text:
\section{Introduction}

\section{Related Work}\label{sec: related_work}
In literature, many different energy functions~\cite{wei2019safe} are proposed to measure safety, including (i) potential function; (ii) safety index;
(iii) control barrier function. Representative methods include potential field method,
barrier function method~\cite{ames2014control}, and safe set algorithm~\cite{liu2014control}.

However, designing such energy functions is difficult and requires great human efforts to find appropriate parameters and function forms.
As for synthesizing \textit{safety index}, the fundamental problem is to ensure nonempty set of safe control for all states.
\cite{zhao2021issa} solves the problem based on worst-case analysis, but it is limited to simple 2D mobile robot dynamics. \cite{wei2022safe} leverages an evolutionary algorithm for safety index parameterization, but the computation time of grid sampling increases exponentially with the dimension of the state space.
As for synthesizing \textit{control barrier functions} (CBFs), safety index is closely related to CBFs as analyzed in~\cite{wei2019safe}. Classical hand-designed CBFs
are hard to scale to nonlinear or high-dimensional systems.
Some automated methods~\cite{ahmadi2016some,clark2021verification} leverage SOSP to synthesize CBFs. However, \cite{ahmadi2016some} is limited to systems with polynomial dynamics.
Existing techniques for synthesizing barrier certificates using SOSP are inapplicable to design CBFs~\cite{clark2021control}. To this end, ~\cite{clark2021verification} develop a SOSP-based program via Positivstellensatz to verify CBFs, but it is also computationally challenging for high-dimensional systems.
To designs certificate functions in complex systems, another line of work~(\cite{liu2022safe, dawson2022safe}) proposes to learn approximate CBFs with neural networks, but these methods lack theoretical guarantees of safety in practice.

\section{Problem Formulation} \label{sec: formulation}

This section introduces system dynamics and safety specification, and formulates the safety index synthesis problem.

\paragraph*{Dynamics}
 Let $x \in \mathcal{X}\subset \mathbb{R}^{n_x}$ be the robot state, where $n_x$ is the dimension of the bounded state space $\mathcal{X}$; $u \in \mathcal{U}\subset \mathbb{R}^{n_u}$ be the control input to the robot, where $n_u$ is the dimension of the control space $\mathcal{U}$, and $\mathcal{U}$ is bounded by a $n_u$-dimensional orthotope, i.e.
\begin{align}
    \forall i = 1,2,\cdots,n_u, \, \mathcal{U}^{[i]} \in [u^{[i]}_{min}, u^{[i]}_{max}],
\end{align}
where $\mathcal{U}^{[i]}$ denotes the $i$-th dimensional control space, and $u^{[i]}_{min}, u^{[i]}_{max} \in \mathbb{R}$. In this paper, we consider a control-affine system, and the system dynamics are defined as:
\begin{equation}\label{eq:dynamics fn}
\begin{split}
    &\dot{x} = f(x) + g(x) u, \\
\end{split}
\end{equation}
where $f: \mathbb{R}^{n_x} \to \mathbb{R}^{n_x}$, $g: \mathbb{R}^{n_x} \to \mathbb{R}^{n_x \times n_u}$ are locally Lipschitz continuous on $\mathbb{R}^{n_x}$. Here we highlight that $f(x), g(x)$ are not necessarily polynomials, and they satisfy the following definition:

\begin{definition}(Substitutable Function)
\label{def: subs}
A function is substitutable if its variables can be projected into the higher dimensional polynomial manifold, such that the function can be rewritten as polynomials.
\end{definition}

\begin{remark}
To better understand \Cref{def: subs}, we give an example of a substitutable function. Consider function
\begin{align*}
    f^*(\theta) := \sin\theta \cos\theta,
\end{align*}
by substituting $\sin\theta$ with $\mathfrak{a}$, and $\cos\theta$ with $\mathfrak{b}$, respectively. $f^*(\theta)$ can be rewritten as following polynomial
\begin{align*}
    f^*(\mathfrak{a},\mathfrak{b}) := \mathfrak{a}\mathfrak{b},
\end{align*}
where $\mathfrak{a}^2 + \mathfrak{b}^2 - 1 = 0$.
\end{remark}

\paragraph*{Safety Specification}
The safety specification requires the system state should be constrained in a closed subset in the state space, called the safe set $\mathcal{X}_S$. The safe set can be represented by the zero-sublevel set of a continuous and piecewise smooth function $\phi_0:\mathbb{R}^{n_x} \rightarrow \mathbb{R}$, i.e., $\mathcal{X}_S = \{x\mid \phi_0(x)\leq 0\}$. $\mathcal{X}_S$ and $\phi_0$ are directly specified by users. The design of $\phi_0$ is straightforward in most scenarios. For example, for collision avoidance, $\phi_0$ can be designed as the negative closest distance between the robot and environmental obstacles.

\paragraph*{Problem}
Since $\mathcal{X}_S$
may contain states that will inevitably go to the unsafe set no matter what the control input is, we need to assign high energy values to these inevitably-unsafe states. Then, safe control can be applied to
ensure \textbf{forward invariance} in a subset of the safe set $\mathcal{X}_S$ by dissipating the energy.
\textit{Forward invariance} of a set means that the robot state will never leave the set if it starts from the set, i.e. when $\phi_0(x(t_0)) \leq 0$, then $\phi_0(x(t)) \leq 0,\ \forall t>t_0$.

In this paper, we adopt safe set algorithm (SSA) ~\cite{liu2014control}, which is an \textit{energy function-based method} for safe control.
SSA has introduced a rule-based approach to synthesize the energy function as a continuous, piece-wise smooth scalar function $\phi:\mathbb{R}^{n_x} \rightarrow \mathbb{R}$. And the energy function $\phi(x)$ is called a safety index. The general form of the safety index is $\phi = \phi_0^* + k_1\dot\phi_0 + \cdots + k_n\phi_0^{(n)}$ where 1) the roots of $1+k_1s+\ldots + k_n s^n=0$ are all negative real (to ensure zero-overshooting of the original safety constraints); 2) the relative degree from $\phi_0^{(n)}$ to $u$ is one (to avoid singularity); and 3) $\phi_0^*$ defines the same set as $\phi_0$ (to nonlinear shape the gradient of $\phi$ at the boundary of the safe set).
In our paper, we assume $\frac{\partial \phi}{\partial x}$ satisfies \Cref{def: subs}.

It is shown in \cite{liu2014control} that if the control input is unbounded ($\mathcal{U} =\mathbb{R}^{n_u}$), then there always exist a control that satisfies the constraint $\dot\phi \leq 0 \text{ when } \phi= 0$; and if the control input always satisfies that constraint, then the set $\{x\mid\phi(x)\leq 0\}\cap \{x\mid \phi_0(x)\leq 0\}$ is forward invariant.
In practice, when $\phi=0$, the safe control $u$ is computed through a quadratic projection of the nominal control $u^r$:
\begin{align}\label{eq: quadratic program for control}
    u =& \argmin_{u\in\mathcal{U}} \|u-u^r\|^2\text{ s.t. } \dot\phi \leq 0.
\end{align}

\begin{figure}[htbp]
    \centering
    \vspace{-20pt}
    \includegraphics[width=\columnwidth]{fig/problem_illustration-crop.pdf}
    \caption{The manifold of safety index and sets of safe control.}
    \vspace{-5pt}
    \label{fig:problem_illustration}
\end{figure}
The issue with \eqref{eq: quadratic program for control} is that this safe controller can saturate as shown in \Cref{fig:problem_illustration} where the sets of safe control may have no intersection with the limited control space. Specifically, saturation occurs when  there does not exist a control input that satisfies the constraint $\dot\phi \leq 0 \text{ when } \phi= 0$,
causing the loss of safety guarantees. This can happen because we have not yet accounted for control limits in our design of $\phi$. Hence, the core problem of this paper is to choose a proper parameterization of safety index that guarantees the existence of safe control (within control limits) to ensure $\dot\phi \leq 0 \text{ for all states where } \phi= 0$. We call this process as \textit{Safety Index Synthesis with Control Limits}. Mathematically, \textit{Safety Index Synthesis} solves the following problem:
\begin{problem}[\textit{Safety Index Synthesis}]
\label{def:problem 1}
Construct safety index as $\phi_\Theta = \phi_0 + k_1\dot\phi_0 + \cdots + k_n\phi_0^{(n)}$, with tunable parameters set $\Theta = \{k_1, k_2, \cdots, k_n\}$ and $\forall i = 1,2,\cdots,n, k_i \in \mathbb{R}^+$, such that
\begin{align}
\label{eq: problem_ori}
    \forall x \in \mathcal{X} \text{ s.t. } \phi_\Theta(x) = 0, \min_{u \in \mathcal{U}} \dot{\phi}(x, u) < 0.
\end{align}
\end{problem}
However, solving Problem \ref{def:problem 1} is not a trivial task. Note that \eqref{eq: problem_ori} contains infinitely many constraints since for every state $x, \text{s.t. } \phi_\Theta(x) = 0$, poses an inequality constraint, hence cannot be directly solved by off-the-shelf nonlinear programming solvers. To deal with this issue, we leverage tools from SOSP (to be reviewed in the following section) to repose the problem as a nonlinear program.

\section{Review of Sum of Square Programming}\label{sec:sos}

Optimization problems with \textit{global positiveness constraints}~\cite{zhao2022provably} in the form of finding a function $F(x), \text{ s.t. } \forall x\in\mathbb{R}^{n_x}, G\circ F(x)>0$, where $G$ is a function,
have been widely studied and successfully applied to many problems~\cite{zhao2022provably}.
SOSTOOLS~\cite{papachristodoulou2013sostools} implements SOSP to solve the problems with \textit{global positiveness constraints}.
However, our problem requires $\min_{u \in \mathcal{U}} \dot{\phi}(x, u)$ to be nonpositive on a limited-size manifold, i.e. $x\in \mathcal{X}, \phi(x) = 0$. Hence, Problem \ref{def:problem 1} cannot be directly solved by SOSTOOLS.
Instead, we will reconsider the theory behind SOSP to solve Problem \ref{def:problem 1}.

