literature.tex

\chapter{A Brief and Opinionated Literature Survey}%
\label{chap:litsurvey}

Teleoperation systems are structurally simple, two connected robotic manipulators, but equally challenging 
systems. This is especially true from a system theoretical point of view. As an example, 
if we just focus on the local and the remote devices that would be used for manipulation, 
we see that they are, whether linear or nonlinear, motion-control systems with well-studied 
properties. Hence, one can view the open-loop teleoperation, i.e., standalone devices without any communication
in between, as a system with a block diagonal structure in which each input to this block effects only one 
of the devices. However, unlike the typical motion-control systems, these two disjoint 
systems must be stabilized simultaneously by the same controller\footnote{We need to emphasize 
that the delayed/undelayed local control loops can be seen as the entries of a structured 
central controller. Thus, there is no reason to distinguish local control methods at this 
point.}, that is performing sufficiently well in order to \enquote{fool} the user such that the 
user feels a force feedback as if s/he is actually operating at the remote medium. With this 
structure, the outputs of either of these subsystems become exogenous inputs of the other 
and these are regulated by the to-be-designed controller. Therefore, it is this controller 
that makes a teleoperation system perform adequately or, as in many cases, drive to instability.


For example, in the case of the so-called \emph{free-air motion}, i.e., the remote device is free 
to roam in the remote site, the human force input to the local device and/or the position of the 
local device should\footnote{Or better, that's what we believe today.} 
be tracked by the remote device. In the case of a hard-contact of the remote device with the environment, however, 
these inputs should be counteracted if the force vector points into the obstacle. Hence, the force signal is 
simultaneously tracked for mimicking the user motion and is defied in case of a resisting force at the remote 
site. As if this is not challenging enough, when the user suddenly decides to release the local device, this 
resistance should die out as soon as possible, preventing a kickback. To sample the artificial nature of such 
a behavior, consider a user who leans to a wall located at position $x_0$ and beyond, applying a horizontal 
force and then retreating after some time. It is not expected that the wall continues to push the user even 
after the user has the position $x<x_0$. Such behavior would not only be unrealistic but also misleading as 
it can be confused with a sticky surface as far as the immersion is concerned. There are many other scenarios 
that would further complicate the requirements but, in short, the user and the environment properties are 
time-varying and make it difficult to design a straightforward control law such that these and many other details 
are handled properly, and more importantly, simultaneously. 


With this short motivation, we can safely claim that looking at the overall system as a typical motion control 
system is not sufficient in terms of complexity (though necessary). In general, motion tracking specifications 
constitute a subset of the general performance requirements of bilateral teleoperation systems. 

%\begin{rem}
%A similar line of reasoning has been given in \cite{buergerhogan1}. But for reasons that are 
%not known to us,  the authors chose to see the classical loop shaping as the sole servo control method. Moreover, modern 
%control theory offers a few options to avoid the obstacles that are given as impossibilities in their paper. Therefore, 
%we emphasize that the difference here is merely about the problem formulation and is not about the 
%limits of performance\footnote{We refer to the elegant book \cite{boydbarratt} for a detailed investigation of the 
%limits of performance.}. 
%\end{rem}


The inception of the bilateral teleoperation technology is often attributed to the work 
of Raymond Goertz in Argonne National Laboratories, \cite{goertz} (in \cite{basanezsuarez}, 
it is traced back to Nikola Tesla and, in \cite{sheridan89}, even some 16th century tools are accepted as precursors 
of the contemporary  teleoperation). The main motivation of Goertz' work (similarly later in Europe 
by Vertut \cite{vertutcoiffet}) was handling and manipulating nuclear material. Thus the very 
first teleoperators were purely mechanical to cope with hostile environment conditions. Though not much 
happened in terms of commercial product realizations, the concept of telemanipulation kept its appeal 
and a large body of research was reported until the 1980s. In that decade, with the help of the 
ever-increasing computational power and the popularity of Virtual Reality (VR), teleoperation 
technology received more attention for a possible use in the space-, underwater-, and medical-related 
tasks. Together with the advances in control theory and network theory (e.g. \cite{miyazaki,furuta}), 
a more systematic control methodology has been adopted. Especially, stability analysis results that can be 
related to design guidelines (physical parameter bounds, bandwidth limitations etc.) were utilized
and limits of performance were explored. A particular phenomenon, namely the destabilizing effect 
of the delays in the teleoperation, lead the experts of the field to delve more into the systematic 
analysis tools and qualitative aspects of teleoperation. Especially, the use of the concepts such as, 
\enquote{passivity}, \enquote{scattering transformations}, and \enquote{wave variables} has become the 
standard methods of analysis and synthesis (see, e.g., \cite{hannaford89,andersonspong,nieslotine}). 
Arguably, this point is where bilateral teleoperation branched off from the general control theory
and became a specialized area of research, specifically dealing with a particular problem that is still
a matter of unresolved debate, as we touch upon later in this chapter.

