text:
\section{Introduction}

Surgical robots have received considerable attention in the context
of minimally invasive surgery, as aids in procedures performed through
small incisions, in systems such as the da Vinci, the RAVEN \cite{lum2009raven},
and the SteadyHand \cite{taylor1999steady}. Their use has been extended
to procedures in restricted workspaces such as deep brain microsurgery
\cite{morita2005microsurgical,UEDA2017110} and endonasal surgery
\cite{li2007spatial}.

In minimally invasive surgery and microsurgery, the surgeon operates
with long thin tools, and views the workspace through an endoscope
or microscope. Their vision is frequently limited to a region near
the surgical tool tips. As the surgeon operates the tools, the tool
shaft may inadvertently collide with neighboring structures. In particular
in microsurgery, the amplitude of hand tremor can be higher than size
of the structures being treated. In this context, surgical robots
are used as equipment to assist the surgeon in order to increase accuracy
and safety, and reduce surgeon's mental load and the effects of their
hand tremor.

To increase accuracy and attenuate hand tremor, surgical robots are
commanded in task space coordinates either through teleoperation or
comanipulation. Joint space control inputs are generally obtained
by using a kinematic control law, which is usually based on the robot
differential kinematics \cite{Siciliano2009}. Kinematic control laws
are valid when low accelerations are imposed in the joint space, and
are ubiquitous in control algorithms designed for surgical robotics
\cite{kapoor2006constrained,li2007spatial,aghakhani2013task,pham2015analysis,marinho2016using},
since low accelerations are expected in such scenarios.

An increase in safety and a reduction in the surgeon's mental load
has been achieved through the generation of active constraints (virtual
fixtures) that can, for instance, act as a layer of safety to prevent
the surgical tool from entering a restricted region, even if this
contradicts the commands given to the robot \cite{li2007spatial,kapoor2006constrained},
or transparently generate constrained motion \cite{marinho2016using,pham2015analysis,aghakhani2013task}.
An in-depth survey of active constraints was presented by Bowyer \emph{et
al}. \cite{bowyer2014active}. More recent papers published after
this survey addressed the use of guidance virtual fixtures to assist
in knot tying in robotic laparoscopy \cite{chen2016virtual}, and
to allow surgeons to feel the projected force feedback from the distal
end of a tool to the proximal end of a tool in a comanipulation context
\cite{vitrani2017applying}.

The generation of active constraints using kinematic control laws
requires obtaining a Jacobian that relates the constraint to the task
at hand. For instance, for the generation of the remote center of
motion in minimally invasive surgery, some groups \cite{aghakhani2013task,pham2015analysis}
developed Jacobians to maintain a low remote center of motion error.
Aghakhani \emph{et al.} \cite{aghakhani2013task} constrained lateral
motion in the pivoting point by using the pivoting point dynamics,
and Pham \emph{et al. }\cite{pham2015analysis} used the so-called
constrained Jacobian to project the desired end effector velocity
into a constrained velocity in the pivoting point frame. Additional
requirements, such as the desired tool tip pose following and obstacle
avoidance are usually projected in the null space of the constrained
task or stacked to form an augmented Jacobian. The main problem from
which both approaches suffer is the difficulty in adding inequality
constraints, which are useful in the design of virtual fixtures \cite{funda1996constrained}.

\section{Related works}

A framework for manipulator control in surgical robotics that can
take inequalities into account was developed by Funda \emph{et al}.
\cite{funda1996constrained} using quadratic programming. Their framework
was extended by Kapoor \emph{et al}. \cite{kapoor2006constrained},
who developed a library of virtual fixtures, including five task primitives
that can be combined into customized active constraints. Li \emph{et
al.} \cite{li2007spatial} used the extended framework to aid operators
to move a surgical tool in a highly constrained space in sinus surgery
without collisions. In all these studies, non-linear constraints were
used. Solving for non-linear constraints requires an initial guess
and takes longer than solving for linear constraints, and therefore,
the authors also considered the use of linear \emph{approximations},
which reduce computation time but may result in errors \cite{kapoor2006constrained}.
This raises the question of whether the constraints can be written
directly in linear form, avoiding approximations. An additional consideration
is that the closed-loop stability of their approach has never been
formally proven.

Gonçalves \emph{et al.} \cite{gonccalves2016parsimonious} developed
a Lyapunov-stable kinematic controller that takes into account both
equality and inequality constraints, which are linear with respect
to the joint velocities. They experimentally validated their approach
using a humanoid robot. Their framework was extended by Quiroz-Omana
and Adorno \cite{Quiroz-Omana2017} to the control of a mobile manipulator
byadding a unilateral equality constraint that pushes the mobile manipulator
out of a forbidden zone. This, however, was a reactive constraint
and did not impede the mobile robot from entering the restricted zone
in the first place. Such reactive behavior is particularly undesirable
in medical robotic applications, in which the patient may be harmed
if a tools enters a forbidden zone.

It is important to note that, in prior approaches \cite{li2007spatial,kapoor2006constrained,funda1996constrained},
when the tool reaches a restricted zone boundary the obstacle constraint
is suddenly activated, which might cause the robot to show acceleration
peaks. Researchers attempted to address this issue. For instance,
Xia \emph{et al.} \cite{xia2008integrated} reduced the proportional
gain in an admittance control law proportionally to the distance between
the robot and the nearest obstacle. This allowed the system to smoothly
avoid collisions, but also impeded motion tangential to the obstacle
boundary. Prada and Payandeh \cite{prada2009study} proposed adding
a time-varying gain to smooth the attraction force of attractive virtual
fixtures.

\subsection{Statement of contributions}

In this paper, we propose a new concept of vector field inequality
applied to active constraints that further extrends the developments
of Gonçalves \emph{et al}. \cite{gonccalves2016parsimonious} and
Quiroz-Omana and Adorno \cite{Quiroz-Omana2017}. The proposed vector
field inequality allows the designer to set a maximum approach speed
in the direction of the restriction boundaries, which guarantees that
the robot can smoothly approach restricted zones without trespassing
on their limits, and that the velocities orthogonal to the restriction
boundaries are undisturbed.

Moreover, in this study we also developed a set of control primitives
for restricted zones, including point, plane, line, and cylinder relations,
which can be customized to model complex environments and interactions.
The effectiveness of the proposed approach is shown in realistic simulated
experiments.

The current work was conducted in the context of \emph{Project 2:
Smart Arm}, part of the\emph{ ImPACT} \emph{Bionic Humanoids Propelling
New Industrial Revolution} project, the objective of which is to develop
robotic systems and control frameworks to allow robot-aided surgical
procedures in constrained spaces, such as those involved in transnasal
pituitary gland resection and surgeries in deep and narrow regions
of the brain.

\section{Mathematical Background}

The proposed virtual fixtures framework extensively uses dual quaternion
algebra because of its several advantages over other representations.
For instance, unit dual quaternions do not have representational singularities
and are more compact and computationally efficient than homogeneous
transformation matrices \cite{Adorno2011e}. Moreover, their strong
algebraic properties allow different robots to be systematically modeled
\cite{yang1963application,Adorno2011e,Perez2004,Quiroz-Omana2017}
and, in addition to representing rigid motions, dual quaternion algebra
is very useful for describing twists, wrenches, and several geometrical
primitives—e.g., Plücker lines and planes—in very straightforwardly
\cite{adorno2017robot}. The next subsection introduces the basic
definitions of quaternions and dual quaternions; more information
can be found in \cite{yang1963application,adorno2017robot,Selig2005}.

\subsection{Quaternions and dual Quaternions}

Quaternions can be regarded as an extension of complex numbers. The
quaternion set is defined as
\[
\mathbb{H}\triangleq\left\{ h_{1}+\hat{\imath} h_{2}+\hat{\jmath} h_{3}+\hat{k} h_{4}\,:\,h_{1},h_{2},h_{3},h_{4}\in\mathbb{R}\right\} ,
\]
in which the imaginary units $\hat{\imath}$, $\hat{\jmath}$, and $\hat{k}$ have the
properties $\hat{\imath}^{2}=\hat{\jmath}^{2}=\hat{k}^{2}=\hat{\imath}\hat{\jmath}\hat{k}=-1$.
The dual quaternion set is defined as
\[
\mathcal{H}\triangleq\left\{ \quat h+\varepsilon\quat h'\,:\,\quat h,\quat h'\in\mathbb{H},\,\varepsilon^{2}=0,\,\varepsilon\neq0\right\} ,
\]
where $\varepsilon$ is the dual (or Clifford) unit \cite{Selig2005}. Addition
and multiplication are defined for dual quaternions analogously to
complex numbers, and hence we need only to respect the properties
of the imaginary and dual units.

Given $\dq h\in\mathcal{H}$ such that $\dq h=h_{1}+\hat{\imath} h_{2}+\hat{\jmath} h_{3}+\hat{k} h_{4}+\varepsilon\left(h_{1}'+\hat{\imath} h_{2}'+\hat{\jmath} h_{3}'+\hat{k} h_{4}'\right)$,
we define the operators
\begin{align*}
\getp{\dq h} & \triangleq h_{1}{+}\hat{\imath} h_{2}{+}\hat{\jmath} h_{3}{+}\hat{k} h_{4}, & \getd{\dq h} & \triangleq h_{1}'{+}\hat{\imath} h_{2}'{+}\hat{\jmath} h_{3}'{+}\hat{k} h_{4}',
\end{align*}
and
\[
\begin{split}\real{\dq h} & \triangleq h_{1}+\varepsilon h_{1}',\\
\imag{\dq h} & \triangleq\hat{\imath} h_{2}+\hat{\jmath} h_{3}+\hat{k} h_{4}+\varepsilon\left(\hat{\imath} h_{2}'+\hat{\jmath} h_{3}'+\hat{k} h_{4}'\right).
\end{split}
\]
The conjugate of $\dq h$ is defined as $\conj{\dq h}\triangleq\real{\dq h}-\imag{\dq h}$,
and its norm is given by $\norm{\dq h}=\sqrt{\dq h\conj{\dq h}}=\sqrt{\conj{\dq h}\dq h}$.

The set $\mathbb{H}_{p}\triangleq\left\{ \quat h\in\mathbb{H}\,:\,\real{\quat h}=0\right\} $
is isomorphic to $\mathbb{R}^{3}$ under the addition operation. Thus,
the quaternion $\left(x\hat{\imath}+y\hat{\jmath}+z\hat{k}\right)\in\mathbb{H}_{p}$
represents the point $\left(x,y,z\right)\in\mathbb{R}^{3}$. The set
of quaternions with unit norm is defined as $\mathbb{S}^{3}\triangleq\left\{ \quat h\in\mathbb{H}\,:\,\norm{\quat h}=1\right\} $,
and $\quat r\in\mathbb{S}^{3}$ can always be written as $\quat r=\cos\left(\phi/2\right)+\quat v\sin\left(\phi/2\right)$,
where $\phi\in\mathbb{R}$ is the rotation angle around the rotation
axis $\quat v\in\mathbb{S}^{3}\cap\mathbb{H}_{p}$ \cite{adorno2017robot}.
Elements of the set $\dq{\mathcal{S}}\triangleq\left\{ \dq h\in\mathcal{H}\,:\,\norm{\dq h}=1\right\} $
are called unit dual quaternions and represent tridimensional poses
(i.e., combined position and orientation) of rigid bodies. Given $\dq x\in\dq{\mathcal{S}}$,
it can always be written as $\dq x=\quat r+\varepsilon\left(1/2\right)\quat t\quat r$,
where $\quat r\in\mathbb{S}^{3}$ and $\quat t\in\mathbb{H}_{p}$
represent the orientation and position, respectively \cite{Selig2005}.
The set $\dq{\mathcal{S}}$ equipped with the multiplication operation
forms the group $\text{Spin}(3){\ltimes}\mathbb{R}^{3}$, which double covers $\mathrm{SE}\left(3\right)$.

