{split}
		\end{equation}
		where the substitution $  e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} +  \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^Te = 2e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} $ was made following the fact that $ e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} $ and $ \dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^Te $ are scalar quantities implying $ e^T(\mathcal{B}_1-\mathcal{B}_2)\dot{q} =(\dot{q}^T(\mathcal{B}_1-\mathcal{B}_2)^Te)^T $.

		The same procedure can be applied on the following quadratic difference form
		\begin{equation} (e-\mathcal{Q})^{T}(e-\mathcal{Q})\ge0 \end{equation}
		which is modified by multiplying $ \mathcal{Q} $ by the $ 2\times2 $ identity matrix $ I_{2\times2} $.
		\begin{equation} (e-I_{2\times2}\mathcal{Q})^{T}(e-I_{2\times2}\mathcal{Q})\ge0 \end{equation}
		which leads to the following after applying the same procedure above
		\begin{equation} e^{T}e + \mathcal{Q}^T\mathcal{Q} \ge 2e^T\mathcal{Q}\end{equation}

		%

redpajama_set_name: RedPajamaArXiv


International Council of Sport Science and Physical Education (ICSSPE), in conjunction with the International Olympic Committee (IOC), the International Paralympic Committee (IPC) and the International Federation of Sports Medicine (FIMS) will host a Pre-Olympic, Pre-Paralympic and Sport Scientific Congress, which was named International Convention on Science, Education and Medicine in Sport (ICSEMIS) in Santos / SP, Brazil under the theme "Saying Yes to Diversity in Sport ". Between August 31st thru September 4th – 2016.
Under the leadership of Maria Dinold IFAPA has submitted a symposium, and we await for confirmation of its acceptance.


text:
Q: What is the direct link to my Symfony App front page? I know this is amateur but I have to ask this.
Basically I am modifying my Symofony based app and I am moving it to another server.
I have already pulled the files from my github repository.
I have my old files on my old server connected with the domain.
What is the direct link to the website on the new server if it is not connected to the domain?
Domain points to the server root on the old server.
What am I supposed to write to enter the website on my new server?
www.mylogin.host.com/XXXX
index.html?
app/ something?
views/indexmain.html?
<?php
ini_set('display_errors', 1);
error_reporting(E_ALL);
use Symfony\Component\HttpFoundation\Request;
#echo phpversion(); exit;
/** @var \Composer\Autoload\ClassLoader $loader */
$loader = require __DIR__.'/../vendor/autoload.php';
if (PHP_VERSION_ID < 70000) {
    include_once __DIR__.'/../var/bootstrap.php.cache';
}

$kernel = new AppKernel('prod', false);
if (PHP_VERSION_ID < 70000) {
    $kernel->loadClassCache();
}
//$kernel = new AppCache($kernel);

// When using the HttpCache, you need to call the method in your front controller instead of relying on the configuration parameter
//Request::enableHttpMethodParameterOverride();
$request = Request::createFromGlobals();
$response = $kernel->handle($request);
$response->send();
$kernel->terminate($request, $response);

redpajama_set_name: RedPajamaStackExchange


In 1969 Harold Baucom bought a Murry Steaks distributorship located on South Tryon Street in Charlotte, North Carolina. Harold immediately began adding to his product line and became the major food service supplier to Eckerd's Drugs snack bars in North and South Carolina.
in 1990, Harold Food Co. was producing millions of pounds a year of a wide range of salad and chili products, along with being a full line food service distributor with over 2,000 items.  Harold Food Co. continued to grow through the nineties. In 2000 Harold Food Co. was purchased by Bost Distributing Co., Inc.  Harold food products continues to be produced in North Carolina with the highest quality standards and level of service that Harold Food Co. customers have learned to expect.
Harold Salads are available for North Carolina institutional and food service accounts through Bost Distributing Co., Inc. of Sanford, NC.  Each item can be purchased in 3 lb., 5 lb., 10 lb, and 30 lb. containers. If  you are looking for a homemade salad, we use only the finest and fresh ingredients. Our salads and chili are simply delicious. We welcome your interest.