To ensure \textit{global positiveness} of a condition, the easiest way is to show that there does not exist any \revision{solution such that} the condition is violated (refute set).
Constructing the refute set and showing it is empty is the core idea behind SOSP~\cite{parrilo2003semidefinite} to ensure \textit{global positiveness}. To show that the refute set is empty, we need to invoke the equivalence conditions in \textit{Positivstellensatz}~\cite{parrilo2003semidefinite}.
Before introducing \textit{Positivstellensatz}, we first review a key concept: ring-theoretic \textit{cone}~\cite{bochnak2013real}.

  \begin{definition}[Cone]
   Denote $\mathbb{R}[x_1,\ldots,x_n]$ a set of polynomials with $[x_1,\ldots,x_n]$ as variables. The cone $\Gamma^*$ of $\mathbb{R}[x_1,\ldots,x_n]$ is defined as a subset of $\mathbb{R}[x_1,\ldots,x_n]$ satisfying the following properties:
1) $a,b \in \Gamma^* \Rightarrow a+b \in \Gamma^*$;
2) $a,b \in \Gamma^* \Rightarrow a\cdot b \in \Gamma^*$;
3) $a\in\mathbb{R}[x_1,\ldots,x_n] \Rightarrow a^2 \in \Gamma^*$.
  \end{definition}

  \begin{definition}[Ring-theoretic cone]\label{prop}
  For a
  set $S = {\gamma_1,\ldots,\gamma_s} \subseteq \mathbb{R}[x_1,\ldots,x_n]$, the associated ring-theoretic \textit{cone} can be expressed as:
  \begin{equation} \nonumber
      \Gamma = \{ p_0 + p_1\gamma_1 + \ldots + p_s\gamma_s + p_{12}\gamma_1\gamma_2 +\ldots + p_{12\ldots s}\gamma_1\ldots\gamma_s\}~,
  \end{equation}
  where
  $\Gamma$ is a subset of cone of $S$ and $p_0, \ldots,p_{12\ldots s}\in\text{SOS}$
  \end{definition}

  Based on the ring-theoretic \textit{cone},  \textit{Positivstellensatz} condition is specified in the following theorem.

  \begin{theorem}[\textit{Positivstellensatz}] \label{theo:posi}
  Let $(\gamma_j)_{j=1,\ldots,s}$, $(\psi_k)_{k=1,\ldots,t}$, $(\zeta_l)_{l=1,\ldots,r}$ be finite families of polynomials in $\mathbb{R}[x_1,\ldots,x_n]$. Let $\Gamma$ be the ring-theoretic cone generated by $(\gamma_j)_{j=1,\ldots,s}$, $\Xi$ the multiplicative monoid~\cite{bochnak2013real} generated by $(\psi_k)_{k=1,\ldots.,t}$, and $\Phi$ the ideal~\cite{bochnak2013real} generated by $(\zeta_l)_{l=1,\ldots,r}$. Then, the following properties are equivalent:
  \begin{enumerate}
      \item The following set is empty
          \begin{align}\label{eq: them1 empty set}
          \begin{cases}
            x \in \mathbb{R}^n \Biggr|\begin{array}{ll}
                                    \gamma_j(x)\geq 0,j = 1,..,s\\
                                    \psi_k(x)\neq 0, k = 1,\ldots,t\\
                                    \zeta_l(x)=0, l = 1,\ldots,r
                                    \end{array}
          \end{cases}.
          \end{align}
      \item There exist  $\gamma \in \Gamma, \psi\in \Xi, \zeta \in \Phi$ such that
      \begin{align}
          \gamma+\psi^2+\zeta = 0,
          \label{eq:pos}
      \end{align}
  \end{enumerate}
  where $\psi = 1$ if
  $t = 0$.
  \end{theorem}

  \revision{We refer the reader to \cite{bochnak2013real} for the proofs of \Cref{theo:posi}.} In summary, \textit{Positivstellensatz} shows that the refute set being empty is equivalent to a feasibility problem in \eqref{eq:pos}.
  Therefore, we can follow the same procedure
  to construct a refute set for
Problem \ref{def:problem 1}
  and then use \textit{Positivstellensatz} to turn the problem into a feasibility problem similar to \eqref{eq:pos}, which can then be formed into an ordinary nonlinear program.

\section{Method} \label{sec: method}
In this section, we introduce the proposed optimization algorithm to efficiently solve Problem \ref{def:problem 1} leveraging \Cref{theo:posi}. The overall procedure of the algorithm contains four main steps. Firstly, to leverage techniques from SOSP, we unroll the analytical forms of $\min_{u \in \mathcal{U}} \dot{\phi}(x, u)$ and construct the \textit{local manifold positiveness problem}. Secondly, we apply \textit{Positivstellensatz} to construct the corresponding \textit{refute problem} \eqref{eq: refute problem} of the \textit{local manifold positiveness problem} \eqref{eq: local manifold problem}. Thirdly, to efficiently find solutions, we simplify the \textit{refute problem} and further transform it to a \textit{nonlinear programming problem} \eqref{probfinal}. Lastly, we solve the \textit{nonlinear programming problem} using off-the-shelf nonlinear programming solvers.  In the following subsections, we will introduce the details of three critical problems in these four steps.

\subsection{The Local Manifold Positiveness Problem}
\label{sec:lmp}
Since \Cref{theo:posi} applies to a set of equalities, inequations, and inequalities, we need to unroll $\min_{u \in \mathcal{U}} \dot{\phi}(x, u)$ and get rid of
$\min$
operator. Firstly, the equivalent form of $\min_{u \in \mathcal{U}} \dot{\phi}(x, u)$ is summarized as following:
\begin{align}
\label{eq:infimum dot phi ori}
    \min_{u \in \mathcal{U}} \dot{\phi}(x, u) &= \min_{u \in \mathcal{U}} \underbrace{\frac{\partial \phi}{\partial x}f(x)}_{L_f} + \underbrace{\frac{\partial \phi}{\partial x}g(x)}_{L_g}u \\ \nonumber
    &= \min_{u \in \mathcal{U}} L_f + \sum_{i = 1}^{n_u} L_g^{[i]}u^{[i]},
\end{align}
where $L_g^{[i]}$ and $u^{[i]}$ denote the $i$-th dimension of $L_g$ and $u$, respectively.
\eqref{eq:infimum dot phi ori} can be rewritten as
\begin{align}
\label{eq:infimum dot phi analytical}
    \min_{u \in \mathcal{U}} \dot{\phi}(x, u) &=  L_f + \sum_{i = 1}^{n_u} L_g^{[i]} \mathbb{I}_{L_g^{[i]}}\{u^{[i]}_{min}, u^{[i]}_{max}\}, \\ \nonumber
\end{align}
where we define operator $\mathbb{I}$ as $\mathbb{I}_{A}\{B,C\} = B$ if $A \geq 0$, and $\mathbb{I}_{A}\{B,C\} = C$ if $A < 0$. Therefore, Problem \ref{def:problem 1} becomes
\begin{problem}[\textit{Local Manifold Positiveness}]
\label{eq: local manifold problem}
Find $\Theta = \{k_1, k_2, \cdots, k_n\}$, such that
\begin{align}
    & \forall_{x \in \mathcal{X}, \phi_\Theta = 0}, L_f + \sum_{i = 1}^{n_u} L_g^{[i]} \mathbb{I}_{L_g^{[i]}}\{u^{[i]}_{min}, u^{[i]}_{max}\} < 0,
\end{align}
where \textbf{local} denotes the state space $\mathcal{X}$ which is a subset of $\mathbb{R}^{n_x}$, and \textbf{manifold} denotes we are considering the states on the manifold defined by $\phi_\Theta = 0$.
\end{problem}

\begin{remark}
The state-of-the-art SOSP-based safety index synthesis methods try to synthesis $\phi_\Theta$, such that $\forall_{x\in\mathbb{R}^{n_x}},\exists_{u\in\mathcal{U}}\dot\phi_{\Theta}(x,u) < 0$, which is a \textit{global positiveness} constraint. On the other hand, our proposed formulation only requires the existence of safe control on a local set of critical states, i.e. \textit{local manifold positiveness} constraint. Therefore, our proposed method considers a larger solution space; hence it is easier to find a solution, and the synthesized safety index is less conservative.
\end{remark}

\subsection{The Refute Problem}\label{subsec:refute_prob}
We can show that the \textit{local manifold positiveness constraint} in Problem \ref{eq: local manifold problem} is satisfied by showing its refute set is empty. The refute set is constructed as:
\begin{equation}
    \begin{split}
    \begin{cases}
      L_f + \sum_{i = 1}^{n_u} L_g^{[i]} \mathbb{I}_{L_g^{[i]}}\{u^{[i]}_{min}, u^{[i]}_{max}\} \geq 0\\
      x \in \mathcal{X} \\
      \phi_\Theta = 0
    \end{cases}
        \label{eq:refute first}
    \end{split}
\end{equation}
Note that the constraint $x \in \mathcal{X}$ can always be represented by a set of inequalities $\{S(x)_i \geq 0\}, i = 1,2,\cdots,n_S$ . For example, suppose $n_x = 1$ and $x \in \mathcal{X} = [-1,1]$, then $x \in \mathcal{X}$ is equivalent to $S(x)_1 = 1 - x^2 \geq 0$.