We start to summarize the advances from this point as this thesis is precisely built on top these systematic 
analysis and synthesis results gathered in the last two decades. However, the reader is referred to 
\cite{hokayemspong,burdea}, and \cite{sheridan89} for a more detailed overview including other practical 
aspects of teleoperation analysis and the hardware developments with a more historical perspective which 
will be omitted here. 


As we keep on narrowing down our focus to the control theoretical parts of this challenging problem, we have to 
note that many parts of the bilateral teleoperation problem can be scrutinized under different frameworks. 
Hence, there is no shortage of techniques for which the bilateral teleoperation problem is an ideal 
test case. In this plethora of methods, for example, the variation of human and environment properties give 
naturally rise to a robust or an adaptive control approach, the hard-contact problem can be analyzed by viewing it
as switched control systems, jump control systems or constrained linear systems etc. Thanks to these advances, 
as we elaborate, the main unsolved problem is not a methodological one but a motivational one. In
other words we are lacking not the solution methods but rather a fundamental understanding of the problem in terms of 
what the requirements are and what a good device is if compared to another. Let us sample a few important and 
successful approaches reported so far together with their shortcomings if any. 

We emphasize that the literature 
covered here is far from comprehensive and deliberately shaped with pragmatic intentions. Hence a large body of 
research is left out. This is certainly not due to their lack of thoroughness or else, but simply due to the irrelevance 
for the purpose of this chapter. In general, the methods that are left out either don't define a performance
objective or only focus on the particular detail about bilateral teleoperation instead of the human perception. 
The reasoning behind this choice should be more apparent after \Cref{chap:perf}.


\begin{figure}%
%http://chat.stackexchange.com/transcript/message/4857161#4857161
\centering
\begin{tikzpicture}[scale=1,manstyle/.style={line width=4pt,line cap=round,line join=round}]
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\begin{scope}[xshift=4.7cm,-stealth,black!20]
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\end{scope}
\begin{scope}[shift={(5.4cm,-0.2cm)},cm={-1,0,0,1,(0,0)},transform shape]
\draw (0,0) -- (0.9cm,0) (0.45cm,0) circle (1.5mm);
\path [fill=white,postaction={pattern=north west lines}] (0cm,0cm) rectangle (0.9cm,-0.16cm);%postaction={pattern=north west lines},
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\node[circle,draw,inner sep=2pt,fill] (merkez) at (0,0) {};
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\draw[thick]
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%0.4cm) --cycle;
%\end{scope}
\end{tikzpicture}
\caption{General Teleoperation System}%
\label{fig:teleop}%
\end{figure}

\section{Modeling of Bilateral Teleoperation Systems}\label{sec:litchapmodeling}

The dominating modeling paradigm of bilateral teleoperation systems is the two-port network approach. Consider the following quote from 
1989:
\begin{displayquote}[{\cite{hannaford89}}][.]
%In this paper, development of the analytical framework is complemented by modeling of an actual teleoperator system.
The modeling approach is to transform the teleoperation system model into an electrical circuit and simulate it using
SPICE, the electronic circuit simulation program developed at UC Berkeley
\end{displayquote}
%\end{mdframed}


As seen from Hannaford's motivation, the computer-based simulation tools are used extensively since then. Arguably,
this is one of the main reasons why network and electrical circuit based modeling dominated the teleoperation literature. 
Reinforced with the circuit simulation tools, experts of the field started to construct analogies that go beyond
a mere mechanical-electrical system analogy. Consequently, the most prominent concept borrowed from these analogies is the two-port 
network view of bilateral teleoperation systems. The reader is referred to \Cref{chap:apdxnetwork} for a short recap of network theory.
Today, the quoted convenience also applies to almost all physical systems, i.e., one can simulate arbitrary models via many 
computational packages. Yet, it's a de facto standard to use the circuit modeling while the teleoperation devices are mostly 
mechanical. Hence, it's not clear whether the benefit of such an artificial step still exists. Once the system is represented 
by a mathematical model, as it is demonstrated in the later sections, the mechanical/electrical analogy is, roughly, an equivalence
based on the resulting model and works in the \enquote{from electrical to mechanical} direction too. Therefore, the 
circuit based modeling approach is merely a convention rather than a requirement.