These bowls are in excellent condition. There are no chips or cracks on five of them. One does have a small chip on its sawtooth edge but it is barely noticeable. This is remarkable considering their age.
One of the many ways to tell original American Brilliant cut glass is to check it under a black light. If it is real it will fluoresce a lime green color. These items have been checked and they do.


Anita Hassan and James Pinkerton June 7, 2014 Updated: June 7, 2014 11:34 p.m.
Rebecca Riley, 63, show the damage to her car after a hit-and-run crash. She got the fleeing car's tag number, but police did nothing in her case.
When Heather Heinke couldn't unlock the front door of her northwest Houston home one spring night and then went around back, child in tow, to open the rear door, she was shocked to find household items strewn all over the place. Then she spotted thieves still at work.
"Three or four figures were in my house," Heinke recalled. "I had my kid with me. I got in the car and drove to the end of the cul de sac and called police."
Houston police officers arrived some 15 minutes later, and after going inside, they told her it was OK to come in, too. The thieves were gone.
The crew had stripped her 3-bedroom home of nearly anything of seeming value, and not just expensive electronics such as a TV, a laptop computer and cameras.
"Everything - all my son's clothes, his cologne, I am just trying to give you a picture of how far they went," Heinke said. "They cut through the bag of my dad's ashes - I don't know what they were looking for. ... They had been in there a long time."
Officers told her there wasn't much they could do, other than to report her crime to the burglary division, Heinke said. They gave her an incident number and quickly left. Although the thieves left behind a set of car keys, officers didn't dust for fingerprints, or collect for DNA testing, Heinke said.
Heinke called police repeatedly, checking for any developments in the 2013 case. She says police never called back. As of Friday, HPD officials were still searching their files and not able to determine whether her case had been assigned for investigation.
Heinke's experience is not unusual in the nation's fourth-largest city. A recent HPD staffing study says that 15,000 burglaries and thefts, 3,000 hit-and-run crashes and 3,000 assaults were simply set aside last year without a follow-up investigation. Houston police commanders told researchers they didn't have enough staff to review the cases, even those with promising leads.
"It seems like (crime has) increased, and to (the point of) not being able to leave your home in a peaceful state of mind," Heinke said. "You kind of feel helpless ... you feel you're out there exposed, like you're out there on your own."
The report's finding that thousands of crimes aren't being fully investigated, although it may not be unique to Houston as a major city, has angered citizens, civil rights groups and victims and surprised some City Council members. The disclosure came shortly after the Police Department in April disciplined eight homicide detectives for either ignoring or conducting shoddy investigations into nearly two dozen deaths.
Last week, Houston Police Chief Charles McClelland defended his officers at a City Council meeting, saying his department was understaffed. Officers cannot and shouldn't investigate every crime reported to them, the chief said. Instead, McClelland said violent crimes are the priority.
They described a daily triage by Houston police lieutenants and sergeants, who review reports of new crimes and determine which have no leads or "solvability factors." Then, the supervisors assign what they consider the most solvable and egregious crimes to investigators. The others, despite having leads, are simply "suspended" and may be investigated if they are linked to another crime.
Mike Knox, a former gang investigator for the Police Department, said he's surprised there were only 15,000 burglary and theft cases that were not investigated.
"I'm sure HPD would love to investigate every single case, but we just don't have the manpower and money to do that. So we go after the ones who are doing the most harm," Knox said.
"There's not enough personnel," he said. "You only have so many investigators in the burglary and theft division."