Additionally, denote $\mathcal{N},\mathcal{M}\subseteq \{1,2,\ldots,n_u\}$, $\mathcal{N} + \mathcal{M} = \{1,2,\ldots,n_u\}$ and $\mathcal{N}\cap\mathcal{M} = \emptyset$. We also denote $\forall i \in \mathcal{N}, L_g^{[i]} \geq 0$, and $\forall i \in \mathcal{M}, L_g^{[i]} \leq 0$. Then, \eqref{eq:refute first} corresponds to $2^{n_u}$ instances of refute set,
since each $L_g^{[i]}$ can be either negative or nonnegative, yielding $2^{n_u}$ combinations of $\{\mathcal{N},\mathcal{M}\}$. Then each instance of \eqref{eq:refute first} can be rewritten as:
 \begin{equation}
    \begin{split}
    \begin{cases}
      \gamma^*_0 := L_f + \sum_{i \in \mathcal{N}} L_g^{[i]}u^{[i]}_{min} + \sum_{i \in \mathcal{M}} L_g^{[i]}u^{[i]}_{max} \geq 0\\
      \gamma^*_i := S(x)_i \geq 0, i = 1,2,\cdots,n_S \\
      \gamma^*_{n_S+i} := L_g^{[i]} \geq 0, i \in \mathcal{N} \\
      \gamma^*_{n_S+|\mathcal{N}|+i} := -L_g^{[i]} \geq 0, i \in \mathcal{M} \\
      \zeta^* := \phi_\Theta = 0
    \end{cases}
        \label{eq:refute second}
    \end{split}
\end{equation}
\Cref{theo:posi} enables us to turn the emptiness problem for \eqref{eq:refute second} to a feasibility problem similar to \eqref{eq:pos}. As a result, we can show \eqref{eq:refute second} is empty using the following condition:
\begin{align}
\label{eq: refute 3rd}
    &\exists q_1 \in \mathbb{R}[x], \exists p_i\in\text{SOS},\forall i, \\ \nonumber
\text{ s.t. } & \gamma = p_0 + p_1\gamma_0^* + \ldots +  p_s\gamma_N^* \\ \nonumber
    & + p_{01}\gamma_0^*\gamma_1^* + \ldots + p_{012\ldots N}\gamma_0^*\ldots\gamma_N^*, \\ \nonumber
     & \zeta = q_1 \zeta^*, \\ \nonumber
     & \gamma+\zeta+1 = 0 ,
\end{align}
where $N = n_S+|\mathcal{N}|+|\mathcal{M}|$.
Therefore, the Problem \ref{eq: local manifold problem} can be turned into the following equivalent problem:

\begin{problem}[\textit{Refute}]
\label{eq: refute problem}
find $\Theta = \{k_1, k_2, \cdots, k_n\}$, such that
\begin{align}
    \forall \mathcal{N,M,} \text{ \eqref{eq: refute 3rd} holds}

\end{align}
\end{problem}

\begin{remark}
It is noteworthy that \Cref{theo:posi} requires $\gamma^*$ and $\eta^*$ to be polynomials, which does not necessarily hold in practice.
Therefore, the state-of-the-art SOSP-based safety index synthesis methods are inapplicable to many practical systems. On the other hand, as long as the system dynamics satisfy \Cref{def: subs}, our proposed method can be applied by projecting non-polynomial terms into a higher-dimensional polynomial manifold, and introducing additional polynomial equality constraints.

\end{remark}

\subsection{The Nonlinear Programming Problem}\label{SDP}
To efficiently search for the existence of SOS polynomials $\{p_i\}$ and polynomials $\{q_i\}$, we set $p_i$ to be a positive scalar $\alpha_i$ and $q_i$ to be a scalar $\beta_i$ for all $i\geq1$.
Hence, a simplified condition of \eqref{eq: refute 3rd} can be defined:
\begin{align}
\label{eq: probsufficient}
    &\exists \beta_i \in \mathbb{R}, \exists \alpha_i \geq 0, \forall i\geq 1, \\ \nonumber
\text{ s.t. } & p_0 = -\alpha_1\gamma_0^* \ldots - \alpha_s\gamma_N^* - \alpha_{01}\gamma_0^*\gamma_1^* -\ldots\\ \nonumber
        &\quad \quad - \alpha_{012\ldots N}\gamma_0^*\ldots\gamma_N^* - \beta_1\zeta^* - 1  \in\text{SOS},
\end{align}

By searching for limited types of SOS polynomials, the existence of $\beta_i, \alpha_i$ satisfying \eqref{eq: probsufficient} is sufficient to satisfy the constraints in \eqref{eq: refute 3rd}.

Denote $\Omega = [\beta_1, \alpha_1, \alpha_2, \ldots , \alpha_{012\ldots N}]$ as the decision vector for \eqref{eq: probsufficient}. To solve \eqref{eq: probsufficient}, suppose the degree of $p_0$ is $2d$, and we first do a sum-of-squares decomposition of $p_0$ such that $p_0 = Y^\top Q^*(\Theta, \Omega) Y$, where $Q^*$ is symmetric and $Y = \begin{bmatrix}1 & x^{[1]} & x^{[2]} & \ldots & x^{[n]} & x^{[1]}x^{[2]} & \ldots & \big(x^{[n]}\big)^d \end{bmatrix}$ and $x^{[i]}$ is the entry in $x$. Specifically, for off-diagonal terms $Q^*_{ij}$ as the element of $Q^*$ at $i$-th row and $j$-th column ($i \neq j$), \revision{assuming that the} coefficient of the term $Y_iY_j$ in $p_0$ is $w_{ij}$, we set $Q^*_{ij} = \frac{w_{ij}}{2}$.
With decomposed $Q^*$, the condition of \eqref{eq: probsufficient} can be rewritten as:
\begin{equation}
    \begin{split}

        & \det(Q^*(\Theta, \Omega)_k) > 0, \forall k,\\

    \end{split}
    \label{eq: sdp}
\end{equation}
where $Q^*(\Theta, \Omega)_k$ denotes the $k \times k$ submatrix consisting of the first $k$ rows and columns of $Q^*(\Theta, \Omega)$.

According to Problem \ref{eq: refute problem}, there are $2^{n_u}$ sets of condition \eqref{eq: refute 3rd} and hence $2^{n_u}$ sets of condition \eqref{eq: probsufficient}. Denote the $\Omega_j = [\beta_1^j, \alpha_1^j, \alpha_2^j, \ldots , \alpha_{012\ldots N}^j]$ as the decision vector for $j$-th set of condition \eqref{eq: probsufficient}. With \eqref{eq: sdp}, Problem \ref{eq: refute problem} can be rewritten as:

\begin{problem}[\textit{Nonlinear Programming}]
\label{probfinal}
Find $[\Theta, \Omega_1, \Omega_2, \cdots, \Omega_{2^{n_u}}]$, such that
\begin{align}
    & \det(Q^*(\Theta, \Omega_j)_k) > 0, \forall k, \forall j, \\ \nonumber
    & \alpha_i^j > 0, \beta_i^j\in\mathbb{R}, \forall i, \forall j.
\end{align}
\end{problem}

Note that Problem \ref{probfinal} is a nonlinear programming problem without any objective. Hence, any arbitrary objective can be added to Problem \ref{probfinal}, and it can be solved efficiently by off-the-shelf nonlinear programming solvers.

\begin{remark}
Note that the state-of-the-art SOSP-based safety index synthesis methods
need to solve multiple sets of SOSP iteratively.
However, Problem \ref{probfinal} demonstrates that our method just needs to solve one nonlinear program problem. Hence, our method is more computationally efficient than the state-of-the-art SOSP-based safety index synthesis method, and is easier to scale to high-dimensional dynamics systems.
\end{remark}

\begin{comment}
\begin{algorithm}
\caption{Joint Tolerance Estimation}\label{alg:sos}
\begin{algorithmic}[1]
\Procedure{JTE}{$\bm{x}^r$, $f(\cdot)$}
\State Construct the lower bound problem \eqref{problowerbound}.
\State Construct the refute  problem \eqref{probafterposi}.
\State Construct the  nonlinear programming Problem \ref{probfinal}
\State Solve \eqref{probfinal} using nonlinear programming solvers.
\EndProcedure
\end{algorithmic}
\end{algorithm}

The integrated algorithm for joint tolerance estimation is summarized in \cref{alg:sos}, which efficiently finds the lower bound solution of \eqref{probori}.
\end{comment}

\section{Properties of General Safety Index Design}\label{sec:proof}
This section proves that we can obtain a feasible safety index design by solving Problem \ref{probfinal}. The main result is summarized in the following theorem.

\begin{theorem}[Feasibility of General Safety Index Design]
The solution of Problem \ref{probfinal}   is also a solution of Problem \ref{def:problem 1}.
\label{theo1}
\end{theorem}

\begin{proof}
Denote the solution of Problem \ref{probfinal} as  $\Theta_3$, the solution of Problem \ref{eq: refute problem} as $\Theta_2$, the solution of Problem \ref{eq: local manifold problem} as $\Theta_1$,
and the solution of Problem \ref{def:problem 1} as $\Theta^o$.

\textbf{Relationship between $\Theta_1$ and $\Theta^o$}: According to \eqref{eq:infimum dot phi ori}, the following condition holds
\vspace{-5pt}
\begin{align}
\label{eq:inf dot phi is what}
    \min_{u \in \mathcal{U}} \dot{\phi}(x, u) &= L_f +  \sum_{i = 1}^{n_u} \min_{u \in \mathcal{U}} L_g^{[i]}u^{[i]},
    \vspace{-5pt}
\end{align}
where $\min_{u \in \mathcal{U}} L_g^{[i]}u^{[i]} = L_g^{[i]}\sup_{u\in\mathcal{U}}u^{[i]}$ if $L_g^{[i]} < 0$, and $\min_{u \in \mathcal{U}} L_g^{[i]}u^{[i]} = L_g^{[i]}\inf_{u\in\mathcal{U}}u^{[i]}$ if $L_g^{[i]} \geq 0$. Hence, the condition $\min_{u \in \mathcal{U}} \dot{\phi}(x, u)$ is equivalent to the condition $L_f + \sum_{i = 1}^{n_u} L_g^{[i]} \mathbb{I}_{L_g^{[i]}}\{u^{[i]}_{min}, u^{[i]}_{max}\}$. So $\Theta_1$ is an instance of $\Theta_o$.