Elements of the set $\mathcal{H}_{p}\triangleq\left\{ \dq h\in\mathcal{H}\,:\,\real{\dq h}=0\right\} $
are called pure dual quaternions and are useful for representing Plücker
lines. More specifically, a Plücker line belongs to the set $\mathcal{H}_{p}\cap\dq{\mathcal{S}}$
and thus is represented by a pure unit dual quaternion such as \cite{adorno2017robot,yang1963application}
\begin{equation}
\dq l=\quat l+\varepsilon\quat m,\label{eq:plucker_line}
\end{equation}
where $\quat l\in\mathbb{H}_{p}\cap\mathbb{S}^{3}$ is a pure quaternion
with unit norm that represents the line direction, and the line moment
is given by $\quat m=\quat p_{l}\times\quat l$, in which $\quat p_{l}\in\mathbb{H}_{p}$
is a point in the line and $\times$ is the cross product. Given $\dq a,\dq b\in\mathcal{H}_{p}$,
the inner product and the cross product are respectively defined as
\cite{yang1963application,adorno2017robot}
\begin{align}
\dotmul{\dq a}{\dq b} & \triangleq-\frac{\dq a\dq b+\dq b\dq a}{2}, & \dq a\times\dq b & \triangleq\frac{\dq a\dq b-\dq b\dq a}{2}.\label{eq:dot and cross products}
\end{align}

The operator $\vecfour{}$ maps quaternions into $\mathbb{R}^{4}$
and $\veceight{}$ maps dual quaternions into $\mathbb{R}^{8}$. For
instance, $\vecfour{\quat h}=\begin{bmatrix}h_{1} & h_{2} & h_{3} & h_{4}\end{bmatrix}^{T}$
and $\veceight{\dq h}=\begin{bmatrix}h_{1} & h_{2} & h_{3} & h_{4} & h_{1}^{'} & h_{2}^{'} & h_{3}^{'} & h_{4}^{'}\end{bmatrix}^{T}$.

Finally, given $\quat h_{1},\quat h_{2}\in\mathbb{H}$, the Hamilton
operators are matrices that satisfy $\vecfour{\left(\quat h_{1}\quat h_{2}\right)}=\haplus{\quat h_{1}}\vecfour{\quat h_{2}}=\haminus{\quat h_{2}}\vecfour{\quat h_{1}}$.
Analogously, given $\dq h_{1},\dq h_{2}\in\mathcal{H}$, the Hamilton
operators satisfy $\veceight{\left(\dq h_{1}\dq h_{2}\right)}=\hapluseight{\dq h_{1}}\veceight{\dq h_{2}}=\haminuseight{\dq h_{2}}\veceight{\dq h_{1}}$
\cite{adorno2017robot}.

It can be shown by direct calculation that, when $\quat a,\quat b\in\mathbb{H}_{p}$,
the inner product can be written as
\[
\dotmul{\quat a}{\quat b}=\left(\vecfour{\quat a}\right)^{T}\vecfour{\quat b}=\left(\vecfour{\quat b}\right)^{T}\vecfour{\quat a}
\]
and the cross product between $\quat a$ and $\quat b$, which is
analogous to the case of vectors in $\mathbb{R}^{3}$, can be mapped
into $\mathbb{R}^{4}$ by using the operator $\crossmatrix{\cdot}$
as
\begin{align*}
\vecfour{\left(\quat a\times\quat b\right)} & =\underbrace{\begin{bmatrix}0 & 0 & 0 & 0\\
0 & 0 & -a_{4} & a_{3}\\
0 & a_{4} & 0 & -a_{2}\\
0 & -a_{3} & a_{2} & 0
\end{bmatrix}}_{\crossmatrix{\quat a}}\vecfour{\quat b}\\
 & =\crossmatrix{\quat a}\vecfour{\quat b}=\crossmatrix{\quat b}^{T}\vecfour{\quat a}.
\end{align*}
Finally, the time-derivative of a quaternion $\quat a\in\mathbb{H}_{p}$
is given by{\small
\begin{align}
\totalderivative{\norm{\quat a}}t=\frac{\dot{\quat a}\conj{\quat a}+\quat a\conj{\dot{\quat a}}}{2\norm{\quat a}}= & \frac{\dotmul{\dot{\quat a}}{\quat a}}{\norm{\quat a}}=\frac{1}{\norm{\quat a}}\vecfour{\quat a^{T}}\vecfour{\dot{\quat a}},\label{eq:quaternion-norm-derivative-1}
\end{align}
}since for any $\quat a\in\mathbb{H}_{p}$ we have $\conj{\quat a}=-\quat a$.

\subsection{Differential kinematics}

Differential kinematics is the relation between task space velocities
and joint space velocities, in the general form $\dot{\myvec x}=\mymatrix J\dot{\myvec q}$,
in which $\dot{\myvec x}$ $\in$ $\mathbb{R}^{m}$ is the vector
of task space velocities, $\dot{\myvec q}$ $\in$ $\mathbb{R}^{n}$
is the vector of manipulator joint velocities, and $\mymatrix J$
$\in$ $\mathbb{R}^{m\times n}$ is the Jacobian matrix.

For instance, given the robot end effector pose $\dq x\triangleq\dq x\left(\myvec q\left(t\right)\right)\in\text{Spin}(3){\ltimes}\mathbb{R}^{3}$,
the differential kinematics is given by $\veceight{\dot{\dq x}}=\mymatrix J_{\dq x}\dot{\myvec q}$,
where $\mymatrix J_{\dq x}$ $\in$ $\mathbb{R}^{8\times n}$ is the
dual quaternion analytical Jacobian, which can be found by using dual
quaternion algebra \cite{Adorno2011e}. Similarly, given the end effector
position $\quat t\triangleq\quat t\left(\quat q\left(t\right)\right)\in\mathbb{H}_{p}$
and the orientation $\quat r\triangleq\quat r\left(\myvec q\left(t\right)\right)\in\mathbb{S}^{3}$,
we have
\begin{align}
\vecfour{\dot{\quat t}} & =\mymatrix J_{t}\dot{\myvec q},\label{eq:translation_jacobian}\\
\vecfour{\dot{\quat r}} & =\mymatrix J_{r}\dot{\myvec q},\label{eq:rotation_jacobian}
\end{align}
where $\mymatrix J_{t},\mymatrix J_{r}\in\mathbb{R}^{4\times n}$
are calculated from $\mymatrix J_{\dq x}$ also by using dual quaternion
algebra \cite{adorno2010dual}. It is important to note that, in general
applications, $\dq x$, $\quat t$, and $\quat r$ may refer to any
relevant coordinate system related to the robot, and not only to the
end effector.

\subsection{Linear programming for differential inverse kinematics }

In closed-loop differential inverse kinematics, first a desired task
space target, $\myvec x_{d}$, and the task error $\myvec{\tilde{x}}=\myvec x-\myvec x_{d}$
are defined. For $\eta$ $\in$ $(0,\infty)$, the minimum-norm analytical
solution of the optimization problem
\begin{align}
\underset{\dot{\myvec q}}{\min} & \norm{\mymatrix J\dot{\myvec q}+\eta\myvec{\tilde{x}}}_{2}^{2}\label{eq:problem_quadratic}
\end{align}
is $\mymatrix J^{\dagger}\eta\myvec{\tilde{x}}=\dot{\myvec q}$, in
which $\mymatrix J^{\dagger}$ is the generalized Moore-Penrose pseudoinverse
of $\mymatrix J$.

Adding inequality and equality constraints in Problem~\ref{eq:problem_quadratic}
turns it into a quadratic programming problem requiring a numerical
solver \cite{Escande2014}. The 1-norm analogue of Problem \ref{eq:problem_quadratic}
is
\begin{align}
\underset{\dot{\myvec q}}{\min} & \norm{\mymatrix J\dot{\myvec q}+\eta\myvec{\tilde{x}}}_{1}.\label{eq:problem_linear}
\end{align}
A Lyapunov stable solution for Problem \ref{eq:problem_linear} that
considers both equality and inequality constraints is given by the
following linear program in canonical form \cite{gonccalves2016parsimonious},
\begin{align}
\underset{\myvec g}{\min} & \begin{bmatrix}-\myvec 1^{T}\mymatrix J & \myvec 1^{T}\mymatrix J & 2\cdot\myvec 1^{T} & \myvec 0^{T} & 0 & \myvec 0^{T}\end{bmatrix}\myvec g-\eta\myvec 1^{T}\myvec{\tilde{x}}\nonumber \\
\text{subject to} & \begin{bmatrix}\mymatrix J & -\mymatrix J & -\mymatrix I & \mymatrix I & \mymatrix 0 & \mymatrix 0\\
\myvec 1^{T} & \myvec 1^{T} & \myvec 0^{T} & \myvec 0^{T} & 1 & \myvec 0^{T}\\
\mymatrix W & -\mymatrix W & \mymatrix 0 & \mymatrix 0 & \mymatrix 0 & \mymatrix I
\end{bmatrix}\myvec g=\begin{bmatrix}-\eta\myvec{\tilde{x}}\\
\beta\norm{\myvec{\tilde{x}}}_{1}\\
\myvec w
\end{bmatrix},\label{eq:problem_linear_final}\\
 & \myvec g\geq0\nonumber
\end{align}
where $\myvec g=\begin{bmatrix}\dot{\myvec q}_{P}^{T} & \dot{\myvec q}_{N}^{T} & \myvec y^{T} & z_{A}^{T} & z_{B} & z_{C}^{T}\end{bmatrix}^{T}$
is the vector of decision variables. Moreover, $\myvec 0^{T}$ and
$\myvec 1^{T}$ are row vectors of ones and zeros of appropriate dimensions,
respectively. In the canonical form, all decision variables should
be non-negative, and hence the joint velocities are split into the
non-negative $\dot{\myvec q}_{P}$ and $\dot{\myvec q}_{N}$ such
that
\begin{equation}
\dot{\myvec q}=\dot{\myvec q}_{P}-\dot{\myvec q}_{N}.\label{eq:control_input_decomposed}
\end{equation}
Moreover, $\myvec y$ is a vector of residuals for the objective function,
and the slack variables $z_{A}$, $z_{B}$, and $z_{C}$ are used
to transform inequality constraints into equality ones.

The matrix of constraints has three horizontal blocks from top to
bottom. The first block concerns error convergence, the second block
guarantees that the joints will stop moving when $\myvec{\tilde{x}}\rightarrow0$
with $\beta$ $\in\left(0,\infty\right)$, and the third block is
a generic block of $r$ constraints, in which $\mymatrix W$ $\in$
$\mathbb{R}^{r\times n}$ and $\myvec w$ $\in$ $\mathbb{R}^{r\times1}$.
The vector field inequalities proposed in this work concern an appropriate
choice of $\mymatrix W$and $\myvec w$ to generate active constraints,
as described in the following section.

\section{Vector field inequality}

\begin{figure}[h]
\centering

\def1.0\columnwidth{1\columnwidth}
\footnotesize\input{vector_field_inequality.pdf_tex}

\caption{\label{fig:vectorfieldinequality}Proposed vector field inequality
method. The vector field is given by an inequality constraint, and
to each point in space is assigned a maximum approach velocity (the
red vector in each vector pair), and a maximum separating velocity
(the green vector in each vector pair) in the direction perpendicular
to the restricted zone boundary. Tangential velocities are unconstrained. }
\end{figure}

The vector field inequality proposed in this paper is a general method
for active constraints. It requires:
\begin{enumerate}
\item A function $d(t)$ $\in$ $\mathbb{R}$ that encodes the (signed)
distance between the two collidable entities, and
\item The Jacobian relating the time derivative of the distance and the
joint velocities in the general form
\begin{equation}
\dot{d}(t)=\underbrace{\partialfrac{d(t)}{\myvec q}}_{\mymatrix J_{d}}\dot{\myvec q}.\label{eq:distance_kinematics_general}
\end{equation}
\end{enumerate}
Using the distance function and the Jacobian, complex restricted zones
can be generated, by either maintaining the distance above a desired
level or keeping the distance below a certain level. The vector field
inequality is illustrated in Fig. \ref{fig:vectorfieldinequality}.

\subsection{Preventing the robot from entering a restricted region\label{subsec:robot_outside_restricted_region}}

We first define a minimum safe distance, $d_{\text{safe}}$ $\in$
$\left[0,\infty\right)$, that delineates the boundary of the restricted
zone. We then define a distance error as
\begin{equation}
\tilde{d}(t)\triangleq d(t)-d_{\text{safe}},\label{eq:distance_error}
\end{equation}
which will be positive when in the safe-zone, zero along the boundary,
and negative within the restricted zone.