\subsection{Two-port Modeling of Teleoperation Systems}

In the teleoperation context, if one uses the \enquote{load-source} analogy for the manipulated environment 
and the human, then the system models all the bilateral interaction between the load and the source 
ports (as in \Cref{fig:portrep}). This modeling view is quite powerful since the components are described via 
their input/output (or external) properties, i.e., across variable/through variable relations (e.g. force/velocity, 
voltage/current etc.). Also, the non/linearity properties of the components are not relevant at the outset if we 
are only interested in energy exchange, which is the basis of the so-called Time-Domain Passivity Methods 
\cite{hannafordryu} which we also mention later in this chapter. Thus, the user, the control system, the 
environment, the remote and local devices and the communication delays are seen as $1$- and $2$-ports exchanging 
energy in time. Since the external behavior of the ports can be characterized completely by the power variables 
associated with the terminals, e.g., voltage drop across the terminals and current flowing through them, it is 
indeed very convenient to model these components with electrical ports as interacting \enquote{black boxes} 
(See \Cref{fig:portrep}).

\begin{rem} We should emphasize here that, in this context, the energy exchange is used as a gauge of a 
potentially unstable behavior. The motivation relies on the fact that in order to classify a system as an 
unstable one, the system should exhibit unstable behavior at its port(s) and to exhibit such behavior 
additional energy is needed. Hence if all the components are incapable of contributing energy in the loop, 
finite energy excitations will eventually decay and the system would reach its steady state after a transient 
response\footnote{For the sake of brevity, marginal stability or limit cycles also require extra energy for 
the sustain.}. As one can directly identify, this is simply the rough sketch of the celebrated passivity theorem. 
Based on this argument, there is a recurring theme in the literature that a teleoperation system should be 
passive in order to have a stable interconnection. This is due to the hypothesis that end terminations are 
also passive. However, in a few studies, this is mistakenly taken as a sufficient and necessary condition 
for stability and hence creates quite some confusion for the non-experts of the field. Passivity is not an 
essential feature of the teleoperation systems but only a convenient shortcut for deriving interconnection 
stability conditions. We should iterate that stability is the top-priority objective and does not require 
passivity by any means even if we do guarantee stability by rendering the sub-components passive. 
\end{rem}

\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\begin{tikzpicture}[>=stealth,baseline=15mm,
every node/.style={draw,minimum size=1cm}]
\node (deltas) at (0,0) {$\Delta_l$};
\node[left=of deltas] (g) {$G$};
\node[left=of g] (deltal) {$\Delta_s$};
\tikzset{every node/.style={draw,circle,inner sep=1pt,fill=white}}
\foreach \y in {s,l}{
    \node at ({$(g)!0.5!(delta\y)$} |- {g.40}) (circ1\y){};
    \node at ({$(g)!0.5!(delta\y)$} |- {g.-40})(circ2\y){};
};
\draw (circ1l) -- (deltal.40) (circ2l) -- (deltal.-40)
(circ1s) -- (deltas.140) (circ2s) -- (deltas.-140);
\draw[->] (circ2s) -- (g.-40) (circ1s) -- (g.40);
\draw[->] (circ2l) -- (g.-140) (circ1l) -- (g.140);
\end{tikzpicture}
\caption{}
\label{fig:portrep}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\begin{tikzpicture}[>=stealth,scale=0.5, transform shape]
\matrix (G) [draw,matrix of math nodes,inner sep=1mm,row sep=1mm,ampersand replacement=\&]{ G_{11} \& G_{12}\\ G_{21} \& G_{22}\\};
\matrix (delta) at (0,3) [outer sep=0,draw,matrix of math nodes,inner sep=0.5mm,ampersand replacement=\&]{\Delta_s \& \\ \& \Delta_l\\};
\draw[->] (G.160) -| ++ (-0.4,0.8) node[draw,circle,fill=white,inner sep=2,label={[inner sep=0]0:$\scriptstyle -$}] {} |- (delta.160);
\draw[->] (G.200) -| ++ (-0.7,0.6) node[draw,circle,fill=white,inner sep=2,label={[inner sep=0]0:$\scriptstyle -$}] {} |- (delta.200);
\draw[<-] (G.20) -| ++ (0.4,0.5) |- (delta.20);
\draw[<-] (G.-20) -| ++ (0.7,0.5) |- (delta.-20);
\end{tikzpicture}
\caption{}
\label{fig:nom_net}
\end{subfigure}
\caption[Two representations of a 2-port network.]{Two representations of a 2-port network. Here $\Delta_s$ and $\Delta_l$ represents
the source and the load immitances respectively. In the teleoperation context, they are the human operator and the explored environment.}
\end{figure}