\textbf{Relationship between $\Theta_1$ and $\Theta_2$}:

  For the constraint $\forall_{x \in \mathcal{X}, \phi_\Theta = 0}, L_f + \sum_{i = 1}^{n_u} L_g^{[i]} \mathbb{I}_{L_g^{[i]}}\{u^{[i]}_{min}, u^{[i]}_{max}\} < 0$, its refute certification is $\forall_{x \in \mathcal{X}, \phi_\Theta = 0}, L_f + \sum_{i = 1}^{n_u} L_g^{[i]} \mathbb{I}_{L_g^{[i]}}\{u^{[i]}_{min}, u^{[i]}_{max}\} \geq 0$.
  By introducing auxiliary variables, the refute certifications of the constraints in Problem \ref{eq: local manifold problem} can be written as  \eqref{eq:refute second}. By \revision{\Cref{theo:posi}}, \cref{prop} for $\Gamma$ and definition for $\Phi$~\cite{bochnak2013real}, we know  \eqref{eq:refute second} is equivalent to \eqref{eq: refute 3rd}. Hence, Problem \ref{eq: local manifold problem} is equivalent to Problem \ref{eq: refute problem}, which indicates $\Theta_2$ is an instance of $\Theta_1$.

\textbf{Relationship between $\Theta_2$ and $\Theta_3$}:
The equivalent condition to \eqref{eq: probsufficient} is that $Q^*$ is positive semidefinite~\cite{parrilo2003semidefinite}. The condition \eqref{eq: sdp} proves that $Q^*$ is positive definite according to Sylvester's criterion.
Hence, the decision vectors $\{\Omega_1, \Omega_2, \cdots, \Omega_{2^{n_u}}\}$ satisfying condition \eqref{eq: sdp} also satisfy condition \eqref{eq: probsufficient}.

The simplest SOS is a positive constant scalar, \textit{i}.\textit{e}.,  $\alpha \geq 0 \in \text{SOS}$. Similarly, the simplest polynomial is a constant scalar, \textit{i}.\textit{e}.,  $\beta \in \mathbb{R}$. Hence, by substituting $p_i$ with $\alpha_i\geq0$ for $i = 1,2,\ldots$, and substituting $q_i$ with $\beta_i \in \mathbb{R}$, condition \eqref{eq: probsufficient} can be rewritten as \eqref{eq: refute 3rd}. Therefore, the existence of $\{\Omega_1, \Omega_2, \cdots, \Omega_{2^{n_u}}\}$ satisfying the conditions in Problem \ref{probfinal} implies the satisfaction of conditions in Problem \ref{eq: refute problem}. Therefore, $\Theta_3$ is an instance of $\Theta_2$.

In summary, $\Theta_3$ is an instance of $\Theta_2$, hence an instance of $\Theta_1$, and hence an instance of $\Theta^o$, which verifies the claim.
\end{proof}

\section{Numerical Study}\label{sec: exp}
In this section, we apply the proposed method to synthesize safety index for robot arms and autonomous vehicles, where second-order systems are considered.
More experiments with higher-order systems are left for future work.

\subsection{Robot Arm Numerical Study Setup}
We evaluate our method in 2D robot arm systems with different degrees of freedom (DOF). The experimental platform is illustrated in \Cref{fig:nDOFS robot arm}. The link length of the robot is 1 meter. The obstacle is set as a half plane
$0.5 \times \mathfrak{n}$
meter away from the robot base with $\mathfrak{n}$DOF.
\begin{wrapfigure}{r}{0.15\textwidth}
    \centering
    \includegraphics[width=0.15\textwidth]{fig/2Drobotarm-crop.pdf}
    \caption{$\mathfrak{n}$DOFs robot manipulator.}
    \label{fig:nDOFS robot arm}
\vspace{-20pt}
\end{wrapfigure}
The state space includes 1) each joint angle and 2) each joint velocity. The control inputs are accelerations of each joint.
The acceleration is limited within $[-1,1]$.
The user-defined safety specification ($\phi_0$) requires the end effector to avoid collision with the wall.

\subsection{Robot Arm Running Example}
In this subsection, we introduce a running example showing 1DOF robot arm safety index design via SOSP.
Consider 1) robot state $x = [\theta, \dot{\theta}]$, where $\theta \in [\pi/3,2\pi/3]$, $\dot\theta \in [-1,1]$;  2) robot control $u = [\ddot{\theta}]$, where $\ddot \theta \in [-1, 1]$.
Consider the user-defined safety index as $\phi_0 =  cos\theta - 0.5$.
Then, safety index becomes $\phi = \cos\theta -0.5 - k\sin\theta\dot \theta$,
with $\dot\phi = -\sin\theta \dot \theta - k \cos\theta \dot\theta^2 - k\sin\theta\ddot\theta$.
Since $\sin\theta > \frac{\sqrt{3}}{2}$, the minimum $\dot \phi$ is achieved when $\ddot \theta = 1$, i.e. $\min_{\ddot\theta} \dot \phi = -\sin\theta \dot \theta - k \cos\theta \dot\theta^2 - k\sin\theta$.
So the fundamental condition for nonempty set of safe control is:
\vspace{-3pt}
\begin{equation}
\label{fundamental}
    \left\{
        \begin{array}{cl}
         & f_1  = -\sin\theta\dot\theta - k\cos\theta\dot\theta^2 - k\sin\theta < 0 \\
         & f_2 = \theta - \frac{\pi}{3} \geq 0\\

         & f_3 = \frac{2\pi}{3} - \theta \geq 0 \quad\quad\quad \\
         & f_4 = 1 - \dot\theta^2 \geq 0 \quad\quad\quad \\
         & h_1 = \cos\theta - 0.5 - k \sin\theta \dot \theta  = 0 \quad\quad\quad

        \end{array}
    \right.
    \vspace{-3pt}
\end{equation}
Next, the equivalent refute set for \eqref{fundamental}  can be constructed by
replacing $f_1 < 0$ in \eqref{fundamental} with $f_1 \geq 0$, which indicates that when $\phi$ hits zero, no $x \in \mathcal{X}$ would satisfy $\min_{\ddot \theta}\dot \phi > 0$. By projecting the non-polynomial terms of \eqref{fundamental} into higher dimensional polynomial manifold, i.e. substituting $\sin\theta = x_1, \cos\theta = x_2, \dot\theta = x_3$,
 the refute set becomes:
 \vspace{-3pt}
\begin{equation}
\label{hiya}
    \left\{
        \begin{array}{cl}
         & f_1 = -x_1x_3 - kx_2x_3^2 - kx_1 \geq 0\\
         & f_2 = x_1 \geq \frac{\sqrt{3}}{2} \\
         & f_3 = 1 - x_1 \geq 0 \\
         & f_4 = 1 - x_3^2 \geq 0 \\
         & h_1 = x_2 - 0.5 - k x_1 x_3  = 0\\
         & h_2 = x_1^2 + x_2^2 - 1 = 0
        \end{array}
    \right.
    \vspace{-3pt}
\end{equation}
According to \Cref{theo:posi}, the emptiness of \eqref{hiya} is equivalent to the existence of $p_0, p_1, p_2, p_3, p_4 \in \text{SOS polynomials}$, $q_1, q_2 \in
\text{ polynomials}$, such that:
\vspace{-2pt}
\begin{align}\nonumber
    p_0 + p_1f_1 + p_2f_2 + p_3f_3 + p_4f_4 + q_1h_1 + q_2h_2 + 1 = 0
    \vspace{-2pt}
\end{align}

By considering a reduced parameter space, the emptiness of \eqref{hiya} in equivalent to the existence of $p_1, p_2, p_3, p_4 \in \mathbb{R}^+$, and $q_1, q_2 \in \mathbb{R}$, such that
\vspace{-2pt}
\begin{align}\nonumber
    p_0 = -(p_1f_1 + p_2f_2 + p_3f_3 + p_4f_4 + q_1h_1 + q_2h_2 + 1)
    \vspace{-2pt}
\end{align}
is a sum of squares polynomial, where $[p_1, p_2, p_3, p_4,q_1, q_2]$ can be searched via nonlinear programming.

\subsection{Robot Arm Results}
To solve for safety index design, we first provide reference $[\Theta, \Omega_1, \Omega_2, \cdots, \Omega_{2^{n_u}}]$ via random sampling, and then solve Problem \ref{probfinal} using MATLAB \verb+fmincon+ function. To evaluate safety index design, we randomly initialize the robot, and use safe set algorithm~\cite{liu2014control} to safeguard the robot for $2000$ time steps, where the robot takes the most dangerous reference control (i.e., increasing $\phi$).
We calculate the following statistics across the trials:
\begin{itemize}
    \item \textbf{Computing Time}: The average time to compute a safety index parameterization.

    All experiments are performed with a 2.3GHz Intel Core i5 Processor.

    \item \textbf{Validness}:
    If safe control can be found over 1000 evaluation runs, we

    regard such parameterization of the safety index as valid.

    \item \textbf{Variance}: The variance of computed safety index parameterization across trials.
\end{itemize}

\begin{table}[t]
\vspace{5pt}
\caption{The performance of safety index design via SOS programming with 1000 random seeds (Robot Arms). Average computing time, validness, and variance are reported.}
\vspace{-10pt}
\begin{center}
\begin{tabular}{c|c|c|c}
\toprule
 &  Time (s) & Validness (\%) & Variance \\
\hline
2DOF & 0.160 & 100 & 0.118 \\
4DOF & 0.180 & 100 & 0.042\\
6DOF & 0.196 & 100 & 0.147\\
10DOF & 0.303 & 100 & 0.247 \\
14DOF & 0.422 & 100 & 0.342\\
\bottomrule
\end{tabular}
\end{center}
\vspace{-25pt}
\label{tab:sos_table}
\end{table}

\begin{figure}[htbp]
    \centering
    \vspace{-5pt}
    \includegraphics[width=0.6\columnwidth]{fig/sos-crop.pdf}
    \caption{SOS running time in comparison with time complexities of $\mathcal{O}(n)$ and $\mathcal{O}(\log n)$. The purple vertical lines represent standard deviation of running time over 5000 random seeds.}
    \label{fig:SOS time}
    \vspace{-10pt}
\end{figure}
The results are summarized in \Cref{tab:sos_table}.
SOS programming achieves 100\% validness (i.e., nonempty set of safe control for all states) across all degrees of freedom.
Moreover, the computing time is less than 1 second for the complex 14DOF robot arm system. As shown in \Cref{fig:SOS time}, the computing time scales logarithmically as the degree of freedom increases, indicating the efficiency of
our proposed method
for safety index design in complex systems.