Assuming a constant safe distance, the distance error dynamics is
given by $\dot{\tilde{d}}(t)=\dot{d}(t).$ A positive $\dot{\tilde{d}}(t)$
means that the system is moving away from the restricted space boundary
and a negative $\dot{\tilde{d}}(t)$ means that the system is moving
closer to the restricted space. The goal is to constrain the distance
dynamics by the inequality
\begin{equation}
\dot{\tilde{d}}(t)\geq-\eta_{d}\tilde{d}(t),\label{eq:lp4_constraint}
\end{equation}
where $\eta_{d}\in\left(0,\infty\right)$.

To understand Constraint~\ref{eq:lp4_constraint}, let us suppose
that $\tilde{d}(t)>0$, which means that the system is outside the
restricted zone. In this situation, \emph{any }increase in distance
is \emph{always} allowed, which implies $\dot{\tilde{d}}\geq0\geq-\eta_{d}\tilde{d}$.
However, when the system is approaching the boundary from the safe
region (i.e., $0\geq\dot{\tilde{d}}\geq-\eta_{d}\tilde{d}$), it can
approach with a maximum velocity that decreases, in the worst case,
exponentially with the distance $\tilde{d}(t)$, because the maximum
decreasing rate is achieved when $\dot{\tilde{d}}(t)=-\eta_{d}\tilde{d}(t)$.
Thus, the closer the object is to the boundary, the slower it can
move in its direction. Any slower motion toward the boundary is also
allowed, and hence $\dot{\tilde{d}}(t)\geq-\eta_{d}\tilde{d}(t)$.
As soon as $\tilde{d}(t)=0$, the restriction becomes $\dot{\tilde{d}}(t)\geq0$
and therefore the system will not enter the restricted zone.

Considering \eqref{eq:distance_kinematics_general}, Constraint~\ref{eq:lp4_constraint}
is written explicitly in terms of joint velocities as
\begin{alignat}{1}
\mymatrix J_{d}\dot{\myvec q}\geq-\eta_{d}\tilde{d}(t) & \iff-\mymatrix J_{d}\dot{\myvec q}\leq\eta_{d}\tilde{d}(t).\label{lp4_constrained_jacobian}
\end{alignat}
To turn Constraint~\ref{lp4_constrained_jacobian} into an equality
and fit it into the linear programming formalism, we introduce the
slack variable $z_{D}$ and decompose $\dot{\myvec q}$ as in \eqref{eq:control_input_decomposed}
to find
\begin{equation}
-\mymatrix J_{d}\dot{\myvec q}_{P}+\mymatrix J_{d}\dot{\myvec q}_{N}+z_{D}=\eta_{d}\tilde{d}(t).\label{eq:lp4_solution}
\end{equation}
Notice that any number of constraints in the form of Constraint~\ref{eq:lp4_solution}
can be found for different interactions within the robot workspace.
\begin{remrk}
Consider that the system is initially inside the restricted zone;
that is, $\tilde{d}(t)<0$. In this case, Constraint~\ref{eq:lp4_constraint}
will be fulfilled only if $\dot{\tilde{d}}(t)$ is greater than zero,
which means that the system will, at least, be pushed back to the
boundary exponentially.
\end{remrk}

\subsection{Maintaining the robot inside a safe region}

Using the same reasoning as in Section~\ref{subsec:robot_outside_restricted_region},
if we want to maintain the robot \emph{inside} a safe region, we must
redefine $d_{\text{safe}}$; that is,
\[
\tilde{d}(t)\triangleq d_{\text{safe}}-d(t),
\]
with the final solution, assuming the desired error dynamics \eqref{eq:lp4_constraint},
given by
\[
\mymatrix J_{d}\dot{\myvec q}_{P}-\mymatrix J_{d}\dot{\myvec q}_{N}+z_{D}=\eta_{d}\tilde{d}(t).
\]

\section{Distance functions and Jacobians for active constraints}

The proposed vector field inequality discussed in the previous section
requires a distance function $d(t)$ and the corresponding distance
Jacobian $\mymatrix J_{d}$ for each collidable pair. In this section,
we find four distance functions and Jacobians for pairs in which one
element is static and the second element is dynamic.

\subsection{Point–static-plane distance Jacobian, $\protect\mymatrix J_{t,\pi}$}

One of the primitives for virtual fixtures is the constraint of a
point such that it is above a static plane. Given a reference frame
$\mathcal{F}$, a plane $\dq{\pi}$ in dual quaternion space is given
by \cite{adorno2017robot}
\[
\dq{\pi}\triangleq\quat n_{\pi}+\varepsilon d_{\pi},
\]
in which $\quat n_{\pi}\in\mathbb{H}_{p}\cap\mathbb{S}^{3}$ is the
normal to the plane and $d_{\pi}$ $\in$ $\mathbb{R}$ is the signed
perpendicular distance between the plane and the origin of the reference
frame. Moreover, the distance $d_{\pi}$ is given by $d_{\pi}=\dotmul{\quat p_{\pi}}{\quat n_{\pi}}$,
where $\quat p_{\pi}$ is an arbitrary point in the plane. If $\mathcal{F}_{\pi}$
is a frame attached to the plane, the signed distance between $\quat t$,
an arbitrary point in the robot kinematic chain, and the plane, from
the point of view of the plane, is given by
\begin{equation}
d_{t,\pi}^{\pi}\triangleq\dotmul{\quat t}{\quat n_{\pi}}-d_{\pi}.\label{eq:point-static-plane-distance}
\end{equation}
The time derivative of \eqref{eq:point-static-plane-distance} is
\begin{alignat}{1}
\totalderivative{d_{t,\pi}^{\pi}}t\underset{\eqref{eq:translation_jacobian}}{=} & \dotmul{\dot{\quat t}}{\quat n_{\pi}}=\underbrace{\left(\vecfour{\quat n_{\pi}}\right)^{T}\mymatrix J_{t}}_{\mymatrix J_{t,\pi}}\dot{\myvec q}.\label{eq:point-static-plane-jacobian}
\end{alignat}

\subsection{Line Jacobian, $\protect\mymatrix J_{l_{z}}$}

The line Jacobian is a pre-requisite for the more complex Jacobians
described in the following sections.

The Plücker line \cite{adorno2017robot} $\dq l_{z}\in\mathcal{H}_{p}\cap\dq{\mathcal{S}}$
collinear to the $z$-axis of a frame represented by $\dq x=\quat r+\varepsilon\left(1/2\right)\quat t\quat r$
is given by
\begin{equation}
\dq l_{z}=\quat l_{z}+\varepsilon\quat m_{z},\label{eq:line_z}
\end{equation}
where $\quat l_{z}=\quat r\hat{k}\conj{\quat r}$ and $\quat m_{z}=\quat t\times\quat l_{z}.$
The derivative of \eqref{eq:line_z} is
\begin{align}
\dot{\dq l}_{z} & =\dot{\quat l}_{z}+\varepsilon\dot{\quat m}_{z}.\label{eq:lz_derivative}
\end{align}
Hence,
\begin{align}
\vecfour{\dot{\quat l}_{z}} & \underset{\eqref{eq:rotation_jacobian}}{=}\underbrace{\left(\haminus{\hat{k}\conj{\quat r}}+\haplus{\quat r\hat{k}}\boldsymbol{C}_{4}\right)\mymatrix J_{r}}_{\mymatrix J_{r_{z}}}\dot{\myvec q}\label{eq:line_direction_jacobian}
\end{align}
in which $\boldsymbol{C}_{4}=\text{diag\ensuremath{\left(1,-1,-1,-1\right)}}.$
In addition,

{\small
\begin{gather*}
\vecfour{\dot{\quat m}_{z}}{=}\underbrace{\frac{\left(\haminus{\quat l_{z}}{-}\haplus{\quat l_{z}}\right)\!\mymatrix J_{t}{+}\left(\haplus{\quat t}{-}\haminus{\quat t}\right)\!\mymatrix J_{r_{z}}}{2}}_{\mymatrix J_{m_{z}}}\dot{\myvec q}.
\end{gather*}
}Finally, \eqref{eq:lz_derivative} can be re-written in term of
joint velocities as
\begin{align}
\veceight{\dot{\dq l}_{z}} & =\begin{bmatrix}\mymatrix J_{r_{z}}\\
\mymatrix J_{m_{z}}
\end{bmatrix}\dot{\myvec q}\triangleq\mymatrix J_{l_{z}}\dot{\myvec q}.\label{eq:robot_line_jacobian}
\end{align}

\subsection{Line–static-point distance Jacobian, $\protect\mymatrix J_{l_{z},p}$}

The line–static-point Jacobian can be used to generate a remote center
of motion. First, we notice that the distance between $\dq l_{z}\in\mathcal{H}_{p}\cap\dq{\mathcal{S}}$
and an arbitrary static point $\quat p\in\mathbb{H}_{p}$ is given
by
\begin{equation}
d_{l_{z},p}=\norm{\quat p\times\quat l_{z}-\quat m_{z}}.\label{eq:line-static-point-distance}
\end{equation}
The derivative of \eqref{eq:line-static-point-distance} is

{\footnotesize
\begin{alignat}{1}
\dot{d}_{l_{z},p} & =\frac{1}{d_{l_{z},p}}\vecfour{\left(\quat p\times\quat l_{z}-\quat m_{z}\right)^{T}}\vecfour{\totalderivative{\left(\quat p\times\quat l_{z}-\quat m_{z}\right)}t}\nonumber \\
 & \underset{\eqref{eq:robot_line_jacobian}}{=}\underbrace{\frac{1}{d_{l_{z},p}}\vecfour{\left(\quat p\times\quat l_{z}-\quat m_{z}\right)^{T}}\left(\crossmatrix{\quat p}\mymatrix J_{r_{z}}-\mymatrix J_{m_{z}}\right)}_{\mymatrix J_{l_{z},p}}\dot{\myvec q}.\label{eq:line-static-point-jacobian}
\end{alignat}

}

\subsection{Point–static-line distance Jacobian, $\protect\mymatrix J_{t,l}$}

The point–static-line Jacobian can be used to keep a point inside/outside
a cylinder. First, we notice that the distance between an arbitrary
line $\dq l\in\mathcal{H}_{p}\cap\dq{\mathcal{S}}$, such that $\dq l=\quat l+\varepsilon\quat m$,
and a point $\quat t\in\mathbb{H}_{p}$ in the robot kinematic chain
is given by
\begin{equation}
d_{t,l}=\norm{\quat t\times\quat l-\quat m}.\label{eq:point-static-line-distance}
\end{equation}
The derivative of \eqref{eq:point-static-line-distance} is given
by
\begin{alignat}{1}
\dot{d}_{t,l} & =\frac{1}{d_{t,l}}\vecfour{\left(\quat t\times\quat l-\quat m\right)^{T}}\vecfour{\totalderivative{\left(\quat t\times\quat l-\quat m\right)}t}\nonumber \\
 & \underset{\eqref{eq:translation_jacobian}}{=}\underbrace{\frac{1}{d_{t,l}}\vecfour{\left(\quat t\times\quat l-\quat m\right)^{T}}\crossmatrix{\quat l}^{T}\mymatrix J_{t}}_{\mymatrix J_{t,l}}\dot{\myvec q}.\label{eq:point-static-line-jacobian}
\end{alignat}

\subsection{Line–static-line distance Jacobian}

The line–static-line Jacobian is particularly useful for avoiding
collisions between a moving cylinder and a static cylinder. In order
to obtain the line–static-line Jacobian, we use the concept of dual
angle between Plucker lines.