The researchers hired to study HPD's staffing noted that while Houston has a lower staffing ratio than many large city police departments, such data is relevant but not all-telling. Major cities in the Northeast that have urban centers that developed "vertically," and are denser, have traditionally had higher ratios of police officers. Southwestern cities that have developed horizontally are less dense and tend to have lower ratios.
Houston's police-to-citizen ratio of 2.3 officers for every 1,000 residents is lower than those of Chicago (4.7), New York City (5.1), Detroit (4.4.) and Washington, D.C. (6.3.).
Rania Mankarious, executive director of Crime Stoppers of Houston, said she is frustrated that more investigators don't call on the nonprofit agency to help them solve cases. HPD is the agency's major sponsor, but she said some investigators don't want to go through the "hassle" of providing information about unsolved cases.
"I'm angry in the sense that we're a free resource, this is all we do, and we need to be utilized," Mankarious said, adding that cash rewards paid for information are funded by probation fees.
Rebecca Riley, 63, said she never heard back from police ab0ut her hit-and-run case. On Riley's way back to her Oak Forest home one afternoon last September, another driver made an illegal turn, causing a collision.
Riley followed the driver, noting the make and model of the vehicle - a Chrysler 300 - as well as the license plate. She filed a report at an HPD substation.
"I thought because I got the license plate number (that) maybe they'll be able to do something," Riley said. Her insurance company was able to track down the driver, so surely police could, Riley thought.
Weeks, then months, went by; she never heard from anyone.
"When I heard about that report, I thought, 'well, I'm one of those 3,000 hit-and-runs,' " Riley said, referring to the city study. "I know that compared to people getting killed and other stuff, it doesn't seem that important to them, but it's important to you."
HPD spokesman John Cannon said investigators sent Riley a letter asking for informatione. Riley said she never received it.
Even though Riley reported her case to police, she felt uncertain they would do much. About five years ago, Riley said, her husband's truck was burglarized in the driveway of their home and the culprit walked away with almost all of his tools.
But the couple then had little evidence to provide to officers when they made their report and it came as no surprise that the case yielded no results. This time, though, Riley noted that she provided a lead on the suspect and a license plate number.
"I really didn't expect that much," she said. "And I got what I expected."
She said she is aware that her hit-and-run case is not of a serious consequence. She was not hurt and the damage to her car was significant - roughly $2100 - but it was not extreme. Although she says she is aware that police are understaffed, she wishes police would have done something more. Even just a phone call letting her know that they were trying to work on her case.
"That would have made me feel better," she said.
Houston police have a crime victim advocate on staff, but they declined a request by the Chronicle to interview him.
The staffing report that two organizations put together for HPD, which was delivered to the City Council's public safety committee on Monday, made suggestions on how several operational divisions could address staffing issues. The report's authors said adding 27 investigators to the burglary and theft division would lead to a 25 percent increase in the number of cases with leads that are investigated. In 2012, HPD reported 26,630 burglaries and 67,978 thefts.
Christine Douglas, 29, would like to see more officers assigned to investigate cases. Last July, cameras that she and her fiancé installed at their southwest Houston town house showed a man stealing a package from the porch.
Douglas said she recognized the thief as a friend of neighbors, and she told the officer who took her report. Even then the officer told her that the likelihood of her case being investigated was slim.
"It appears an investigator did not get assigned to this case," Cannon said..
She said the experience was frustrating. She said it wouldn't bother her if small thefts like hers were not investigated by police as long as their time was being devoted to more severe crimes. But in reading about the study, Douglas doesn't feel assured.
"If there are thousands of (uninvestigated) cases, I'm sure they are not all minor like mine," she said. "That's just ridiculous."