We further evaluate the tightness of our method under the evaluation setting.
The results show that the robot is able to get extremely close to the obstacle where the minimum distance is 0.001 meter while being able to avoid colliding with the wall. This indicates the safety index designed by SOS programming is not conservative and enables the robot to manipulate in a large safe set.

\subsection{Autonomous Driving Results}
\begin{wrapfigure}{r}{0.17\textwidth}
\vspace{-32pt}
    \centering
    \includegraphics[width=0.16\textwidth]{fig/unicycle-crop.pdf}
    \caption{Unicycle robot.}
    \label{fig:unicycle}
\vspace{-10pt}
\end{wrapfigure}
In addition to robot arm problems, we also tested our method in autonomous vehicles for collision avoidance. The experimental platform is illustrated in \Cref{fig:unicycle}. The vehicle is a 4-state unicycle model, and the control inputs are acceleration and angular velocity.

The average computing time, validness and variance are summarized in \Cref{tab:sos_table_drive}, which indicates that our proposed method can efficiently and robustly synthesize safety index for vehicle systems.
In fact, the same safety index synthesis problem is studied in~\cite{zhao2021issa}, where the authors provide an analytical safety index design rule for 2D mobile robots.
As for our setting, a valid safety index needs $k_1 > 1$ according to the design rule~\cite{zhao2021issa}. In our experiments with 1000 random seeds, the minimum synthesized $k_1$ is $1.0001$, which again indicates good optimality of solutions of our proposed method.

\begin{table}[t]
\vspace{5pt}
\caption{The performance of safety index design via SOS programming with 1000 random seeds (Autonomous Driving).}
\vspace{-10pt}
\begin{center}
\begin{tabular}{c|c|c|c}
\toprule
 &  Time (s) & Validness (\%) & Variance \\
\hline
4-State Unicycle & 0.066 & 100 & 0.011 \\
\bottomrule
\end{tabular}
\end{center}
\vspace{-20pt}
\label{tab:sos_table_drive}
\end{table}

\section{CONCLUSIONS}
This paper proposed a framework for synthesizing the safety index for general control systems under control limits using sum-of-squares programming. Our approach leverages Positivstellensatz theorem to ensure the non-emptiness of safe control. The experimental results show that the synthesized safety index guarantees safety and our method is effective even in high-dimensional robot systems.

\bibliographystyle{IEEEtran.bst}

\section{Introduction}
Energy-function-based algorithms~\cite{wei2019safe} have been widely studied as appealing tools for safe control. When properly designed, the energy function (also called the safety index or barrier function) maps danger states to high values and safe states to low values. Then safety can be ensured by finding a control that dissipates the energy. However, when control limits exist, the desired safe control may not be realizable, which breaks down the safety guarantee. Therefore, it is important to account for control limits.

Designing safety index is challenging. Because we have to ensure the existence of safe control for infinitely many states. The presence of control limits makes the problem even harder where the complex constraints depend on system dynamics, control limits, and safety requirements. Many existing works leverage special structures of systems to guarantee safety, such as kinematic bicycle~\cite{zhao2021issa}, Euler-Lagrange system~\cite{cortez2020safe}, and polynomial systems~\cite{ames2014control}. Another approach is to formulate
safety index synthesis
as an optimization problem of a parameterized safety index. However, these methods do not scale well to high dimensional systems~\cite{clark2021control,wei2022safe}.

Sum-of-Squares Programming (SOSP) is a powerful tool and has been applied to safety index synthesis~\cite{clark2021verification, clark2021control, ames2019control} since it is a widely adopted approach to formally deal with optimization problems with infinitely many constraints. Existing SOSP-based safety index synthesis approaches try to design safety index such that there exists safe control for all states in the set of real numbers~\cite{ames2019control}. This type of constraint is known as \textit{global positiveness} constraint~\cite{zhao2022provably}.

In this work, we highlight that instead of satisfying \textit{global positiveness} constraints, safety index synthesis simply needs to satisfy \textit{local manifold positiveness} constraints, which enables the corresponding safety index design much less conservative. Subsequently, we propose a general, scalable, efficient SOSP-based method to transform
safety index synthesis problem into a nonlinear programming problem that can be solved efficiently by off-the-shelf solvers.
The experiments show that the synthesized safety index ensures the existence of safe control, and our method is time-efficient, robust, and optimal even in high-dimensional robot systems.

\textbf{Our contributions} We propose a novel SOSP-based method that can efficiently synthesize the safety index. The state-of-the-art SOSP-based methods are (i) limited to polynomial control systems~\cite{ames2019control}; (ii) hard to scale to high-dimensional systems (e.g., iteratively solving $\mathcal{O}(2k)$ SOS programs at $k$-th iteration~\cite{clark2021control}); (iii) limited to conservative solutions due to considering \textit{global positiveness} constraints. Our algorithm (i) is applicable to general nonlinear control systems; (ii) scales well to high-dimensional systems (solving only one nonlinear programming problem) and (iii) gives nonconservative solutions via solving \textit{local manifold positiveness} constraints.
We show how the theoretical results transfer to a practical algorithm with safety guarantees and apply it to safety index synthesis for complex robot systems.
\section{Introduction}
Energy-function-based algorithms~\cite{wei2019safe} have been widely studied as appealing tools for safe control. When properly designed, the energy function (also called the safety index or barrier function) maps dangerous states to high values and safe states to low values. Then, safety can be ensured by finding a control input that dissipates the energy. However, when control limits exist, this desired safe control may not be realizable and thus the safety guarantee may be broken. Therefore, it is important to account for control limits in energy function design.

Accounting for control limits in safety index design is challenging, because we have to ensure the feasibility of safe control at \textit{infinitely many} states. This feasibility constraint itself can be quite complex, as it is a function of the control limits, system dynamics, and safety requirements. There have been various approaches to safety index synthesis under control limits. Some works assume that the dynamical system has a special structure (i.e. kinematic bicycle, Euler-Lagrange), which enables hand-derivation of a safety index~\cite{zhao2021issa, ames2014control, cortez2020safe}. Another approach is to formulate synthesis as optimizing parameters within some safety index function form~\cite{clark2021verification,wei2022safe}. However, these methods do not scale well to high-dimensional systems.

Sum-of-Squares Programming (SOSP) has been applied to safety index synthesis~\cite{clark2021verification,ames2019control}, as it is a powerful tool for dealing with  optimization problems with infinitely many constraints. The issue with existing SOSP-based synthesis techniques is that they try to enforce existence of safe control for \textit{all states in the state space}~\cite{ames2019control,clark2021verification}. This is otherwise known as enforcing a \textit{global positiveness} constraint~\cite{zhao2022provably}.

In this work, we highlight that instead of satisfying \textit{global positiveness} constraints, it is sufficient for the safety index to satisfy \textit{local manifold positiveness} constraints, which enables the corresponding safety index design to be much less conservative. That means we can find solutions (a valid safety index) more often. Subsequently, we propose a general, scalable, efficient SOSP-based method to transform
safety index synthesis problem into a nonlinear programming problem that can be solved efficiently by off-the-shelf solvers.

The experiments show that the synthesized safety index ensures the existence of safe control and that our method is time-efficient, even for high-dimensional robotic systems.

\textbf{Our contributions} We propose a novel SOSP-based method that can efficiently synthesize a safety index. The state-of-the-art SOSP-based methods are (i) limited to systems with polynomial or sinusoidal nonlinearities~\cite{ames2019control}; (ii) hard to scale to high-dimensional systems (e.g., iteratively solving $\mathcal{O}(2k)$ SOS programs at $k$-th iteration~\cite{clark2021verification}); (iii) limited to conservative solutions due to considering \textit{global positiveness} constraints. Our algorithm (i) is applicable to polynomial substitutable (Def.~\ref{def: subs}) control systems; (ii) scales well to high-dimensional systems (solving only one nonlinear programming) and (iii) gives nonconservative solutions via solving \textit{local manifold positiveness} constraints.
We show how the theoretical results transfer to a practical algorithm with safety guarantees and apply it to safety index synthesis for high-dimensional robot systems.

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from provider chr3:124881193-124903541, - strand. Annotation of mouse strain WSB/EiJ genome assembly provided by GENCODE consortium. Distributed via Ensembl Release 92. Gene type: protein coding gene; Gene Name: Mettl14.


text:
More from NEA's ESEA Committee
By Stephen Sawchuk — July 06, 2009 1 min read
I had a nice chat this weekend with Christy Levings, the chair of the NEA's Committee on ESEA (the Elementary and Secondary Education Act, which is now called the No Child Left Behind Act).
Levings took the job after the former committee head, Becky Pringle, advanced to the NEA Secretary-Treasurer spot. She hails from Kansas, where she headed up the state affiliate for two terms.
We had a chance to talk a bit about her work for the committee, which is now working on helping state leaders prepare for the reauthorization of the federal education law, primarily through the NEA's Positive Agenda for NCLB (find it here).
In Levings' view, NCLB's testing requirements have gotten away from the philosophy of the ESEA. The original law, she said, was designed to expand opportunities for poor kids and widen the curriculum. Instead the law has done the opposite by squeezing out other subjects and focusing on narrow test prep, she contended.
Levings did praise the law's focus on subgroups of students, such as low-income children, children with disabilities, and students from certain racial and ethnic groups. "It's the only strong point of NCLB, in my opinion, the conversations around these children," she said. "It's unfortunate that the obsession with testing drove that to a poor spot," she added, referring to the law's requirement that a school must meet testing benchmarks for each subgroup in order to be deemed as making "adequate yearly progress."
She said she thinks the economic-stimulus bill, including the Race to the Top competitive grants to be distributed by Education Department, will help the administration take some new ideas "for a test drive," after which they might be included in the NCLB reauthorization. She thinks it's too early to tell what the administration means by the teacher-evaluation language in the stimulus guidance.
Stephen Sawchuk
Stephen Sawchuk covers district leadership and management, school safety, and civics education for Education Week.
Unions Federal Policy
A version of this news article first appeared in the Teacher Beat blog.
Bet Noire/Getty
Federal Biden Picks San Diego Superintendent for Deputy Education Secretary
Evie Blad, January 18, 2021