\subsubsection{Inner product Jacobian, $\protect\mymatrix J_{\protect\dotmul{\protect\dq l_{z}}{\protect\dq l}}$}

The dual cosine between Plücker lines $\dq l,\dq l_{z}\in\mathcal{H}_{p}\cap\dq{\mathcal{S}}$
is obtained by using the inner product \cite{yang1963application}
\begin{gather}
\dotmul{\dq l}{\dq l_{z}}=\underbrace{\norm{\dq l}}_{1}\underbrace{\norm{\dq l_{z}}}_{1}\cos\left(\phi+\varepsilon d_{l}\right)=\cos\phi-\varepsilon d_{l}\sin\phi,\label{eq:dual_distance}
\end{gather}
where $d_{l}\in[0,\infty)$ and $\phi\in[0,2\pi)$ are the distance
and the angle between the lines, respectively.\footnote{Given a function $f\,:\,\mathbb{D}\to\mathbb{D}$, where $\mathbb{D}\triangleq\left\{ \dq h\in\mathcal{H}\,:\,\imag{\dq h}=0\right\} $,
it is possible to show that $f\left(a+\varepsilon b\right)=f\left(a\right)+\varepsilon bf'\left(a\right)$.
For more details, see \cite{adorno2017robot}.}

Since the line $\dq l$ is static, $\dot{\dq l}=0$ and thus the derivative
of \eqref{eq:dual_distance} is given by
\begin{align}
\veceight{\totalderivative{\dotmul{\dq l_{z}}{\dq l}}t} & =\underbrace{-\frac{\left(\hapluseight{\dq l}+\haminuseight{\dq l}\right)}{2}\mymatrix J_{l}}_{\mymatrix J_{\dotmul{\dq l_{z}}{\dq l}}}\dot{\myvec q}\nonumber \\
\begin{bmatrix}\vecfour{\getpdot{\dotmul{\dq l_{z}}{\dq l}}}\\
\vecfour{\getddot{\dotmul{\dq l_{z}}{\dq l}}}
\end{bmatrix} & =\begin{bmatrix}\mymatrix J_{\getp{\dotmul{\dq l_{z}}{\dq l}}}\\
\mymatrix J_{\getd{\dotmul{\dq l_{z}}{\dq l}}}
\end{bmatrix}\dot{\myvec q}.\label{eq:dotprodjacobian}
\end{align}

\subsubsection{Cross product Jacobian, $\protect\mymatrix J_{\protect\dq l_{z}\times\protect\dq l}$}

Given the dual angle $\dq{\phi}=\phi+\varepsilon d_{l}$, where $d_{l}\in[0,\infty)$
and $\phi\in[0,2\pi)$, and the Plücker line $\dq s\in\mathcal{H}_{p}\cap\dq{\mathcal{S}}$,
with $\dq s=\quat s+\varepsilon\quat m_{s}$, the cross product between
$\dq l_{z}$ and $\dq l$ is given by \cite{yang1963application}
\begin{align}
\dq l_{z}\times\dq l & =\underbrace{\norm{\dq l_{z}}}_{1}\underbrace{\norm{\dq l}}_{1}\dq s\sin\dq{\phi}\nonumber \\
 & =\left(\quat s+\varepsilon\quat m_{s}\right)\left(\sin\phi+\varepsilon d_{l}\cos\phi\right)\nonumber \\
 & =\quat s\sin\phi+\varepsilon\left(\quat m_{s}\sin\phi+\quat sd_{l}\cos\phi\right),\label{eq:line-dual-sine}
\end{align}
in which $\dq s$ is the line perpendicular to both $\dq l_{z}$ and
$\dq l$. The time derivative of the cross product between $\dq l_{z}$
and the static line $\dq l$ is
\begin{align}
\veceight{\totalderivative{\dq l_{z}\times\dq l}t} & =\underbrace{\frac{\left(\haminuseight{\dq l}-\hapluseight{\dq l}\right)}{2}\mymatrix J_{l}}_{\mymatrix J_{\dq l_{z}\times\dq l}}\dot{\myvec q}\nonumber \\
\begin{bmatrix}\vecfour{\getpdot{\dq l_{z}\times\dq l}}\\
\vecfour{\getddot{\dq l_{z}\times\dq l}}
\end{bmatrix} & =\begin{bmatrix}\mymatrix J_{\getp{\dq l_{z}\times\dq l}}\\
\mymatrix J_{\getd{\dq l_{z}\times\dq l}}
\end{bmatrix}\dot{\myvec q}.\label{eq:crossprodjacobian}
\end{align}

\subsubsection{Distance Jacobian between non-parallel lines}

The distance $d_{\quat l_{z},\quat l\not\parallel}$ between $\dq l_{z}$
and $\dq l$ when they are not parallel (i.e., $\phi\in(0,2\pi)\setminus\pi)$
can be obtained as
\begin{equation}
d_{\quat l_{z},\quat l\not\parallel}=\frac{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}{\norm{\getp{\dq l_{z}\times\dq l}}}=\frac{\norm{d_{l}\sin\phi}}{\norm{\quat s_{l}\sin\phi}}=d_{l}.\label{eq:line-static-line-nonparallel-distance}
\end{equation}

Noting that both the numerator and denominator of \eqref{eq:line-static-line-nonparallel-distance}
are real numbers, we can find the derivative of \eqref{eq:line-static-line-nonparallel-distance}
as
\begin{multline}
\dot{d}_{\quat l_{z},\quat l\not\parallel}=\underbrace{\frac{1}{\norm{\getp{\dq l_{z}\times\dq l}}}}_{a}\totalderivative{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}t\\
\underbrace{-\frac{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}{\norm{\getp{\dq l_{z}\times\dq l}}^{2}}}_{b}\totalderivative{\norm{\getp{\dq l_{z}\times\dq l}}}t.\label{eq:non-parallel-squared-distance-jacobian}
\end{multline}
We obtain the derivative of the norm of $\getd{\dotmul{\dq l_{z}}{\dq l}}$
using \eqref{eq:quaternion-norm-derivative-1} and \eqref{eq:dotprodjacobian}
as

{\footnotesize
\begin{align}
\totalderivative{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}t & =\underbrace{\frac{1}{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}\vecfour{\getd{\dotmul{\dq l_{z}}{\dq l}}^{T}}\mymatrix J_{\getd{\dotmul{\dq l_{z}}{\dq l}}}}_{\mymatrix J_{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}}\dot{\myvec q}\label{eq:nonparalleldual}
\end{align}
}and similarly we obtain the derivative of the norm of $\getp{\dq l_{z}\times\dq l}$
using \eqref{eq:quaternion-norm-derivative-1} and \eqref{eq:crossprodjacobian}
as

{\footnotesize
\begin{align}
\totalderivative{\norm{\getp{\dq l_{z}\times\dq l}}}t & =\underbrace{\frac{1}{\norm{\getp{\dq l_{z}\times\dq l}}}\vecfour{\getp{\dq l_{z}\times\dq l}^{T}}\mymatrix J_{\getp{\dq l_{z}\times\dq l}}}_{\mymatrix J_{\norm{\getp{\dq l_{z}\times\dq l}}}}\dot{\myvec q}.\label{eq:nonparallelprim}
\end{align}
}Finally, we substitute \eqref{eq:nonparalleldual} and \eqref{eq:nonparallelprim}
in \eqref{eq:non-parallel-squared-distance-jacobian} to find
\begin{align}
\dot{d}_{\quat l_{z},\quat l\not\parallel} & =\underbrace{\left(a\mymatrix J_{\norm{\getd{\dotmul{\dq l_{z}}{\dq l}}}}+b\mymatrix J_{\norm{\getp{\dq l_{z}\times\dq l}}}\right)}_{\mymatrix J_{\quat l_{z},\quat l\not\parallel}}\dot{\myvec q}.\label{eq:nonparalleldistancejocobian}
\end{align}

\subsubsection{Parallel distance Jacobian, $\protect\mymatrix J_{\protect\quat l_{z},\protect\quat l\parallel}$\label{subsec:parallel-distance}}

In the degenerate case in which $\dq l_{z}$ and $\dq l$ are parallel
(i.e., $\phi\in\left\{ 0,\pi\right\} $), the distance between them
can be retrieved as
\begin{equation}
\squaredistance{\quat l_{z}}{\quat l\parallel}{}\triangleq\norm{\getd{\dq l_{z}\times\dq l}}=\norm{\quat sd_{l}}=d_{l}.\label{eq:line-static-line-parallel-distance}
\end{equation}
Resorting once more to \eqref{eq:quaternion-norm-derivative-1}, we
find the derivative of \eqref{eq:line-static-line-parallel-distance}
as{\small
\begin{align}
\dotsquaredistance{\quat l_{z}}{\quat l\parallel}{} & =\underbrace{\frac{1}{\norm{\getd{\dq l_{z}\times\dq l}}}\vecfour{\left(\getd{\dq l_{z}\times\dq l}\right)}^{T}\mymatrix J_{\getd{\dq l_{z}\times\dq l}}}_{\mymatrix J_{\quat l_{z},\quat l\parallel}}.\label{eq:paralleldistancejacobian}
\end{align}
}

\subsubsection{Manipulator line-to-line distance Jacobian, $\protect\mymatrix J_{\protect\quat l_{z},\protect\quat l}$\label{subsec:manipulator-line-line-distance}}

The line-to-line distance Jacobian is given by combining \eqref{eq:nonparalleldistancejocobian}
and \eqref{eq:paralleldistancejacobian} as
\begin{alignat}{2}
\mymatrix J_{\quat l_{z},\quat l} & =\begin{cases}
\mymatrix J_{\quat l_{z},\quat l\not\parallel} & \phi\in(0,2\pi)\setminus\pi\\
\mymatrix J_{\quat l_{z},\quat l\parallel} & \phi\in\left\{ 0,\pi\right\}
\end{cases} & ,\label{eq:manipulator-line-line-distance-jacobian-residual}
\end{alignat}
in which $\phi$ can be obtained as $\phi=\arccos\getp{\dotmul{\dq l}{\dq l_{z}}}$.

The distance between lines can be given by the composition of the
distance \eqref{eq:line-static-line-nonparallel-distance} and \eqref{eq:line-static-line-parallel-distance}
as
\begin{equation}
d_{\quat l_{z},\quat l}=\begin{cases}
d_{\quat l_{z},\quat l\not\parallel} & \phi\in(0,2\pi)\setminus\pi\\
d_{\quat l_{z},\quat l\parallel} & \phi\in\left\{ 0,\pi\right\}
\end{cases}.\label{eq:line-line-distance}
\end{equation}

\section{Simulations}

\begin{figure}
\centering

\def1.0\columnwidth{1\columnwidth}
\footnotesize\input{constraints.pdf_tex}

\caption{\label{fig:setup}Task and constraints used in the simulations. The
task T consists of moving the end effector, $\protect\dq x\left(t\right)$,
of one of the tools in a given trajectory $\protect\dq x_{d}(t)$.
The constraints are: C1, a line-static-line distance constraint, to
avoid collisions between tool shafts, where the shaft of the moving
tool has a center-line given by $\protect\dq l_{z}(t)$ and the static
shaft has a center-line $\protect\dq l$; C2, a line-static-point
distance constraint to generate the entry sphere with the center given
by $\protect\quat p_{\text{rcm}}$; C3, a point-static-line distance
constraint to prevent the lower part of the moving tool given by $\protect\quat t_{6}(t)$
from colliding with the cylindrical workspace with center-line $\protect\dq l_{c}$;
and C4, a point-static-plane distance constraint to prevent the end
effector from crossing the lower bound of the cylindrical workspace
given by the plane $\protect\dq{\pi}$.}
\end{figure}

In order to validate our approach,\footnote{See accompanying video.}
we developed a realistic simulated version, using V-REP\footnote{http://www.coppeliarobotics.com/}
and DQ Robotics for MATLAB\footnote{http://dqrobotics.sourceforge.net},
of a deep neurosurgical robotic system that we previously developed
in close cooperation with neurosurgeons \cite{UEDA2017110} (see Fig.~\ref{fig:setup}).
In deep neurosurgery, the workspace is highly constrained in the microscopic
view. The diameter of the tools is $3.5\,\unit{mm}$, and the radius
and depth of the workspace cylinder are $2.8\,\unit{cm}$ and $8\,\unit{cm}$,
respectively.

In this preliminary evaluation, one of the tools was kept fixed while
the second tool moved. Five simulations (S1-S5) with increasing complexity
were run, as follows. S1: the task T consisted of moving the end effector
along the desired trajectory with no virtual fixtures; S2: constraint
C1 iwas added to prevent the shafts from colliding with each other
using $\mymatrix J_{l_{z},l}$ from \eqref{eq:line-line-distance}
and $d_{l_{z},l}$ as in \eqref{eq:line-line-distance}; S3: in addition
to C1, a spherical entry-point constraint C2 was added to avoid collisions
with the top part of the cylinder using $\mymatrix J_{l_{z},p}$ from
\eqref{eq:line-static-point-jacobian} and $d_{l_{z},p}$ as in \eqref{eq:line-static-point-distance};
S4: in addition to constraints C1 and C2, a point line distance constraint
C3 was added to avoid collisions of the lower part of the shaft with
the workspace boundary, which is represented by a cylinder, using
$\mymatrix J_{t,l}$ from \eqref{eq:point-static-line-jacobian} and
$d_{t,l}$ as in \eqref{eq:point-static-line-distance}; and finally,
S5: in addition to constraints C1, C2, and C3, a point–static-plane
distance constraint C4 was added to prevent the end effector from
crossing the plane $\dq{\pi}$, using $\mymatrix J_{t,\pi}$ from
\eqref{eq:point-static-plane-jacobian} and $d_{t,\pi}$ as in \eqref{eq:point-static-plane-distance}.