text:
Q: React Native iOS Error: undefined is not an object (evaluating 'allRowIDs.length') I have been trying React Native and Firebase with the iOS steps mentioned in this link https://firebase.googleblog.com/2016/01/the-beginners-guide-to-react-native-and_84.html
I want to first follow Static Mockup steps mentioned in this link. I have followed the steps and provided the code as it is, but when I run the app in Xcode.
2017-10-17 01:25:19.743 [fatal][tid:com.facebook.react.ExceptionsManagerQueue] Unhandled JS Exception: TypeError: undefined is not an object (evaluating 'allRowIDs.length')

This error is located at:
        in ListView (at index.js:64)
        in RCTView (at View.js:113)
        in View (at index.js:60)
        in ReactFirebaseSample (at renderApplication.js:35)
        in RCTView (at View.js:113)
        in View (at AppContainer.js:102)
        in RCTView (at View.js:113)
        in View (at AppContainer.js:126)
        in AppContainer (at renderApplication.js:34)

Please advise, how to fix this error?
Here is the index.JS code:
'use strict';

import React, {Component} from 'react';
import ReactNative from 'react-native';
const firebase = require('firebase');
const StatusBar = require('./components/StatusBar');
const ActionButton = require('./components/ActionButton');
const ListItem = require('./components/ListItem');
const styles = require('./styles.js')

const {
  AppRegistry,
  ListView,
  StyleSheet,
  Text,
  View,
  TouchableHighlight,
  AlertIOS,
} = ReactNative;

// Initialize Firebase
const firebaseConfig = {
apiKey: "AIxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx",
authDomain: "fir-realtime-dc900.firebaseapp.com",
databaseURL: "https://fir-realtime-dc900.firebaseio.com",
storageBucket: "fir-realtime-dc900.appspot.com",
};
const firebaseApp = firebase.initializeApp(firebaseConfig);

AppRegistry.registerComponent('ReactFirebaseSample', () => ReactFirebaseSample);

class ReactFirebaseSample extends Component {

  constructor(props) {
    super(props);
    this.state = {
      dataSource: new ListView.DataSource({
        rowHasChanged: (row1, row2) => row1 !== row2,
      })
    };
  }
    componentDidMount() {
        this.setState({
                      dataSource: this.state.dataSource.cloneWithRows([{ title: 'Pizza' }])
                      })
    }

    _renderItem(item) {
      return (

        <ListItem item={item} onpress={() => {}}/>
      );
    }

    render() {
        return (
          <View style="{styles.container}">

            <StatusBar title="Grocery List"/>

            <ListView datasource="{this.state.dataSource}" renderrow="{this._renderItem.bind(this)}" style="{styles.listview}/"/>

            <ActionButton title="Add" onpress={() => {}} />
          </View>
        );
      }
    }


Hereby I would like to start a small series of articles describing the elements of the optimization of Solr instances. At first glance I decided to describe the parameter that specifies the data fetch window size – the query result window size. Hopefully, this article will explain how to use this parameter, how to modify and adapt it to your needs.
To start talking about this configuration parameter, we must first say how Solr fetches results from the index. When passing the rows parameter to Solr, with the value of 20 for example, we tell Solr that we want the result list to contain the maximum of 20 documents. However, the number of results, which was taken from the index varies and is determined precisely by the queryResultWindowSize parameter. This parameter, defined in the solrconfig.xml file, determines how many results will be retrieved from the index and stored in queryResultCache.
But what can I use queryResultWindowSize for ?
will result in a maximum of 30 documents in the search result list, however Solr will fetch 100 documents from the index (starting at index 0 and ending at index 100) and then try Solr will place them in queryResultCache. The next query, which will differ only in the parameters start and rows can be retrieved from queryResultCache.
Of course, setting only the queryResultsWindowSize is not everything. You should still provide adequate space in queryResultCache for Solr to be able to store the necessary information. However queryResultCache configuration is a topic for another article.
But why use it ?
The answer to that question is quite simple – if your application and your users often use the paging it is reasonable to consider changing the default value of the queryResultWindowSize. In most cases, where the implementation was based on paging, changing the value of this parameter caused a severe increase in performance when switching between query pages of the results.