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Ever since birth, I have lived with a blocked nose and explored endless 'cures', even surgery, for some relief especially at night. In February, I had an attack of hives on my legs and my dermatitis had spiralled out of control, not to mention my nose was so blocked, I was breathing through my mouth alone. On top of this, my immunity was down and my thyroid was above average levels. I had felt like the energy had drained from my body and fatigue set in. I had no desire to go out and going to work became a struggle. I had hit an all-time low, thinking I would never recover.
After all of the steroid creams, cotton gloves, urine therapy (which helped to some extent), and countless antibiotic courses, mum and I had come to the conclusion that common methods of treatment and medicine was not the answer to my health problems. We had to think outside the box and seek out an alternative method of treatment that would not only improve my condition, but also allow me to continue pursuing my dream job as a hairdresser. That's when we discovered a form of natural healing through Zazeh and in the form of The Body Code. To be honest, I was a little sceptical as to what was involved, but I was at my wit's end and decided to keep an open mind.
My first Body Code session included an evaluation, clearing of negative trapped emotions and some underlying imbalances. At this point, through the nutritional evaluations, I was advised of a few remedies needed to clear up viral & fungal pathogens as well as some toxicity in my body. Included in that Zazeh suggested a simple 3-day liver cleanse. I started all the remedies, including the liver cleanse, and within those 3 days, the itchiness of my dermatitis disappeared and within a week, my hands were cured. In two weeks the hives were gone and my eczema cleared. My acne started to reduce and my skin has been the clearest it's ever been.
I'm five months into the process with Zazeh and my thyroid is functioning normally again, (which I continue to keep a check through regular blood tests), my energy is back, I am no longer falling asleep on the couch after a short day of work, my eczema has cleared and I am breathing so much better.
My diet in general has changed, reducing processed foods and eating as natural as possible, which in turn has helped spike my energy levels and waking up in the morning is a breeze. I also continuing doing a cleanse one day every week to keep my health on track.
I look forward to going to work each day, knowing that the events of the day will not be have an emotional affect as they used to and my health won't be compromised.
It's a working progress and thanks to my Emotion and Body Code practitioner, Zazeh Morfitis, whom I would highly recommend to anyone who has exhausted all medical avenues for their health and wellbeing.
I really didn't know what to expect when I first asked you to work your Emotion Code skills on my 12-year-old white cat, Precious, who for about a year prior had started peeing very inappropriately around my house. However, you have transformed my beautiful girl back into one who uses the litter tray again. Since around December 2014 we have only had one stray pee, every other time she has used the litter trays. I simply cannot believe it! Not only has this behaviour been modified but her general demeanour is now more calm and she seems very contented. Along with all of this her general condition has improved; she has gained some weight and her coat is soft and gorgeous. Thank you so much for your regular updates, including the dietary advice for her, and your continued confidence that you would be able to help her. She is happy and I am happy and we are both so very grateful to you for all your knowledge and caring.
Thank you for your ongoing support of my cat Precious. You had suggested that working with animals, there may come a need for some reinforcement sessions. This, in fact, proved to be the case. After a few incidences of Precious reverting to her inappropriate peeing earlier this year, you very quickly addressed the issue and with just 2 sessions, she again started exclusively using the litter tray. She is still a very happy and content cat exhibiting much more affection than she ever did prior to her first sessions with you in 2014. Thanks again for your care of Precious and also your dedication to healing her and keeping her happy.


text:
A Survey of Manuscripts Illuminated in the British Isles (HMMSBI 4.2)
N. J. Morgan
Early Gothic Manuscripts (2)
374 p., incl. 472 ills., with 8 in colour, 235 x 335 mm, 1988
This is a survey of early Gothic manuscripts illuminated in the British Isles, setting them in a historical framework, describing the influences at work, defining the types of books and their decoration, and discussing evidence for dating and localization.
In England the art of illumination flourished widely in the second half of the 13th century - a time in which the connections with the continent, particularly France, where strong. Artists moved away from the monastic scriptorium to professional workshops in urban centres, and the rise of the Universities resulted in the production of new types of illustrated text. As with cross-fertilization of ideas from Paris to London, Oxford, and Cambridge, so with the styles and techniques of illumination. The magnificent examples catalogued here include the Lambeth, Metz, Douce and Trinity Apocalypses; their relationships are examined and detailed iconography is described. Many of the Bibles and - notably - the Amesbury, Oscott and Rutland Psalters are some of the greatest works of the period, and among surviving religious manuscripts the earliest examples of Books of Hours reveal the increasing range of devotional interests of lay people. Astrological, legal, medical, topographical and historical works are also included, and the volume ends with the remarkable Hereford Mad which summarizes many of the ideas of Earth and Heaven prevailing at the time. The Introduction sets the manuscripts in a historical framework, describes the influences at work, defines the types of books and their decoration, and discusses evidence for dating and localization.
Book History, Manuscript Studies & Palaeography
Codicology
Illumination & illustration of books
Fine Arts & Performing Arts
Medieval painting
Medieval & Renaissance History (c.400-1500)
Medieval European history (400-1500) : local & regional history
This publication is also distributed by: ISD, Marston
The Psalter of Robert de Lisle in the British Library
Sandler 'Item a Guillaume Wyelant aussi enlumineur'
Bousmanne Monarchy and Consent
O'Meara The Lichtenthal Psalter
Freeman-Sandler The Masters of the Dark Eyes
Broekhuijsen The Regal Image of Richard II and the Wilton Diptych
Gordon et al. Later Gothic Manuscripts 1390-1490


"Well, there were actually two stations owned by Czechs. One was the Czech Antarctic station named after Johan Gregor Mendel, ran by the Czech Antarctic programme. The second case was a little bit peculiar. It was a private station, which is rather unique, established a long time before the Czech Antarctic law was adopted.
The station you are talking about was built by a Czechoslovak traveller Jaroslav Pavlíček. Can you tell us a bit more about the history of this station?
"The EcoNelson station, which was a name Mr Pavlíček used for the facility, was built 30 years ago and at the time, ad at the time, the Czech Antarctic Law didn't exist.
"He was responsible for the operation of this station for 30 years and after 30 years he devoted this facility to the Czech Antarctic Foundation, a private body established to support Czech Antarctic and Artic Research.
Can you tell us a bit more about the Nelson Island? In what way is it different from the James Ross Island and in what way will the research be different?
"From the climatological point of view, and climatology is the most important field of our scientific work in Antarctica, these two islands are in very different parts of Antarctica.
"Nelson Island is part of the Archipelago of South Shetlands and it is very humid, because it lies in the warmest part of Antarctica, on its northern edge.
"James Ross Island is on the other side of Antarctic peninsula, on the transition to the continental climate. It means it is much less humid and also the temperatures are different.
How long did you spend on the island and what was the main purpose of your visit?
"The main goal of our expedition to the Nelson Island was to clean the site. After 30 years of operation there was a large waste deposit from its operation and also from the sea.
Pavel Kapler, photo: archive of Masaryk University in Brno "The sea currents in this part of the world are very strong and they carried a lot of garbage to the beaches of the island. The Southern Ocean is the cleanest of all oceans but it is still part of the larger oceanic system. It serves as a very good record of how the human civilization is polluting the globe.
"The previous users of EcoNelson collected all the garbage from the sea and from the coast and the deposit driven here by the sea currents was really impressive.
"So our main goal was to clean the site as a first step of the reconstruction of the base. In the future we will probably continue in this reconstruction to establish a basis for scientific research and for accommodating scientists.
What does the waste that you collected what did it reveal about the global ocean pollution?
"Unfortunately we don't have a record about the precise amount or time or place of origin of the waste, so we cannot really say too much. However, the amount of the waste we removed from the island was almost 11 cubic metres, which was a lot of mess. Therefore we can say that the Earth is really endangered.
As always, Masaryk University has also dispatched an expedition to the James Ross Island. What was their main focus of research?
"We already have 15 years of scientific work on the James Ross Island and long-term monitoring has always been the main purpose of our expeditions there.
"That's true. This year was very specific for the Czech Republic and its Antarctic research. Members of our programme took part in the Turkish and US Antarctic programme and they were also part of the Chilean expedition and many others. So the collaboration on the Antarctic research is getting wider and wider.
"There is no coincidence that this year Prague will be hosting the 42nd annual meeting of the Antarctic treaty, the biggest Antarctic event of all times.
You travel to Antarctica during the polar summer. How long does it last there and what were the conditions like there this year?
"The Antarctic summer is very short. It usually lasts from December to February. This year was a bit colder than usually. This is not contrary to the Global Warming. It is an evidence of extremes brought by the Climate Change.
Gregor Mendel station, photo: Kamil Láska / Masaryk University in Brno It was your ninth expedition to Antarctica. Does it still surprise you? Are you still looking forward to visiting that part of the world?
"Absolutely. On one hand of course after nine expeditions it is starting to be just work, but it is still the most amazing work in the world.


Join Trish Wheaton and Marge Watters, for a unique and innovative program designed for late-career senior professional women who want to stand on the shoulders of their professional selves to purposefully envision and plan their next phase — and make it their 'Primetime'.
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What's included in the LEANING OUT Primetime Package?
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In 1995, Marge and her partner, John Knebel, founded and led KWA Partners, now known as Optimum Talent. From start-up, the firm grew to be ranked among the top five organizations in the career management industry. Marge knows what it's like to take a huge personal risk to start a business, manage every aspect of building it into a respected nationwide organization, and sell it successfully.
Working as a career consultant, Marge coached individuals from all levels of the organization to help them move successfully through the challenging process of career transition. As an executive coach, she helped her clients maximize their strengths and understand their development opportunities to advance their leadership potential. Among Marge's achievements while leading KWA Partners is the authoring of It's Your Move, a best-selling book about career transition published by HarperCollins in 2001, and now in its fourth edition.