The robot has eight degrees-of-freedom in which the first, second,
and fifth joints were prismatic and the remaining joints were revolute.
The robots' initial configurations were such that the shafts were
at a long distance from each other. The trajectory, which was the
same for all simulations, was the screwlinear interpolation of four
poses. The final form of the linear program, when all constraints
were used, was

{\small
\begin{align*}
\underset{\myvec g}{\min} & \left[\begin{array}{ccccc|c|cccc}
-\myvec 1^{T}\mymatrix J_{\dq x} & \myvec 1^{T}\mymatrix J_{\dq x} & 2\cdot\myvec 1^{T} & \myvec 0^{T} & 0 & \myvec 0^{T} & 0 & 0 & 0 & 0\end{array}\right]\myvec g\\
\text{s.t.} & \left[\begin{array}{cccccccccc}
\mymatrix J_{\dq x} & -\mymatrix J_{\dq x} & -\mymatrix I & \mymatrix I & \mymatrix 0 & \mymatrix 0 & \mymatrix 0 & \mymatrix 0 & \mymatrix 0 & 0\\
\myvec 1^{T} & \myvec 1^{T} & \myvec 0^{T} & \myvec 0^{T} & \begin{array}{c}
1\end{array} & \myvec 0^{T} & 0 & 0 & 0 & 0\\
\hline \mymatrix W_{l} & -\mymatrix W_{l} & \mymatrix 0 & \mymatrix 0 & \mymatrix 0 & \mymatrix I & \mymatrix 0 & \mymatrix 0 & \mymatrix 0 & 0\\
\hline \mymatrix J_{l_{z},l} & -\mymatrix J_{l_{z},l} & \myvec 0^{T} & \myvec 0^{T} & 0 & \myvec 0^{T} & 1 & 0 & 0 & 0\\
-\mymatrix J_{l_{z},p} & \mymatrix J_{l_{z},p} & \myvec 0^{T} & \myvec 0^{T} & 0 & \myvec 0^{T} & 0 & 1 & 0 & 0\\
-\mymatrix J_{t,l} & \mymatrix J_{t,l} & \myvec 0^{T} & \myvec 0^{T} & 0 & \myvec 0^{T} & 0 & 0 & 1 & 0\\
\mymatrix J_{t,\pi} & -\mymatrix J_{t,\pi} & \myvec 0^{T} & \myvec 0^{T} & 0 & \myvec 0^{T} & 0 & 0 & 0 & 1
\end{array}\right]\myvec g=\begin{bmatrix}-\eta\veceight{\tilde{\dq x}}\\
\beta\norm{\veceight{\tilde{\dq x}}}_{1}\\
\hline \myvec w_{l}\\
\hline \eta_{l_{z},l}\tilde{d}_{C1}\\
\eta_{l_{z},p}\tilde{d}_{C2}\\
\eta_{t,l}\tilde{d}_{C3}\\
\eta_{t,\pi}\tilde{d}_{C4}
\end{bmatrix}\\
 & \myvec g\geq0\\
 & \myvec g=\begin{bmatrix}\dot{\myvec q}_{P}^{T} & \dot{\myvec q}_{N}^{T} & \myvec y^{T} & z_{A}^{T} & z_{B} & z_{l}^{T} & z_{C1} & z_{C2} & z_{C3} & z_{C4}\end{bmatrix}^{T},
\end{align*}
}in which $\mymatrix W_{l}$ and $\myvec w_{l}$ are constraints
for the joint limits \cite{Quiroz-Omana2017}, $z_{l}^{T},z_{C1},z_{C2},z_{C3},z_{C4}$
are slack variables, $\tilde{d}_{C1}=d_{\text{safe},C1}-d_{l_{z},l}$,
$\tilde{d}_{C2}=d_{l_{z},p}-d_{\text{safe},C2}$, $\tilde{d}_{C3}=d_{t,l}-d_{\text{safe},C3}$,
and $\tilde{d}_{C4}=d_{\text{safe},C4}-d_{t,\pi}$.

For all simulations, the trajectory tracking gain was $\eta=50$,
and $\beta=40$. Whenever present in a given simulation, vector field
inequality gains were set at the same value $\eta_{l_{z},l}=\eta_{l_{z},p}=\eta_{t,l}=\eta_{t,\pi}=0.5$.
The safety distances for each constraint were $d_{\text{safe},C1}=5\,\unit{mm}$,
$d_{\text{safe},C2}=d_{\text{safe},C3}=14\,\unit{mm}$, and $d_{\text{safe},C4}=0$.

\section{Results and discussion}

\begin{figure*}
\centering

\def1.0\columnwidth{2.0\columnwidth}
\small\input{final_distances_reviewed.pdf_tex}

\caption{Distance between collidable entities. Only S5 provided a collision-safe
trajectory, since S4 violated the plane, S3 violated the plane and
the lower part of the cylinder, S2 violated the plane and both the
upper and lower parts of the cylinder, and S1 violated all constraints.
\label{fig:results}}
\end{figure*}

\begin{figure}
\centering

\def1.0\columnwidth{1.0\columnwidth}
\small\input{final_dt_reviewed.pdf_tex}

\caption{Trajectory tracking error (\emph{bottom}) and 2-norm of joint velocities
(\emph{top}). Only S1 could closely follow the prescribed trajectory
as it had no active constraints. In all other simulations, some trajectory
tracking error was allowed in order to avoid collisions. \label{fig:results-1}}
\end{figure}

\begin{figure}
\centering

\def1.0\columnwidth{1.0\columnwidth}
\small\input{tooltiptrajectory.pdf_tex}

\caption{Tool tip trajectories for all experiments. Notice that S1 is effectively
the same as the desired trajectory, and S5 was the only collision-free
trajectory. \label{fig:results-2}}
\end{figure}

The results of the simulation are shown in Fig.~\ref{fig:results}
in terms of the distance between shafts, the distance between the
entry point and tool shaft, the distance between the lower point of
the shaft and the cylinder centerline, and the distance between the
tool tip and the lower plane.

Simulation S1, in which no virtual fixtures were applied caused collisions
between the shafts and between the moving tool and the workspace boundaries.
Constraining only the shafts in S2 was sufficient to avoid collisions
between the shafts, but tool-boundary collisions still occurred. Constraining
both the shafts and the entry point in S3 provided a collision-free
shaft motion and prevented collisions with the uppermost part of the
cylinder. Simulation S4 violated only the plane constraint, and S5
provided a collision-free path. From the point of view of trajectory
error, as shown in Fig.~\ref{fig:results-1}, naturally S1 was able
to follow the trajectory closely since no active constraints were
applied. All other simulations presented tracking errors given the
gradual imposition of restrictions. The trajectories for all the experiments
are shown in Fig.~\ref{fig:results-2}.

In S3, the robot showed a low amplitude vibration near the end of
the trajectory, which may occur when the trajectory tracking error
is unable to stabilize at the origin. It can be somewhat reduced by
tuning $\beta$; however, other solutions, such as regularizing the
objective function, are being analyzed in ongoing research.

\section{Conclusions}

In this paper a novel method for applying active constraints using
vector field inequalities was proposed. The method can be used to
prevent a robot from entering a restricted zone or for maintaining
its location inside a safe area, and is based on a Lyapunov stable
solution that can handle both equality and inequality constraints.
The vector field inequalities limit the velocity of the robot in the
direction of the forbidden zone's boundary while leaving tangential
velocities undisturbed. To use the method, a Jacobian describing the
relation between the time derivative of the distance and joint velocities
is required. In this work, four of the Jacobians are obtained for
relations in which one of the entities is static. Simulation results
for two eight-DOF neurosurgical arms showed that, by using the vector
field inequalities and relevant primitives, all collisions could be
avoided and the robot safely navigated through a restricted workspace.

The proposed method does not require a high-accuracy model of the
environment, but the environment must be decomposed in sufficient
primitives. However, it still requires high-accuracy calibration between
the primitives and the robot. A practical methodology for achieving
this is the topic of ongoing work. An additional topic of ongoing
work is the tracking of a user generated trajectory, which raises
points of discussion such as gain tunning for minimum trajectory tracking
error and latency analysis. The implementation on the real platform
is currently in progress.\bibliographystyle{ieeetr}

redpajama_set_name: RedPajamaArXiv


text:
Tim  Fry (25 August 1935 – 17 May 2004), was an automotive engineer who, whilst in his twenties and working for the Coventry-based Rootes Group, designed the Hillman Imp, in conjunction with Mike Parkes. He subsequently became chief engineer at Chrysler Rootes. He left Rootes and set up the industrial design consultancy Smallfry in 1971.

Fry and Anthony Smallhorn were awarded the Prince Philip Designers Prize for the Sea Truck In 1979.

Life
Fry died of cancer on 17 May 2004 in Rugby, Warwickshire. He was married to Karin and had two daughters.

References

External links
Obituary, The Times (via The Imp Site)

British automotive engineers
1935 births
2004 deaths
Deaths from cancer in England
20th-century English businesspeople

redpajama_set_name: RedPajamaWikipedia


Mica Scraps is the nomenclature given to the irregular lumps of mica as received from the mines having an area between 1/2" and 1 1/2" sq. We export mica scrap mainly to plants all over the world to be made into a pulp used for the manufacturing of mica paper and reconstituted mica plates. It is also used as a raw material for the manufacture of mica flakes and mica powder. There are two forms which are in demand generally one is the tested/ ruby mica scrap (good for calcinations) & another is the untested/white mica scrap (not good for calcination).
We offer Ruby Mica Scrap to our clients. Ruby scrap is generally of Bihar/Jharkhand origin. These are the best known ruby mica scrap from all over the world because of its color, high tensile strength & best known for calcination due to its sustainability at more than 900 degree celsius.
Size: There are mainly 2 sizes available: 1/2" & above(JMS1171) and 1" & above(JMS1024).
White mica Scrap is generally found in Bihar region in mines like Bhana & Chatkari. White Mica is Best suited for the production of high quality Mica Paper & Board. This white mica scrap is not good for calcination as it cannot sustain its properties at a temperature of more than 650 degree Celsius. But its Shine & Colour makes it apt for the production of Paper & Board. Most regions in India besides Bihar have deposits of only Untested Mica, like Bhilwara (Rajasthan), Gudur (Andra Pradesh) but their colour is generally Ruby or green. White Mica Scrap is a specialty of Bihar/ Jharkhand.
We offer Natural Mica Scrap to our clients. This is the basic raw material in the form of lumps dug out from the mines, used to produce mica scrap.
Looking for Mica Scraps ?


text:
Does Photo Show Jackie Kennedy Reaching For Help After JFK's Assassination?
A photograph purportedly taken during the JFK assassination is actually from a re-enactment of that event staged for a 1977 television movie.
Dan Evon
A photograph shows Jackie Kennedy reaching for help in the moments after President John F. Kennedy was assassinated in 1963.
Since at least 2014, several Twitter accounts that specialize in old photographs, such as @OldPicsArchive and @HistoryInPics, have been sharing the above-displayed image as an authentic photograph of former first lady Jackie Kennedy reaching for help in the moments after her husband, former President John F. Kennedy, was assassinated in Dallas in November 1963.
This image is realistic in depicting how Jackie Kennedy reached out across the back of the presidential limousine as her husband was shot (while Special Agent Clint Hill rushed to her aid), as captured in the famous Zapruder film of the event:
Although the picture at the head of this page may be realistic, it isn't real: No reporter or onlooker was in a position to have taken a shot like that, from the vantage point of the roadway, right alongside the presidential limousine. This photograph was taken in conjunction with a reenactment of the assassination staged for "The Trial of Lee Harvey Oswald", a two-part ABC television movie produced in 1977.
The three people seen in this photograph are not JFK, the former First Lady, nor Hill, but rather actor Don Gazzaway, actress Christine Rose, and stuntman Larry Hill, as described in a press release issued to promote the 1977 TV movie:
Secret Service agent Clinton Hill (stuntman Larry Hill) rushes to get Jacqueline Kennedy (actress Christine Rose) back into the car after her husband President John F. Kennedy (actor Don Gazzaway) was shot by an assassin. The action was all part of a made for TV reenactment movie being filmed in Dallas. (UPI Photo/Files)
This article was updated to meet Snopes' current formatting standards on Sept. 1, 2022.
By Dan Evon
Dan Evon is a former writer for Snopes.
The Arts john f. kennedy lee harvey oswald kennedy assasination
Did Hunter Biden Say Controversial Laptop 'Could Be' His?
Does 'The Wizard of Oz' Include a Munchkin Suicide?