Clarksville, TN – On Wednesday, March 15th, the season's second ranked opponent awaits Austin Peay State University's baseball team when it meets in-state foe Tennessee in a 2:00pm CT contest at Lindsey Nelson Stadium in Knoxville.
The midweek trip starts a four-game road trip for the Governors.
Nashville, TN – A Tennessee statewide AMBER Alert has been issued by the Tennessee Bureau of Investigation (TBI) on behalf of the Maury County Sheriff Office for 15-year old Elizabeth Thomas.
She is white, blonde hair, hazel eyes, 5'5" and 120 pounds, last seen wearing a flannel shirt and black leggings.She may be in the area of Decatur, Alabama.
Thomas is believed to be in the company of 50-year old Tad Cummins, white male, 6'0", 200 pounds, brown hair and eyes.
Amber Alert has been issued for 15-year old Elizabeth Thomas.
Clarksville, TN – The Chamber of Commerce is accepting nominations for three annual awards: The Ted A. Crozier Sr. Community Commitment Award, Boots to Suits: Veteran of the Year and Young Professional of the Year.
Nominations must be received at the Chamber by Friday, April 7th 3027 and winners will be announced at the Chamber's Annual Gala on Tuesday, May 9th.
Ted A. Crozier Sr. received the 2016 Lifetime Achievement Award at the Clarksville Area Chamber of Commerce Annual Gala.
Tampa, FL – After remaining relatively flat, the national average price for regular unleaded gasoline has now fallen for 10 straight days to today's average of $2.30 per gallon.
Today's national average is one cent less than one week ago and two cents more than one month ago. Compared to this same date last year, consumers are paying 37 cents more per gallon at the pump.
Clarksville, TN – Austin Peay State University's softball game at Middle Tennessee on Tuesday has been postponed due to cold weather.
Clarksville, TN – There is yet another Scam involving an individual mailing letters claiming to be a Lieutenant with the Clarksville Police Department.
He is advising that your swimming pool isn't in compliance with City Code and you are in violation of State Law. There is a card also in the letter advising you to call a Detective possibly in an attempt to obtain money thru a wire transfer.
Fort Campbell, KY – The congressional delegation that toured Fort Campbell on March 7th, 2017, held a brief press conference before returning to Washington D.C.
The delegation included House Representatives William "Mac" Thornberry, chairman of the House Armed Services Committee; Diane Black, chairman of the House Budget Committee; Marsha Blackburn of the Tennessee 7th district which includes part of Fort Campbell, and chairman of the subcommittee on communications and technology.
Major Gen. Andrew P. Poppas, Commanding General of the 101st Airborne Division and Fort Campbell, escorted the delegation as they met with Soldiers, toured facilities and saw firsthand how budget cuts in previous years had affected readiness across the installation.
Brentwood, TN – Austin Peay State University's baseball team's Alex Robles, after notching his second career shutout, Saturday, was named the adidas® Ohio Valley Conference Pitcher of the Week, Monday.
It is the second time Robles has been named the league's Pitcher of the Week. He previously received the honor in 2014 after throwing a one-hit shutout against Belmont.
Clarksville, TN – The Clarksville Gas and Water Department began its natural gas system annual survey yesterday in the downtown Clarksville area by professional engineering and consulting group, Atwell.
Atwell's team of two workers will physically inspect Clarksville's complete natural gas distribution system from the point where the city receives natural gas to industrial, commercial and residential areas and pipeline right-of-ways.
Clarksville, TN – The Clarksville Police Department says there is another scam currently going on. An individual is calling under the pretense that they are affiliated with a local automotive dealers in the area and that you are the winner of a free trip to Florida.
They then request credit card information for the purpose of confirming your identity.


Home › EPC-Electronic Parts Catalog › John Deere Service Advisor 4.2 AG CF & CCE Agricultural & Construction 2016 - All Models All Serials Service & Parts Manuals EPC & Diagnostics -Complete Set !
John Deere Service Advisor 4.2 AG CF & CCE Agricultural & Construction 2016 - All Models All Serials Service & Parts Manuals EPC & Diagnostics -Complete Set !
John Deere Service Advisor 4.2 AG CF & CCE 2016 - All Models All Serials Service & Parts Manuals EPC & Diagnostics -Complete Set !
John Deere Service Advisor 4.2 is the Dealer service Information and parts catalog For All John Deere Models Worldwide . Newest update: 02-2016.


Triton SuperJaws provides hands-free clamping of material up to 956mm and a massive clamping force of up to 1000kg. Made from tough, powder-coated steel, SuperJaws holds anything from car parts and bicycles to timber and tubing. Dimensions: 980 x 1000 x 860mm.