Melissa strives "to be a superfit, well-fed, dressed-to-kill, glossy-haired, rock-n-roll, tart-tongued detective." It's only fitting that her Paleo cookbook, Well Fed, is equally glossy and rock-n-roll.
Well Fed is not just a cookbook–yes it does include over 115 original recipes, and beautiful full-page pictures of every single dish, but it's much more than a cookbook. It's a comprehensive guide to successfully running a Paleo kitchen.
As a holistic nutrition counselor, I understand the importance of giving people the tools they need to succeed in changing their diets and their lives. The proverb "give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime" definitely applies to making diet and lifestyle changes.
One needs to an action plan–a method to implement in order to make sustainable change. Well Fed gives you a stress-free plan for turning your kitchen into a smoothly running paleo machine. Give a man (or woman) a recipe and you feed him for a day. Give a man Well Fed and you feed him for a lifetime.
In Well Fed, Melissa gives step by step instructions on how to realistically approach a week of cooking Paleo meals. For those of us who are not the best at organizing our food shopping trips, she gives formulas to easily calculate how much food we'll need for the week and a How To guide to the "Weekly Cookup". Your grocery shopping will be more efficient and you'll cut prep time in half with her Sunday Cooking WOD.
For those of us who don't have a lot of time on our hands to prepare elaborate dinners every night of the week, have no fear: there is a whole section on how to turn raw ingredients into delicious "Hot Plates"meals made up of veggies, beef, or chicken and spices– in about 10 minutes! There are over 30 different ideas for "Hot Plates" categorized by geographical region (French, Greek, Moroccan and Thai for example),and even a section on how to put together filling salads.
Well Fed will "tempt you with recipes that will make you want to smash in your face with joy." This cook book gives you all of the tools that you'll need to transform your kitchen into a lean, mean, Paleo meal making machine. The book comes in both a printed, soft-cover version (175 pages) and an eBook (PDF) version. Want a sneak peek? You can download a free 30 page PDF sampler on Melissa's website.
I seriously love this cookbook! If you're looking for even more reasons why this book is great, check out this post on 5 Cool Things About Well Fed.
FRISKY! Thank you so much for this kickass review. I'm so glad you like it! I was thinking about people like you the whole time I was writing — the people I love to read around the internet and how I couldn't let you all down. Thanks for the enthusiasm and glowing review. It means a lot to me!
You're so welcome! It was my pleasure! It's such a beautiful ( and practical) book that everyone should know about it.


text:
IHBC Conservation Wiki Part of Designing Buildingswww.designingbuildings.co.uk
Site info / tools
See full history
Local Authority Conservation Services continue to decline
This article was created by The Institute of Historic Building Conservation (IHBC) and originally appeared in the IHBC NewsBlogs
IHBC research, funded and published by Historic England, shows the nation's local authority conservation services have continued to suffer in what is now (2016) a decade of cuts, as conservation capacity reduced by a further 0.5% in the last year, with a shocking cumulative decline of 36% since 2006.
James Caird, IHBC Chair said: 'The continuing decline in local authority conservation staffing is disappointing. This is not least as it ignores substantial evidence that local conservation services are the best way - and best value - for LPAs to carry out their statutory duties under the Planning Acts as well as promoting heritage as a driver for urban and community renewal.'
'The IHBC is doing what we can to mitigate impacts that are otherwise outside our control. We collaborate sector-wide and in partnerships with like-minded bodies, such as the CIfA and the RTPI; offer national and local training; publish guidance, such as our Research and Guidance notes in our Toolbox, as well as offer advocacy resources, such as our 'Why Local Authorities need conservation skills' web site.'
IHBC President David McDonald, said: 'Our evidence, both statistically and anecdotally shows that local authorities are increasingly unable to cope with the pressure of applications and therefore not able to play a full part in meeting the government's planning objectives.'
IHBC Director Sean O'Reilly said: 'It is especially unfortunate too that much the same long-term, sorry story is also told by the parallel archaeology service data collected by ALGAO, in work also funded by Historic England, which shows a structurally damaging 13.5% fall in local authority archaeological specialists in the last year alone.'
Fiona Newton, who led the IHBC's role in the wider project, said: 'This Eighth Annual Report on Local Authority Staff Resources captures the dramatic impact of local government cutbacks on heritage management. Whilst the scale of the decline in conservation specialists has reduced to 0.5% in the last year, it still continues to fall each year and since 2006 the number of conservation specialists has fallen by 35.8%.'
'At the same time, in the last 12 months alone the number of planning application decisions increased by 3.6% and Listed Building Consent decisions increased by 0.62%. Whilst not the whole picture of the workload of conservation staff ,these increases illustrate the picture of rising workload and decreasing resources.'
For more recent background on conservation services see for example: Richard Bate in Context at http://ihbconline.co.uk/context/144/#33/z and the IHBC's website on LA skills at http://www.ihbc.org.uk/skills/england/index.html.
Download the Eighth report http://ihbc.org.uk/resources_head/SelectPapers/files/8LAresources.pdf
See more resources on our listing of papers at http://www.ihbc.org.uk/resources_head/SelectPapers/index.html and on our Toolbox at http://ihbconline.co.uk/toolbox/
Related articles on Designing Buildings Wiki
Historic England
Local authority conservation
IHBC
RTPI
IHBC http://ihbc.org.uk
Retrieved from "https://www.designingbuildings.co.uk/wiki/Local_Authority_Conservation_Services_continue_to_decline"
IHBC NewsBlog
IHBC Marsh Awards 2022 launched - Submissions Close 31 March
A chance to nominate retired IHBC members and/or successful learners in heritage skills, with prizes that include £500 and a free place at the IHBC's Aberdeen 2022 Annual School in June.
IHBC-SAHGB announce 2021 annual research award winner
Donald Insall Associates has been announced as the winner of the new joint award with the Society of Architectural Historians of Great Britain (SAHGB). The award celebrates the quality of architectural-historical research produced as private consultants or for public bodies etc.
IHBC at COP26 - Conserving our Places Conserves our Planet 1-12 Nov 2021
The IHBC seeks to raise awareness and understanding of how building conservation philosophy and practice contributes towards meeting the challenge of climate change.
IHBC's latest CONTEXT explores 'Conservation Leadership by the People'
From Amenity Societies and Wentworth Woodhouse to Kurt Schwitters, Scotland's Towns, Chester and more...
Edinburgh's Royal High School to become '£55M national centre for music'
The former Royal High School building in Edinburgh is to be transformed into a £55 million national centre for music after the City of Edinburgh Council agreed to the lease of the historic property.
IHBC, CIfA, IEMA launch Principles of Cultural Heritage Impact Assessment
The joint-institute document aims to help maintain cultural heritage by providing a consistent framework across different sectors & geographies
Enter Now to Win: IHBC's GASA Awards 2021
IHBC's Gus Astley Student Awards 2021: Win £500 and a place on IHBC's 2022 Aberdeen School with your built environment/heritage coursework, closes 31/07!
Former Harris meat factory mansion in Calne, Wilts, to be restored
The last remaining buildings on the site of the Harris meat factory family's historic mansion are being restored to their former glory and converted into new homes.
CICV Forum introduces professional indemnity insurance guide
The Construction Industry Coronavirus Forum (CICV Forum) has unveiled a new guide to the crucial and increasingly complex issue of professional indemnity insurance (PII).
The World Heritage Committee(WHC): Liverpool should lose its World Heritage status.
ICOMOS has advised that the new football stadium proposal, if implemented, would have a completely unacceptable major adverse impact its authenticity and integrity.


Ministers from 80 countries underlined the importance of the "The Hague Roadmap for Action" at It's Down 2 Earth, the international conference on agriculture, food security and climate change. This plan is a roadmap towards climate smart agriculture. The Netherlands has vowed to make sure that the actions in this roadmap will stay on the international agenda. Minister Bleker for Agriculture and Foreign Trade is very pleased with the results. "This is not just an intention. This is a task the ministers have put upon themselves."
How are we going to feed 9 billion people in 2050? How are we going to make sure that agriculture becomes a carbon sink instead of a carbon emitter? How are we going to make agriculture climate change resiliant? These were some of the questions financial institutions, scientists and ministers of agriculture spoke about at the conference. But also high profile guests like HRH Prince Willem-Alexander, HRH Princess Maxima, The Prince of Wales, Kofi Anan and Andrew Steer (World Bank) gave their vision on these matters. The outcome of these talks is the The Hague Roadmap for Action.
At this unique gathering, this roadmap started to achieve the 'triple win' of improving agricultural productivity and food security, helping to address climate change, and improving the lives an livelihoods of rural populations that live in poverty. The roadmap is a living process and document. It contains a list of actions that countries and organizations should take to stimulate this new and climate smart agriculture so more people can be fed and less carbon is emitted. The countries have given their commitment towards these actions. The conference is only the starting point. Other stakeholders can sign in to initiatives in the list or broaden the activity.
The Netherlands has vowed to make sure these actions stay on the international agenda. Minister Bleker: "I will make sure that climate smart agriculture that can feed more people and is better for the environment will stay on the agenda. I will present the 'The Hague Action Plan' myself at the climate conference in Cancun, Mexico."
Agriculture has not been on the political agenda for decades in many organisations or countries as a vital part of the economy or as a base for food security. Only recent it is known that agriculture offers a chance to help reduce the emissions of greenhouse gasses. Worldwide there are about 2,2 billion farmers. They will have to feed 9 billion people in 2050. The lack of attention for agriculture has lead to lower yields, lack of knowledge and fewer investments. Poverty is common for smallholder farms in many developing countries.
To keep up with a growing global population in order to feed the growing amounts of people in a climate smart way, the 80 countries, financials like the World Bank, Civil Society Organisations and scientists have come up with a range of solutions. To finance these solutions many possibilities have been looked at. The market in carbon credits is also promising. This way, farmers can have extra income by making their land actual carbon sinks while having higher yields at the same time. During the conference the World Bank, the Rockefeller Foundation and the ministry of Economic Affairs of The Netherlands have signed commitments towards existing projects for a total amount of 2 million euro. And this is just the beginning. Many contacts were made to start a large amount of climate smart projects all over the world.
After The Hague, agriculture, food security and climate change will be back on the political agenda in many countries.