redpajama_set_name: RedPajamaCommonCrawl


text:
Congress Clears Bill To Ease Flood Insurance Hikes
Filed Under:Flood Insurance, homeowner's insurance, Rate Hikes
(Photo credit: Thinkstock)
WASHINGTON (AP) — Less than two years after passing major legislation aimed at reforming the government's much-criticized flood insurance program, Congress on Thursday sent President Barack Obama a bill to scale back many of the resulting big flood insurance premium increases faced by hundreds of thousands of homeowners. The measure would also allow below-market insurance rates to be passed on to people buying homes with taxpayer-subsidized policies.
The measure breezed through the Senate and on to Obama's desk by a 72-22 vote. The House passed the measure last week.
The legislation significantly rewrites a major overhaul of the flood insurance program that passed almost unanimously in 2012. Those 2012 changes were aimed at weaning hundreds of thousands of homeowners off of subsidized rates and required extensive updating of the flood maps used to set premiums. But its implementation has stirred anxiety among many homeowners along the Atlantic and Gulf coasts and in flood plains, many of whom are threatened with unaffordable rate increases.
Sen. Mary Landrieu, D-La., said the White House has indicated Obama will sign the measure into law despite earlier administration reservations about a Senate-passed bill that would have delayed implementation of the 2012 law.
"While it is important to put this program on sound financial footing, middle-class families should be able to afford the insurance they need to stay in their homes," White House spokesman Bobby Whithorne said.
Thursday's bill was written by House Majority Leader Eric Cantor, R-Va., and Rep. Michael Grimm, R-N.Y., with input from Democrats like Rep. Maxine Waters of California, whose votes were critical to House passage last week.
The bill would repeal a provision in the 2012 law that threatens hundreds of thousands of homeowners with huge premium increases under new and updated government flood maps. Their properties were originally built to code but were subsequently found to be at greater flood risk. Such "grandfathered" homeowners currently benefit from below-market rates that are subsidized by other policyholders, and the new legislation would preserve that status and cap premium increases at 18 percent a year. The 2012 reforms required premiums increases to actuarially sound rates over five years.
Another provision, eagerly sought by the real estate industry, would allow home sellers to pass taxpayer-subsidized policies on to the people buying their homes instead of requiring purchasers to pay actuarially sound rates immediately, as required by the 2012 law. The new rates are particularly high in older coastal communities in states like Florida, Massachusetts and New Jersey, and have put a damper on home sales as prospective buyers recoil at the higher, multifold premium increases.
Anger over the higher rates has fueled a bipartisan drive to delay or derail many of the 2012 changes and the political urgency to provide relief from higher premiums was especially felt in states like Louisiana, Florida, and New York.
"Affordable flood insurance is about more than just actuarial numbers on a page. It is about protecting our unique culture, a treasured way of life and preserving the historic coastal communities that are the engine's our of nation's economy," Landrieu said. Landrieu's potential GOP opponent in the fall, Rep. Bill Cassidy, was a key force with Grimm in advancing the bill through the House over objections from key lawmakers like Financial Services Committee Chairman Jeb Hensarling, R-Texas, whose turf got trampled in the process. Senate Banking Committee Chairman Tim Johnson, D-S.D., also opposed the bill after backing the earlier Senate measure.
The measure steamrolled to passage over objections from defenders of the 2012 reforms who said Thursday's bill continues to subsidize home ownership in risky, flood-prone areas and increases the odds of a future taxpayer bailout while requiring people in safer areas who carry flood insurance to subsidize below-market rates for homeowners benefiting from grandfathered rates.
"Reducing rates does not reduce risk. In fact, it reduces incentives to mitigate that risk. And they extended the subsidies on the backs of policyholders paying full freight," said Steve Ellis, vice president of Taxpayers for Common Sense, a Washington-based watchdog group. "While politically expedient today, this abdication of responsibility by Congress is going to come back and bite them and taxpayers when the next disaster strikes. Everyone knows this program is not fiscally sound or even viable in the near term."
The measure would also give relief to people who have bought homes after the changes were enacted in July 2012 and therefore face sharp, immediate jumps in their premiums; they would see those increases rolled back and receive rebates.
"We've solved a very short-term problem and made it a long-term problem," said Sen. Tom Coburn, R-Okla. "We didn't really do our work because we were in such a hurry to take the political pressure off of the increases in the flood insurance rates."
But people whose second home is in a flood zone and those whose properties have flooded repeatedly would continue to see their premiums go up by 25 percent a year until reaching a level consistent with their real risk of flooding.
Sen. Mike Lee, R-Utah, opposed the measure but allowed Thursday's speedy vote in exchange for separate passage of a measure to repeal premium rebates for people who have bought beach houses and other second homes since the 2012 law was passed. He said Cantor has promised it will be voted on by the GOP-controlled House.
The Senate took a different approach in January, passing a bill to delay the changes, which were aimed at putting the flood insurance program on sound financial footing. The flood program is presently $24 billion in the red, mostly because of huge losses from Hurricane Katrina and Superstorm Sandy.
The White House did not issue an official policy statement on the measure but said during debate on the Senate bill that it still supported a phased-in transition to risk-based flood insurance rates to help ensure that the federal flood insurance program has adequate resources to pay future claims. Democrats supporting the measure fumed and the White House didn't weigh in on the subsequent House measure.
Some environmental groups also opposed the bill, saying climate change has increased the risk of flooding in coastal areas, making it illogical to continue to rebuild in flood zones.
The measure also would give relief to people who have bought homes after the 2012 overhaul and therefore face sharp, immediate jumps in their premiums. Those homeowners would see rate increases capped at an average of 15 percent, with a maximum of 18 percent a year.
The Federal Emergency Management Agency would retain the ability to increase premiums each year, but the increases wouldn't be as steep as required under the 2012 law. A $25 surcharge on each of 5.6 million policyholders would offset the cost of continued subsidies for about 1.1 million homeowners. Owners of second homes would pay a $250 surcharge.
(Copyright 2014 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.)


text:
История Якутии — события на территории современной Якутии с момента начала расселения там людей и до сегодняшнего дня.

Доисторический и дороссийский периоды

Человек заселил территорию современной Якутии в глубокой древности. Археологами установлено, что древний человек заселил Якутию уже в раннем палеолите. Артефакты аллалаевской культуры среднего палеолита на северной стороне острова Столбовой имеют ориентировочный возраст 250—200 тыс. лет назад.

Фрагмент лопатки мамонта со стоянки Мунгхарыма I на реке Мунгхарыма (приток Вилюя) датируется радиоуглеродным методом (СОАН-6361) возрастом >41 310 лет назад. Кварцитовые изделия древнего и среднего палеолита и кремнёвые и халцедоновые изделия позднего палеолита со стоянки Мунгхарыма II могли быть смешаны за счёт склоновых процессов и переотложены в отложениях стоянки Мунгхарыма I. По материалам стоянки выделена среднепалеолитическая протодюктайская кызылсырская культура. К наиболее известным и хорошо изученным стоянкам в Якутии относится стоянка Диринг-Юрях дирингской культуры (находки якутских археологов под руководством Ю. А. Мочанова), которая расположена в среднем течении реки Лены.

К стоянкам раннего этапа верхнего палеолита относятся Эжанцы, Ихине 2, Усть-Миль и стоянка Бунге-Толля/1885 (45 тыс. л. н.) на Яне, местонахождение Буор-Хая/Орто-Стан (27 тыс. л. н.) на полуострове Буор-Хая. На стоянке Бунге-Толля/1885 (BT-1885, урочище Юнюген в долине Яны) обнаружили плечевую кость плейстоценового волка с прижизненным повреждением от какого-то орудия, возможно копья. Животное продолжало жить после травмы. Радиоуглеродный возраст находки — 47 тыс. лет назад.

Позднепалеолитическая Янская стоянка (Yana RHS) датируется возрастом ~31,6 тыс. л. н. Оба человеческих образца (Ancient North Siberians, ANS) с Янской стоянки принадлежат к Y-хромосомной гаплогруппе P1 (предковой для Y-хромосомных гаплогрупп Q и R) и митохондриальной гаплогруппе U2.

26 тыс. л. н. (финал каргинского межледниковья), когда Котельный был не островом, а соединялся с материком, люди разделали мамонта Павлова.

На озере Никита в 40 км от участка Урез-22 (MKR/U22, 71° 42' северной широты) в долине реки Максунуоха к югу от полуострова Широкостан на участке NKL (~13,800-13,600 лет назад) в ребре мамонта был обнаружен фрагмент каменного инструмента. Возраст Урез-22 можно оценить по временному интервалу от 14 900 до 13 900 лет назад.

Опорный разрез верхнеплейстоценовых отложений Колымской низменности Дуванный яр находится в крутой излучине на правом берегу Колымы в 35—43 км ниже устья Омолона. У образца Колыма-1 со стоянки Дуванный яр (9769 лет до настоящего времени) определена Y-хромосомная гаплогруппа Q1a1-F746>Q1a1b-YP1500/B143>Q-Y222276>Q-Y222276* и митохондриальная гаплогруппа G1b. Генофонд популяции палеосибирцев из Дуванного Яра складывался из двух компонентов — восточно-азиатского и древнего северосибирского (Ancient North Siberians, ANS), близкого к тому, который был обнаружен у представителей Янской стоянки и стоянки Мальта́ в Прибайкалье. Смешение восточно-азиатской и древней северосибирской (Ancient North Siberians, ANS) популяций привело к возникновению палеосибирской популяции и предков коренных американцев, впоследствии заселивших Америку.

10 000 лет назад пришла другая группа людей, вытеснив палеосибирскую. Эта группа намного больше похожа на сегодняшних китайцев, японцев и корейцев.

На острове Жохова люди жили 9 тыс. лет назад. У обитателей Жоховской стоянки определена митохондриальная гаплогруппа K и, предположительно, определены митохондриальные гаплогруппы W и V. Они разводили собак и привозили обсидиан с берегов озера Красное на Чукотке.

У ранненеолитического образца N2a (6845—6675 л. н.) из могильника Матта (Matta) на озере Матта на юго-западной окраине села Матта определили митохондриальную гаплогруппу F1d, у средненеолитического образца N5a из могильника Onnyos (6290—6186 л. н.) определили митохондриальную гаплогруппу D4b1c, у поздненеолитического образца N4a1 (4782—4424 л. н.) со стоянки Кёрдюген 2 определили Y-хромосомную гаплогруппу N* и митохондриальную гаплогруппу A12a, у образца N4b2 (4348—4091 л. н.) определили Y-хромосомную гаплогруппу N1a1a1a1a-L392/Z1979>Z1979* и митохондриальную гаплогруппу A12a2a2.

Ымыяхтахская культура эпохи неолита и бронзового века существовала в интервале 2200—1300 гг. до н. э.

Начиная с середины I тысячелетия н. э. на территории Якутии появились предки эвенов и эвенков. Продвижению древних тунгусоязычных охотников и оленеводов из Забайкалья и Приамурья на север дали толчок появившиеся в Восточной Сибири племена тюркоязычных скотоводов. К XIII веку тунгусские племена расселились на Средней Лене, Вилюе, Олёкме. Приход предков якутов в Ленский край заставил их отойти к западу и востоку от Лены. Часть тунгусских родов, оттесненная к Охотскому морю, расселилась по бассейну Колымы, Индигирки и Яны. Смешавшись с юкагирами и коряками, они образовали новый народ — эвенов.