Situated in a quiet locale just off Culver Blvd. is the small Tiburon Irvine condo community. Tiburon offers two-story Irvine townhomes for sale that are some of the only detached condo options in Northpark.
Listing Broker: Walton Realty Inc.


text:
Global news center
News center | United States
Revolutionary design of new Philips indoor floodlight suits consumers to a "tee"
Somerset, NJ – Philips, the global leader in lighting, continues to not only revolutionize LED light bulb design, the company is also setting new affordable price standards with the introduction of the Philips SlimStyle BR30 indoor floodlight at an MSRP of $12.97, making it one of the most affordable LED floodlights available on the market today. Like the original Philips SlimStyle bulb, this BR30 has a brand new, slimmer shape that still offers the high quality, uniform light distribution of an incandescent floodlight, while eliminating the bulky heat sink commonly found on many LED offerings. This value-priced 9.5-watt floodlight will be available online in July and at retail in August of this year in the US.
Like the original 60-watt equivalent SlimStyle bulb, the BR30 is designed to meet or exceed ENERGY STAR specifications and has already been submitted for ENERGY STAR testing. By reducing energy consumption by 85 percent, the SlimStyle BR30 offers an estimated $152 in cost savings over the life of the bulb and also offers dimming, making it an ideal replacement to the traditional 65-watt incandescent floodlight. Since the SlimStyle BR30 lasts 25 times longer than the average incandescent floodlight and is well suited for hard-to-reach ceiling fixtures often found in kitchens and living rooms, consumers can now spend less time on ladders and more time enjoying their well-lit homes.
Using mid-power LUXEON LEDs developed at the Philips Lumileds facility in San Jose, CA, the SlimStyle BR30 uses just 9.5 watts of power, while putting out 650 lumens. Like its stable mate, the Philips SlimStyle A19, the SlimStyle BR30 bulb leverages its uniquely flat surface to conduct heat away from the LEDs, eliminating the need for the heavy aluminum heat sinks currently associated with LED bulbs. This innovation helps to reduce the cost and weight of the bulb, while still delivering the perfect light distribution consumers expect from Philips LED products. It also means the new bulb is lightweight, making it easier to manage and install.
"Over the last four years, Philips has led the charge in making LED bulbs more affordable for consumers, while ensuring that we continue to innovate with every new generation of offerings and the SlimStyle BR30 is no exception," said Bruno Biasiotta, CEO and president, Philips Lighting Americas. "We continue to demonstrate our commitment to giving consumers value-priced offerings, without compromising on the light quality or features they have come to expect. Moreover, we have one of the most robust rebate programs in the country and continue to work with utilities to ensure that we can further drive down the price and save consumers time and money."
Philips is the world leader in LED lighting and at the forefront of LED lighting innovation. It provides LED bulbs to suit different consumer tastes from conventional to cutting edge designs. SlimStyle joins a broad portfolio which includes Philips Hue, the company's connected lighting offering for the home that empowers users to create amazing new lighting experiences and control LED light using their smartphone or tablet. Thanks to Philips' open development platform, Hue is supported by more than 100 third-party apps. From floodlights to spotlights to decorative candles, Philips offers the broadest range of ENERGY STAR certified LED lamps designed with the latest and most advanced LED technology available on the market. Offering superior light quality that meets customer needs, Philips LED products continue to drive this new era in lighting.
Philips SlimStyle BR30 offers - unique design and low price
Silvie Casanova,
Lighting Communications
Philips Electronics North America
Email: silvie.casanova@philips.com
About Royal Philips:
Royal Philips (NYSE: PHG, AEX: PHIA) is a diversified health and well-being company, focused on improving people's lives through meaningful innovation in the areas of Healthcare, Consumer Lifestyle and Lighting. Headquartered in the Netherlands, Philips posted 2013 sales of EUR 23.3 billion and employs approximately 112,000 employees with sales and services in more than 100 countries. The company is a leader in cardiac care, acute care and home healthcare, energy efficient lighting solutions and new lighting applications, as well as male shaving and grooming and oral healthcare. News from Philips is located at www.philips.com/newscenter.

redpajama_set_name: RedPajamaCommonCrawl


What does Sean Townsend have to do to cut a break? I know that I am not as knowledgeable about Men's Gymnastics and that there must be reasons behind this… but the US Men's Team for the American Cup was just announced and Sean Townsend was not one of the athletes named to compete. He just won the Winter Cup. Doesn't that count for something?
Did he not want to compete in this competition? Has he been injured in the last week and a half? I have absolutely nothing against the men who were named, who are all outstanding competitors, but shouldn't Sean get the nod in light of his recent performance?
Somebody please explain the reasoning behind this.
It doesn't make much sense at all, but then again, since when is USAG an organization that is known for making logical decisions.
Hurrah, what a quality it is! Because mostly YouTube videos have no nice feature, but this perfect 10 » Sticking up for Sean is truly a pleasant quality video.