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RRR Movie New Release Date and Updates 2022
Leave a Comment	/ Entertainment / By Patil7274
A lot of people were eagerly waiting for the RRR movie. More recently, the movie was about to be released on 7th January 2021. But due to the increasing number of covid, the lockdown is being imposed again in India. And all theaters are being closed. That's why the RRR movie Release Date has been postponed. Today we are going to know about rrr release date in this post.
All of you are watching the path of RRR movie, but due to the lockdown, this movie has been posted. Due to which people are searching on the internet about rrr release date, rrr movie new release date, rrr release date OTT. Through this post, you are going to get complete information about RRR movie.
RRR Full Movie in Hindi 720p Download Filmyzilla
1 RRR Movie New Release Date
2 Is RRR Movie Release On OTT Platform?
2.1 RRR Movie Trailer Hindi
RRR Movie New Release Date
Source: RRR Movie Trailer/Youtube
The release of SS Rajamouli's mega-budget film 'RRR' has been postponed due to the lockdown. The film was scheduled to release in theaters on January 7. But the release date of this movie is being pushed forward. SS Rajamouli's film 'RRR' is very special for South Indians as well as North Indians. Apart from Tamil, Telugu, Kannada, and Malayalam, the film will also release in Hindi.
So far, no information has been given by the RRR team and director Rajamouli about the new release date of the film, but according to some reports, it has been learned that if the theaters are opened then you will get to see the film soon. And somewhere else this film will be released in June, it is also being said that.
Pushpa Movie Download in Hindi HD 720p – Filmymeet
Is RRR Movie Release On OTT Platform?
Let me tell all the people that RRR movie will not be released on any OTT platform. This movie is going to be released officially in theatres. A few days after its release in theaters, this movie can be released on any one OTT platform.
RRR Movie Trailer Hindi


I am bathing the girls. They are in the tub, each wearing a pair of swimming goggles, and each armed with a small water pistol that they are spraying, gleefully, into the others face. I am sheltering behind the shower curtain, which is partially extended to stop more water from going onto the floor - a floor which has long since edged out of 'damp', through 'wet' and is now heading firmly into 'soaking'.
"Try not to get your hair wet," I say, pointlessly, for the ninth time. "Mummy won't like it".
"But you gave us the water pistols..." she says.
"Pistols!" shrieks her sister, excitedly.
"Yes, yes..." I agree. "But I didn't think you'd shoot each other."
"But you said: Why don't you shoot each other..."
"Pistols! WATER PISTOLS!" cries Youngest.
"Yes, you said: Don't you dare shoot me, but you can shoot each other..."
"Ah, yes. Yes, I did. But I didn't really mean in the face..."
"In the face!" adds Youngest, in voice wheeling with joy.
"But you specially went and got our swimming goggles..." continues her sister, coolly returning fire, "...to stop it going in our eyes."
"That was a precautionary measure," I say. "It wasn't meant to encourage you to get your hair wet. Mummy won't like it if you get your hair wet."
"She won't like the water on the floor, either..." she observes.
I decide I will distract them from the water pistols, and think I may have had a brainwave as to how. I start knocking gently on the side of the tub, out of their line of sight.
"What's that knocking?" asks Eldest.
"Oh, that?" I say, casually. "That's Grumblemouth. He comes out from behind the bath panel if the floor gets too wet."
Both stop dead in their tracks. I continue knocking.
"What is Grumblemouth?" asks the first.
"Is he nice?" asks the second.
"Sort of..." I say, reaching for the toothpaste and playing for time. "Sometimes."
"Is he...a monster?" she asks.
"Yes," I say, not really listening while I go to work. "That's right. He is a monster..." With my left hand, I make a fist, tucking my thumb in under the fingers. By turning my hand sideways and flapping my thumb up and down, I can make a monster's mouth. With the toothpaste I squeeze a blob of white either side of the knuckle of my first finger, forming two large gelatinous eyes. I am delighted with the result.
Ha, I think, this is brilliant! They will love this!
"I don't like monsters..." announces Youngest.
"Oh, you'll be fine " I say airily, still knocking on the side of the bath, and reaching for their blue 'Winnie the Pooh' toothpaste, which is mean to tastes like bubblegum to encourage them to clean their teeth, but which both find revolting and will go to quite remarkable lengths to avoid using at all. With this second toothpaste I add a pair of blue irises to the wobbly white eyes of Grumblemouth, and a streak of a nose.
Wow, I think, that looks amazing. They will laugh like drains!
"Is he going to come out? asks the Eldest, her voice quivering a little with what I think is excitement (but now, in retrospect, I understand was fear).
"Yes, I think he might..." I say, suppressing a chuckle. I start knocking louder, as Grumblemouth gets nearer.
They will remember this forever, I think. It will be a cherished childhood memory. They will pester me every bathtime to bring Grumblemouth out!
"I don't want Gumballmouth..." says Youngest, in a tiny voice.
"HERE HE IS!" I cry, and pop my hand up from of the side of the bath, ready for the gales of laughter.
The reaction is immediate. Both scream in genuine terror, far louder than I have heard either before. The youngest bursts into immediate tears.
"Nooooooo!" she shrieks. "No No No!"
"It's OK! It's OK!" I soothe."It's just Daddy being silly..." I reach to cuddle her, which of course brings Grumblemouth closer, with his mouth gaping wide open as if about to bite. She howls, and lashes out with a shampoo bottle.
"Make him GO AWAY!" she screams.
I quickly duck my hand in the bath and swoosh it around to wash the toothpaste off. This doesn't work as well as it might, and when I bring my hand out of the water the toothpaste has simply partially dissolved and run a little way down my hand, making Grumblemouth's face even more disturbing.
"His face is melting!" shrieks eldest in sheer terror, forcing herself into the corner of the bath and trying to climb out backwards using the taps as handholds. She is right: Grumblemouth looks like that French bloke who opened the Ark of the Covenant in the first Indiana Jones movie.
At this point my wife enters the room. She surveys the carnage with an open mouth.
"We were just playing," I try and explain, "and it all went wrong..."
Grumblemouth has not made an appearance since. I would like to particularly stress that fact to anyone reading who might work for Social Services.
:) Always enjoy your posts, but this one is special! Why don't we get this quality of entertaining column in one of the week end broadsheets, instead of the same old hacks?
PG is correct of course.
I'm looking forward to the next post where you, in intimate detail, let us know exactly how long before you are permitted back into the marital bed.
Hehehe, when I was a kid my Dad hung a toy parrot over my bed and then told me that it would crap on me if I got up in the middle of the night for any other reason other than to go to the toilet!! I was pretty certain he was lying but not certain enough to try out the theory!!


Last modified on 2/4/2011 10:52 AM by User.
ClockRing is a small utility for iOS, that makes your iDevice ring every hour. It's GPL v2, which means it's free.
ClockRing is both free/gratis (as in free beer) and Free Software (as in free speech), as stated in the GPL v2 license.
Who can I blame for that perpetration against good programming taste and such horrible code?
My name is Diego Freniche. You can find me in my blog page (in Spanish only, sorry), and in Twitter.


text:
Francisco de Gálvez · Orchestra Conductor
Born in Málaga, Francisco de Gálvez is one of the leading Spanish conductors of his generation. He obtained international recognition when he was awarded First Prize and Gold Medal at the 1994 Tokyo International Conducting Competition(Min-On), among 194 conductors of 36 countries. Dr. De Gálvez is Director of Orchestral Studies at Weber State University, Utah, USA. As guest conductor, Francisco has offered concerts with Orchestras like Berliner Symphoniker, Zagreb Philharmonic, New Japan Philharmonic, Tokyo City and Osaka Philharmonics, Saint Petersbourg Philharmonic, Filarmonica di Russe Orchestra, Postdam Philharmonic, Sinfonica di Sofia, Fort Worth Symphony, Danish Radio Orchestra, Flanders Opera, Spanish National Orchestra, Madrid Symphony, Málaga, Seville and Granada Symphony Orchestras, among many others. He has been Music Director of University of Malaga Chamber Orchestra and of Malaga Symphony Orchestra between 1998 and 2011. Also, he was Artistic Director of Contemporary Music Cycles organized by the Malaga Philharmonic Orchestra between 1995 and 2000. He was director and founder of the Montreal Chamber Players Orchestra in 1991, with whom he performed extensively in North America and Spain. With this orchestra made its first CD, which others have followed, as well as numerous recordings for radio and TV.
He studied Conducting, Composition, Orchestration and Analysis in Canada and graduated with a Master in Orchestral Conducting at McGill University, Montreal. His main teachers in conducting have been Timothy Vernon and Ferdinand Leitner. He also studied conducting at Vienna, Hochschule für Musik, and Siena, Accademia Chigiana, Italy. In 1993 he continued his studies with Carlo Maria Giulini in Milan, Italy. He has an extensive musical background as a violinist (Título Superior-Madrid; Orquesta Nacional de España; Performing Diploma-Royal College, London, England; Master-McGill, Montreal, Canada). Between 1995 y 1998 lives in Berlín, where he performs with various orchestras (it is worth mentioning his conducting of the World´s Premiere of Carlo Domeniconi's El trino del diablo at the Philharmonie, later presented at the Istambul International Music Festival and recorded on CD). De Gálvez obtained a PhD "cum laude" in Music and Comunication at Malaga University. Dr. De Gálvez now lives with his family in Ogden, Utah, teaches at Weber State University and develops an intensive and creative pedagogical and artistic musical program.
Francisco de Gálvez
Bienvenida / Welcome
BIO Intro
Biografía / Biography
Repertorio / Repertoire
Fotos / Photos
Discografía / Discography
Montreal Chamber Players, Francisco de Gálvez
© Francisco de Gálvez


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