У образца Yana_young с Янской стоянки (766 л. н., конец XII века) определили Y-хромосомную гаплогруппу N1a1a1a1a4a1-M1993>N-M1991* и митохондриальную гаплогруппу D4o2a

Крупнейший из коренных народов республики, давшей ей название — якуты — (самоназвание саха) занимают весьма своеобразное положение, которое характеризуется особенностями языка, традиционной культуры и физического облика. Большинство учёных полагает, что в XII—XIV веках якуты несколькими волнами мигрировали из области озера Байкал в бассейн Лены, Алдана и Вилюя, где они частично ассимилировали, а частично вытеснили эвенков (тунгусов) и юкагиров (одулов), живших здесь ранее.

Принято считать, что этнос приспосабливается к определённому ландшафту в период своего формирования. С этой точки зрения, якуты как народность сформировались в бассейне Средней Лены, то есть на территории Центральной Якутии. Здесь произошло окончательное формирование народа на основе смешения пришлых тюркоязычных племен с местными палеоазиатскими родами, а также с пришлыми монголоязычными хоринцами и тунгусами.

Распространение скотоводства внесло значительные изменения в хозяйственную жизнь региона. Предки якутов сумели сохранить и развить коневодческо-скотоводческую структуру своего хозяйства в экстремальных условиях Севера (якутская корова), привнесли в регион ремесленное производство (кузнечное, ювелирное, гончарное и др.), строительство жилищ постоянного типа. Археологические памятники кулун-атахской культуры XIV—XVI веков отражают материальный комплекс скотоводов. Уже к началу XVII века якутские роды жили в бассейнах Индигирки и Яны, продвинув в арктические районы Якутии культуру разведения крупного рогатого скота и табунного коневодства. Параллельно шло формирование особой этнографической группы северных якутов-оленеводов, перенявших от местных тунгусских и юкагирских племён способы ведения хозяйства в условиях тундры и лесотундры.

В якутских преданиях XVII век определяется эпохой Тыгына и тыгынидов. Академик А. П. Окладников называл его «Якутским царём». «Фигура Тыгына, — писал А. П. Окладников в первом томе "Истории Якутии", — мудрого старца, владыки и грозного воина, избранника самого Улуу Тойона, каким представляли его сородичи, уже при жизни сливалась на этом фоне с величественными образами эпических богатырей и божеств… Наконец и сама по себе гибель Тыгына связана была с крупнейшим историческим переломом в жизни якутов, появлением русских на севере и обрисована в величественных чертах эпической драмы».

Якутия в составе Российского государства

Первым из русских проникли в Якутию казаки Пянды в 1623, спустившись на стругах по реке Лене. В 1632 году другой казак Петр Бекетов основал на территории Якутии первый русский город Ленский острог, названный позже Якутском. Эта дата считается датой вхождения Якутии в состав Российского государства. В январе 1634 г. большой отряд (до 1000 якутов) осадил Якутский острог, где находилось около 200 казаков, но последние отбили атаки, и якуты вскоре прекратили нападения, но ещё долго сопротивлялись либо уходили в отдаленные районы. В 1638 году было образовано Якутское воеводство.

С деятельностью православного духовенства связаны развитие народного образования и просвещения народов Якутии, появление литературы на национальных языках, углубление процессов межэтнических взаимодействий.

Русские крестьяне-переселенцы положили начало северному земледелию, приобщая к ним северные народы.

В 1646 году якутские предводители обратились к воеводе В. Н. Пушкину с просьбой поехать в Москву к царю для улучшения жизни в их регионе. Но эта поездка состоялась только в декабре 1676 года, когда они наконец прибыли в Москву. Делегация состояла из трех князцов: Нохто Никина, Мазары Бозекова и Трека Осюркаева с их «кашеварами». Их торжественно принял царь Федор Алексеевич. Результаты были в общем положительные.

Отдельной страницей в истории Якутии является ссылка. Ссылка началась ещё с 1640-х годов. В начале большинство ссылалось «в пашню», «в службу», «в посад». Начиная с XIX века якутская ссылка по большей части становится политической. В ссылке в Якутии побывали декабристы, участники польского восстания 1863 года, народники, социал-демократы. Именами многих из них названы улицы городов и посёлков Якутии. В верховьях Вилюя отбывал ссылку Н. Г. Чернышевский; его именем ныне назван посёлок Мирнинского района.

31 октября 1857 года в состав Приморской области из Якутской области был передан Удский (Охотский) край.

В 1916 году Якутия избежала указа императора Николая II от 25 июня 1916 года о «реквизиции инородцев» в возрасте от 19 до 43 лет включительно, на тыловые работы в прифронтовых районах первой мировой войны. Во исполнение этого указа 30 июня 1916 года штаб Иркутской местной бригады потребовал от якутского уездного воинского начальника Н. Д. Попова набрать «бродячих
и всех обитающих в… Верхоянском, Среднеколымском, Вилюйском округах Якутской области» в возрасте от 19 лет до 31 года. Попов сообщил, что по приблизительному подсчету будет призвано около 20 тысяч инородцев. Их должны были отправить 19 партиями пароходами с 24 июля по 24 августа, но 17 июля 1916 года якутский губернатор де Витте сообщил, что «призыв инородцев Якутской области на работы в тылу действующей армии отменен». Отказ от мобилизации якутов произошел благодаря ходатайству Ленского золотопромышленного товарищества.

Якутия в годы Гражданской войны

После Октябрьской революции антибольшевитскими силами Якутии был создан Комитет защиты революции, выступавший в поддержку идеи созыва Учредительного собрания. 1 июля 1918 года красногвардейский отряд А. С. Рыдзинского занял Якутск. В городе был создан исполком Совета Рабочих депутатов, во главе с М. К. Аммосовым. Органы советской власти также были образованы городе Вилюйске, Нюрбинском, Сунтарском и других улусах. В ноябре 1918 советская власть была ликвидирована и Якутия попала под власть правительства Колчака. Известно о создании в 1918 году белыми властями Якутской местной команды (около 200 чел.), которая принимала ограниченное участие в боях против «красных» партизан в Якутии, но затем попала под влияние большевиков, в ночь с 14 на 15 декабря 1919 года подняла восстание в Якутске и установила в городе Советскую власть.

В результате разгрома белогвардейских войск в Сибири в конце 1919 — начале 1920 в Якутии была восстановлена советская власть.

В сентябре 1921 года в Якутии вспыхнуло антисоветское восстание. Восставшие обратились за помощью к русским эмигрантским кругам в Харбине, откуда им на помощь был послан крупный белогвардейский отряд. В марте 1922 в Чурапче было создано Временное якутское областное народное управление. Повстанцы осадили Якутск. 27 апреля 1922 года большевиками была создана Якутская Автономная Советская Социалистическая Республика (ЯАССР) в составе РСФСР. Летом повстанцы были выбиты из центральных улусов Якутии, однако продолжали сопротивление ещё около года и лишь в июне 1923 Якутия полностью перешла под контроль советской власти.

Якутская АССР

В 1924 году в Якутии началось восстание, которое было вызвано неоправданными действиями местных властей: закрытием портов для иностранной торговли, ограничением торговли, перебоями с завозом товаров с материка, конфискацией оленей у частных владельцев, изъятием обширных пастбищ под промышленные новостройки. В 1925 году восставшие заключили перемирие с советскими властями и сложили оружие.

Однако в 1927 году началось новое восстание под руководством якута, юриста Павла Ксенофонтова (выпускника юридического факультета Московского универистета, работника МинФина РСФСР). После его подавления в 1928 году были расстреляны 128 человек, 130 получили различные тюремные сроки, из них часть не была связана с восстанием. Среди репрессированных были видные представители интеллигенции, ничего не знавшие о восстании или даже осуждавшие его.

Советский период истории Якутии связан с широкомасштабным промышленным освоением её природных богатств, начало которому положила разработка в 1920-е годы золотоносных алданских месторождений. В 1930-е годы началась эксплуатация Северного морского пути, в устье реки Лена был построен морской порт Тикси; судоходные и воздушные трассы связали ранее труднодоступные районы республики. В 1950-е годы с открытием алмазоносных месторождений на западе республики была создана мощная алмазодобывающая промышленная инфраструктура.

Республика Саха (Якутия)

Новый этап в истории Якутии начался 27 сентября 1990 года, когда при активной поддержке населения республики была провозглашена Декларация о государственном суверенитете. В октябре 1991 года был учрежден пост президента республики. Первым президентом в декабре 1991 года стал якут Михаил Ефимович Николаев. Тогда же было изменено название республики на Республика Саха (Якутия).

С начала 1990-х годов в республике осуществлялся переход к рыночной экономике в русле общероссийских процессов. В результате разделения собственности, либерализации цен, приватизации предприятий произошли кардинальные изменения в социально-экономическом устройстве и производственных отношениях. В настоящее время развитие экономики республики все больше определяется разными формами собственности, воздействием рыночных регуляторов, внедрением рыночных механизмов.

В начале 1990-х годов были приняты нормативные акты, сыгравшие значительную роль в жизни республики:

 Постановление Совета министров РСФСР от 23 октября 1990 года № 603 «О расширении самостоятельности Якутской-Саха ССР в решении социально-экономических проблем в условиях перехода к рыночным отношениям»;
 Декларация о государственном суверенитете республики;
 Конституция Республики Саха (Якутия);
 Федеративный договор, экономическое соглашение и Соглашение по разграничению собственности;
 Указ Президента РСФСР № 277 от 11 декабря 1991 года «О полномочиях Якутской-Саха ССР в распоряжении природными ресурсами республики».

См. также
 История Сибири
 Сахаляр

Примечания

Ссылки

 «История Якутии» на Официальном web-сервере органов государственной власти Республики Саха (Якутия)


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Photo by Gerry Neely
Tak It Hame
Planning and consultation responses
Respecting Scotland's Mountains
Our conservation strategy
Windfarms statement
Hilltracks statement
Hydropower statement
Hill tracks
Mountain wind farm research
Mountain memorials
Protecting Scotland's mountains
Scotland's mountains are globally renowned for their natural beauty, shaped by geology and nature. We campaign to protect them, not just for hill walkers, mountaineers and climbers, but for all those who cherish them.
Working with partners, we takes a leading role in challenging insensitive developments in Scottish mountain landscapes – developments that can damage their unique character, their natural habitats, the views to and from them, and our freedom to enjoy them. Experience shows that often one development leads to another, resulting in progressively more damage. We want to halt this trend.
In 2015 we launched our vision for the future of Scottish mountains, a vision focused on safeguarding natural beauty, capitalising on leisure and tourism for sustainable economic development, and community-led small-scale developments in renewable energy.
> Read our vision: Respecting Scotland's Mountains
We want the Scottish Government to enshrine protection for 'wild land' (as defined by their own conservation body, Scottish Natural Heritage) in planning legislation, to ensure speculative industrial-scale wind farm applications in mountain areas are stopped for good.
> Find out more about our landscape campaigning
Our responses to planning proposals
We respond to planning authorities on specific development proposals which we feel damage Scotland's wild land. We have strict criteria for whether we will object to a planning application. Though we respond to very few, the ones we do oppose could have a huge impact on some iconic mountain landscapes.
> Take a look at our planning responses
Campaign priorities
Hill roads
Poorly constructed hill roads and 'tracks' are a growing concern in Scotland's mountains. Many of these are bulldozed through some of our most valued wild land with little regard for build-quality, appearance or effective drainage, causing scars that can be seen for miles around.
> Find out more and how you can help
Mountain wind farms
We have published a report into the impact mountain wind farms have on the behaviour of mountaineers and hill walkers as part of the evidence base we use when opposing the small number of wind farm planning applications that we believe would cause irreparable damage to Scottish mountain landscapes if allowed to go ahead.
> Check out our research
We believe mountains should be protected from man-made intrusion and remain in as wild and natural a state as possible. For this reason, while we sympathise with the grief and loss that bereaved friends and relatives feel, we believe that permanent memorial artefacts should not be a feature of the mountain landscape.
> Our views and alternative options
National parks position statement
Mountaineering Scotland supports the introduction of new national parks where precedence is given to the Sandford Principle in which the conservation and enhancement of the natural and cultural heritage of an area is given precedence over economic and social development
We thank the BMC for their financial contribution to our landscape work.