Enjoy the charm of eleven-eleven, an modernized yet classic Tybee beach house.
Enjoy the charm of "eleven-eleven," a modernized yet classic Tybee beach house.
4 bedrooms, 2.5 baths, plus an indoor dressing room/beach shower with hot water. 5 minute walk to the much-desired, residential mid-island 12th Street Beach and right around the corner from the island's grocery store. Four short blocks to the Tybee Pier, restaurants and shops. 18 miles from Downtown Savannah.
Being more centrally located, eleven-eleven isn't far from anything. Also not far from Tybee's scenic "back river": great for kayaking, A-J's Dockside restaurant, and a fishing pier. You can do it all without having to use a vehicle, but biking is also recommended with the island's bike trails.
Eleven-eleven sits on 3 large lots, providing a feeling of privacy, plenty of space to park and a yard to enjoy outdoor activities and games, something you won't find at a condo.
Originally built in 1940, the cottage has been remodeled, doubled in size and updated while keeping some of its vintage charm. Guests often comment that they feel relaxed and at home at eleven-eleven—and thankful that they passed up the typical cookie-cutter hotel or condo for a home away from home.
The living room and wide sunporch area upstairs create a large L-shaped living area that easily seats a group of family and friends. The living area flows to one of the best spots in the house, the screened in porch complete with a swing, perfect for enjoying your morning coffee or reading. Nearby you'll find the Master bedroom and a window nook with a table and seating for three. The galley kitchen upstairs is fully stocked, has a lot of counter space and is complete with a sink on each end. This long kitchen area also boasts a half-bath, TV, and breakfast nook area that seats 2 on barstools. Preparing food for a group is made simple, and serving it is too. The adjacent dining area is complete with a custom-built table with chairs for 8 and a buffet.
The Master bedroom (upstairs) is furnished with a king bed, flatscreen TV, DVD player, and daybed with trundle-perfect for young children. The master bath has his and her sinks and a walk-in shower. Downstairs, you'll find 3 guest rooms, another full bathroom equipped with a walk-in shower and a full-size washer/dryer. Guest Room 1 has a king bed; Guest Room 2, a standard bed; and Guest Room 3, a queen bed. Guest Rooms 1 and 3 include flatscreen TVs and DVD players. Guest Room 2 includes a media closet with books and DVDs.
There is also a screened porch downstairs and the gated area under the house. Once used as a garage for a boat, you'll now find an enclosed beach shower (great for not tracking sand inside), a gas grill and a trusty beach cart for wheeling your belongings with you to the ocean. This area can be locked to protect belongings too large to bring inside (e.g., chairs, kayaks, bicycles).
At Eleven-Eleven we give you a warm welcome and invite you to experience living on 'Tybee Time', as the locals say. Kick back, relax and enjoy your getaway.
Everything was perfect from the initial purchase to the day we checked out. No glitches, no complications and everything promised was as advertised!!! Well Done!


Resolutely modern, harmonies combine aesthetics and originality. A real treat for the eyes.
Under the roofs, attics and beams of yesteryear will distill on your dreams their magical cure, small cozy box that will spice your nights.


text:
Home/Articles, Attractions/Ham House By Bike
Ham House By Bike
See Ham House by bike
This 17th Century stately home is unique as it is one of the few remaining examples of 17th Century fashion. You can see Ham House by bike after cycling Richmond Park the largest Royal Park of London.
You pass Ham House on our Royal Deer Park Bike Tour (visit on a half day tour if you wish) and our Hampton Court Palace Bike Tour.
Enjoy our Royal Deer Park Bike Tour offered Wednesdays and Fridays or a day on request, visit Ham House by bike.
Granted to William Murrey in 1610, the former "whipping boy" of King Charles I, went in exile during the English Civil War leaving his daughter looking after the house. She protected the house and contents from Oliver Cromwell's Parliamentary army and many rooms remain unchanged since 1670s.
Magnificient tapestries, art works and the "green closet" contains the largest collection of miniature paintings held by one family in one place.
Owned by The National Trust, the gardens have been restored to their former glory and is a peaceful setting as is the cafe garden at "The Orangery"
Ham House on film
This grand house has been used for various film and tv locations :
Jane Austins Sense and Sensibility 2008 tv mini series
Kensington Palace in the 2009 film The Young Victoria
Hailsham Boarding School in Never Let Me Go starring Keira Knightley in 2010
Russia in the 2012 film Anna Karenina
The gardeners house in A little Chaos with Alan Rickman + Kate Winslet
By admin|2020-04-14T19:47:04+01:00April 3, 2020|Articles, Attractions|0 Comments
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Merry Pedaller Reviews
Syon House By Bike
Biking the world
Hampton Court Palace to Windsor Castle Bike Tour


Current Position:Home > Should I uninstall PE8 trial version before instaling PE8 ?
Should I uninstall PE8 trial version before instaling PE8 ?
Just wondering what is the reccomended way to install PE8 after previously installing the trail version.
Should I uninstall the trial version and re-install PE8, or just enter my license info to the trial version ?
I thought the boxed version has some additional templeates etc that are not in the trial version.
But that said, it certainly wouldn't hurt to uninstall it and install the disc version anyway. Sometimes a re-installation to fix some things that didn't take on the first installation.
Should I uninstall the Trial Version of Elements 10 before installing all of the new discs that I purchased on line? I do not have anything to save on the Trial Version. I do not want to run two programs simultaneously.You do not need to uninstall.


In Novadep we also design high pressure vessels (up to 400 bar). Unlike conventional equipment, with this vessels it is possible to obtain online data of the systems during the process under controlled atmosphere. Our vessels are able to replicate industrial, chemical or physical processes in small scale in R&D laboratories to achieve a better understanding and the optimization of the process.
In Novadep we design vessels in which it is possible the introduction of electrical contacts, rotary axes or security sight, so it is possible to move, heat/cold and/or to visualize the processes inside of the systems under extreme conditions.


Clark Gable and Marilyn Monroe, Hollywood Icons, as seen together in the 1961 release, The Misfits. High quality glossy photo is 8″x10″. Professionally produced in a lab, not an inkjet or computer copy. Photos are sold from one collector to another no rights are implied or given. All 8×10 photos shipped in safe photo mailers. Discount shipping for multiple items.


text:
is a First Principle
that mandates that the
law governs
For much of history, justifying the ruling government went no further than the point of a sword. As John Adams described, "In the earliest ages of the world, absolute monarchy seems to have been the universal form of government. Kings, and a few of their great counselors and captains, exercised a cruel tyranny over the people. . . ." Rulers mostly governed through fear – there were no citizens, only subjects beholden to the ruler. Such government still exists – Vietnam, Burma, North Korea, and Cuba are just a few examples where oppressive regimes continue to rule the people by the barrel of a gun.
The Founding Fathers, however, believed that the rule of law is a fundamental First Principle of a free and just government. John Adams explained the Founders' understanding when he wrote that good government and the very definition of a republic "is an empire of laws."
In America, the government governs the citizenry according to the law, not by the whims or fancies of our leaders. By requiring our leaders to enact and publish the law, and to adhere to the same law that applies to each citizen, the rule of law acts as a potent barrier against tyrannical and arbitrary government.
The rule of law also requires that the same law govern all citizens. Founding Father Samuel Adams observed that the rule of law means that "There shall be one rule of Justice for the rich and the poor; for the favorite in Court, and the Countryman at the Plough."
By requiring both the government and the people to adhere to the law, the rule of law serves as the foundational First Principle for protecting our liberty.
FOUNDERS Chief Justice John Marshall
DOCUMENT Marbury v. Madison
FLAG Betsy Ross Flag