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The Nestene Consciousness
Nestene Consiousnous
The Nestene Consciousness, also known as the Nestenes and as the Mother Consciousness to its Autons, (Brave New Town) was a disembodied life force, a hive mind existing as energy that could possess plastic.
The Nestene Consciousness was a being of pure psychic energy, although it generally maintained a physical, somewhat cephalopod-like appearance. (Doctor Who and the Terror of the Autons, Synthespians™) The Nestenes' molecular structure had a close affinity to plastic, polymers of high molecular weight, some partially amorphous. (Autonomy) The polymer chains of both thermosetting and thermoplastic resonated along their psychic wavelength. (Synthespians™)
Because of its nature, "humanly speaking" a Nestene manifestation served as the brain or nervous system, (Spearhead From Space) although it could also be defined as the heart, or appendix, or duodenum, or other terms. (Autonomy) This body would be made out of plastic, allowing it to manipulate its surroundings, (Spearhead From Space) though a Nestene's original body was large enough to dwarf a radio telescope. (Doctor Who and the Terror of the Autons) After the stresses of the Last Great Time War, the Consciousness manifested as a more liquid form with a humanoid face. (Rose)
Physical manifestation for the Consciousness was conceptual, being more than just a brain in flesh; it could exist in various forms, various places, and even various times. (Autonomy) Because of their molecular affinity, the Nestenes had the ability to possess plastic, animating it and making it quasi-organic. It existed within everything it controlled, with the more significant of these manifestations including the energy units which transported the Consciousness, the Auton robots used as soliders, and a Nestene creature that served as the central brain/nervous system of the Autons, created from the collected intelligence of the energy units.
While they were fragments of the Consciousness, Autons had degrees of individuality depending on the portions of intelligence given to them. Channing, a Swarm Leader, could directly control another Auton and was unaware of another's destruction. (Spearhead From Space) The Nestenes could speak through their Autons (Terror of the Autons) which they directed via the astral plane, (Synthespians™) although to control larger numbers the Nestenes required a relay station or transmitter. (Rose) Over the course of its invasions it possessed shop-window dummies, the more advanced Autons (Spearhead From Space), inflatable chairs, dolls, telephone cords, plastic daffodils designed by the Master, (Terror of the Autons) and rubbish bins (Rose).
While the Consciousness was a group mind, it could transport sections of its mind through radio waves, (Doctor Who and the Terror of the Autons) separating parts of the Nestene Mind for invasions so as not to risk the entire Consciousness. With this, if an invasion failed the rest of the Consciousness would know about it. (Doctor Who and the Terror of the Autons)
Nestene matter may have been capable of performing Block Transfer Computations, as they required morphologically unstable living organic matter to do so. (Cat's Cradle: Witch Mark)
at the beginning of the universe, Shub-Niggurath came into N-Space from the previous universe, pregnant. She was able to deliver her child on Polymos, who infused into the entire planet and became the Nestene Consciousness. (Synthespians™) The Nestenes' original appearance was that of a cephalapod, before transcending their bodies to become pure energy. (Doctor Who and the Auton Invasion) The Nestenes proceeded to send out swarms to other planets for colonisation.
The Consciousness, lead by the Nestene High Command, (Doctor Who and the Terror of the Autons) had been invading planets for billions of years. They used these planets for sources of plastic and proteinat first they sent several plastic meteors to a planet. These asteroids could telepathically control specially created plastics, but were more effective when combined. The Nestene Consciousness would next control people to create Auton forces to help with further waves of invasions. Some of the planets which the Nestene Consciousness took over includes Cramodar and Plovak 6. (Synthespians™)
On some worlds, its takeover strategy was as follows: portions of the Consciousness would travel through space in hollow spheres known as energy units, making planetfall in formation but appearing like normal meteorites to indigenous lifeforms. When they became active, the spheres would pulse with light and ring with electronic beeps. The first spheres could place a local being under mental control. The Swarm Leader unit either created a facsimile of a native or an original native-form from scratch, (Synthespians™) and would proceed to construct other Autons and collect the other spheres. The Autons could replace key authority figures. Once the Consciousness within them recombined in an environment tank, the Consciousness would take on a form that it believed perfectly adapted for conquest of the planet on which it had landed. (Spearhead From Space)
In 102, the Nestene were among the factions who joined the Alliance. they came to Stonehenge and helped imprison the Eleventh Doctor in the Pandorica in order to save the universe. As well as the obvious spaceships, they also created a legion of Autons disguised as Roman soldiers. These Autons, believing themselves to be genuine Roman soldiers, tricked The Doctor, who realised too late what they were, and imprisoned him in the Pandorica. (The Pandorica Opens) When the plan failed, the Autons and the Nestene were turned to dust and removed from the universe along with the rest of creation, except for an Auton copy of Rory Williams, who rebelled against his nature and tried to help The Doctor. they were returned when The Doctor managed to "reset" the universe. (The Big Bang)
The Nestene's first attempt to invade the Earth in the 20th century, using human-looking Autons to replace key government figures, was thwarted by UNIT with the help of the Third Doctor. The remains from this attempt were stored by UNIT. (Spearhead From Space)
In their next attempt, the Master obtained one of the Nestene meteors and took over a plastics company. Using the company to create fake daffodils, disguised Autons would hand out these flowers for free. Eventually, once millions had been distributed, the Nestene would activate them by remote radio signals, causing them to suffocate and kill millions of people. The Master also created dolls, inflatable chairs and telephone cords to kill difficult individuals. the Third Doctor was able to figure out this plan. He convinced the Master that the Nestenes were too dangerous to be reliable allies, and They reversed the radio beam the invasion force was coming in on, sending it back into space. (Terror of the Autons)
the Sixth Doctor helped UNIT to defeat another Nestene invasion. This did not appear to tax him overly, as he found time to take dance lessons during a lull in the crisis. (Teach Yourself Ballroom Dancing)
In 1989, the Nestene Consciousness tried to take over Earth using the computer company SenéNet. the Sixth Doctor, Mel and the Brigadier prevented the Consciousness from taking over the Earth. (Business Unusual) at the same time, it established a front operation called Galaxy Plastics Inc, and was still operational and undiscovered in August 1990. (Business as Usual)
Circa the 1990s an Auton replica known as Alisha Hammerson set up the Hammerson Plastic PLC company. On New Year's Eve, 1999, she planned to replace several key figures in the plastic industry with Auton replicas, in order that the Earth could be dominated and absorbed. the Seventh Doctor and Mel, gate-crashing the party, broke the Nestene link which de-activated her. (Plastic Millennium)
In the late 1990s, Doctor Sally Arnold whilst attempting to discover an energy unit's function reactivated it using UNIT's deep space satellite, it in turn communicated with the Nestene Consciousness and provided it power to activate several Autons from the Nestene's first invasion attempt. The energy unit was forced to adapt to the new environment allowing the energy unit to move around the facility and locate the remnant of the Nestene body from its first Earth invasion attempt. It eventually escaped the UNIT warehouse and attempted to invade Earth again, this was thwarted by UNIT's cybernetically enhanced agent Lockwood and his team. (Auton, Auton 2: Sentinel, Auton 3: Awakening)
In 2005, the Nestenes infiltrated Earth once more, using warp shunt technology. The Nestenes had lost their food planets in a war, so their intention was to overthrow and destroy the human race, as Earth was ideal for their consumption needs, being filled with smoke, oil and various pollutants. Rose Tyler eventually defeated them by knocking a vial of anti-plastic solution that one of the Autons had confiscated from the Ninth Doctor into the vat of molten plastic which housed the main bulk of the Consciousness, causing it to explode. (Rose)
Part of the Nestene Consciousness from the 1980s resurfaced barely eight years later, taking over through Hyperville, but was once again foiled by the Tenth Doctor (Autonomy)
In the 101st century, the Nestene Consciousness was attempting to completely leave Polymos after an invasion by the Time Lords. It chose the New Earth Republic and attempted to transport its entire body to Reef Station One with the help of Walter J. Matheson. It was stopped by the Sixth Doctor, who trapped it in the body of an advanced Auton. (Synthespians™)
When they were first introduced, they were referred to only as "the Nestenes" in Spearhead From Space. The actual name "Nestene Consciousness" seems to originate from the novelisation Doctor Who and the Auton Invasion.
The Nestene Consciousness are one of the only two villains, along with the Master, that appeared in two post-regeneration stories.
According to non-narrative material in The Doctor Who Annual 2006 which this does not consider a valid source, the Time War caused the Nestene Consciousness to lose all of its protein planets, and it found itself mutating under temporal stress. This verifies a comment the Ninth Doctor makes in Rose, in which he states the Nestene lost all its protein planets during a "war".


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House Considers Foreign Relations Bill
House lawmakers are considering a 22-billion dollar spending bill to fund the State Department, and other foreign relations spending needs over the next two years. Key portions aim to help protect U.S. diplomatic interests abroad, strengthen efforts to prevent nuclear and missile technology from falling into the hands of terrorists, and promote democracy around the world.
The Foreign Relations Authorization Act provides money for the State Department, U.S. international broadcasting, refugee assistance, international organizations, and aid to key nations over the next two years.
In line with a Bush administration request, it includes funds to upgrade and protect U.S. embassies and facilities overseas, and help with strengthening U.S. diplomatic missions in Iraq and Afghanistan.
The legislation aims to strengthen U.S. public diplomacy efforts, including U.S. international broadcasting, which includes the Voice of America and other government-funded radio and television stations.
Lawmakers gave particular attention to helping U.S. government efforts against terrorism.
Democrat Tom Lantos describes on of the key provisions which aims to disrupt or eliminate black markets for transfer of nuclear materials: "The Nuclear Black Market Elimination Act, which is included in our bill, will authorize sanctions against individuals and companies that provide nuclear enrichment technology to countries which do not have it, or have not signed the additional IAEA protocols relating to verification. Our initiative will help prevent nuclear weapons technology from getting into the hands of terrorists and rogue states."
On bilateral aid issues, the bill would begin a process of making non-military assistance to Egypt subject to benchmarks for democratic and human rights reforms beginning in 2007.
Congressman Earl Blumenauer is an Oregon Democrat and a key House critic of U.S. aid policy toward Egypt: "This bill starts to move us away from the preponderance of military aid and move that same dollar amount but to humanitarian assistance."
Minority Democrats complained bitterly that majority Republicans blocked amendments that would have made wider debate possible on issues ranging from the war in Iraq, to Sudan and U.S. counter-narcotics assistance to Colombia.
Florida Democrat Alcee Hastings was infuriated by a Republican-backed amendment declaring opposition to any premature withdrawal of U.S. forces from Iraq until it is clear that doing so would not threaten stability: "Republicans are pulling out every political maneuver they can to regain the support of the American people for a war that has not uncovered any weapons of mass destruction in Iraq and certainly not made us any safe today than we were three years ago."
Republicans countered by saying all of these issues had been thoroughly debated at the committee level.
Consideration of the bill revisited two issues the House had already previously voted on.
Trying to ensure eventual approval by the whole Congress of a U.N. Reform bill the House passed in June, Republican Henry Hyde included it in the foreign relations bill to improve chances it will become law. He also re-introduced and the House approved by voice vote, an amendment rejected by the House last week, aimed at deterring European Union arms sales to China.
The Foreign Relations Act includes sense of Congress declaration supporting what it calls transparent full democracy in Iran. And it requests reports to Congress on U.S. counter-terrorism aid to African nations, U.S. efforts to support stability in Haiti, and steps to counter internet jamming by repressive foreign governments.
Discussion in Burmese (Real Player) Download - 1.00MB.


A short film, written and directed by Upperchurch actress and playwright Aine Ryan, is part of a cohort of films set for a special screening at the SGC Cinema, Dungarvan on April 19.
Áine, whose family run the 'Jim O' the Mill' pub in Upperchurch has written a beautiful children's short drama 'Feadóg' as part of Waterford Institute of Technology's Higher Diploma in Television and Digital Media Production.
The film tells the story tells of a magic tin whistle and a young boy called Eoin who is obsessed with playing it. On his way home from school one afternoon, Eoin is devastated to find out that his tin whistle has gone missing and the audience is immediately taken on a journey to find it. With his mother, he searches high and low for his beloved whistle. Eoin is played by Áine's young cousin, Canice Ó Riain aged 12, and the his mother is played by Joanne Woodlock, who is very well known in the local drama circles.
This short film is another string in Áine's bow as her quirky dramas have already gained her great critical acclaim, including five star reviews at the Edinburgh Fringe Festival.
Short films and documentaries from students of the Television and Digital Media Production course will be screened at the SGC Cinema, Dungarvan on April 19th at 1.30pm. This event is open to family, friends and the public.