Clearly, thanks to this modeling method, we don't even need to know exactly what $\Delta_l,\Delta_s$ blocks are, except 
their class (e.g. linear/nonlinear, time invariant/time varying etc.) to analyze the interconnection via $G$ and its port 
behavior.  Thus, the problem of modeling of the human arm or of the uncertain environment is circumvented. However, with 
the same reasoning, the passivity property does not distinguish particular systems as long as they are passive. That is to 
say, some of the crucial information is lost about these specific ports; we discard any impedance or admittance relations 
shared by the port variables.

Energy based modeling is also the natural basis of bond-graphs. Bond-graphs, much like port representations, are graphical tools 
to model the dynamical systems via energy balancing between subcomponents (See \cite{gawthrop} for an introduction). In other words, 
the bond-graphs are built on top of the notion of bonds representing the instant energy or power exchange between nodes via edges 
drawn between them. Therefore, bond-graphs already present a powerful framework for the abstraction of the bilateral interaction between 
the local and the remote site. For a classical use of bond-graphs in impedance control, the reader is referred to Hogan's trilogy 
(\cite{hogan:1,hogan:2,hogan:3}). There are also many studies with application focus, e.g., \cite{krishnaswamy} using hydraulic 
systems for bilateral teleoperation, among many others. 



\subsection{Assumptions on the Local and Remote \enquote{Ports}}

As mentioned above, network theory offers a great opportunity for modeling teleoperation systems, or better, avoiding 
a refined modeling. Still, to invoke the stability analysis and synthesis results of network theory, there is a need 
to distinguish $\Delta_s,\Delta_l$ further in the universum of $1$-ports. Otherwise, with no additional assumptions
there is not much we can conclude from such an interconnection since they can be any arbitrary model with arbitrary 
behavior set as long as they respect the port condition. This is obviously a crude approximation of the real physical 
interaction that teleoperation systems exhibit. 

In the teleoperation and haptics literature, it is customary to assume the load and the source terminations as \enquote{passive}
mathematical operators (see \Cref{chap:apdxnetwork}). Starting with this hypothesis, the stability problem can be converted to 
a typical energy dissipation problem. Hence the view of the designer is tuned to look out for the energy sources and interaction 
between two distant media. This approach treats the human and the environment as passive $1$-port circuit elements together with 
additional voltage and/or current sources modeling the intentional force input to the system. The controller(s) act as the energy regulator 
preventing excess energy generation to avoid a possible instability even in the cases where extra energy does not endanger 
stability or in fact needed by the user to accomplish certain task or stabilize the system. 


Additionally, as summarized in \Cref{chap:apdxnetwork} and in \Cref{chap:analysis}, one can use the 
network theory based conditions to assess the stability and performance conditions thanks to this hypothesis.

This brings us to the discussion of the justification of the assumption as it is generally not given in full generality in the
literature. Is it indeed valid to assume that the human can be modeled as a passive system? If one scans through the literature 
about the passivity of human operators, it is the Hogan's paper \cite{hogan89}
that is almost universally cited. The striking detail is, however, that Hogan never claims that the human hand/arm is a passive
system. Instead he clearly shows that under very specific conditions, human behavior is indistinguishable from that of a 
passive system: 
\begin{displayquote}[{\cite{hogan89}}][.]
Thus, despite the fact that the
limb is actively controlled by neuro-muscular feedback, its apparent stiffness is equivalent to that of a completely passive
system. In the light of Colgate's recent proof [3]\footnote{Reference \cite{colgatehogan88} of this thesis. However exactness 
of stability characterization for two LTI passive complex uncertainty blocks was already well-known in SSV theory (e.g., \cite{packdoyle}) 
and also in the classical network theory works at the time of writing. Therefore it is a misattribution.} that an apparently 
passive impedance is the necessary and sufficient condition for a stable actively-controlled system to remain stable on contact 
with an arbitrary passive environment, this experimental result strongly suggests that neural feedback in the human arm is carefully 
tuned to preserve stability under the widest possible set of conditions
\end{displayquote}
Moreover, the task reported in the paper that is given to human operators and analyzed afterwards, can be considered as a biased one because the success 
of the test is related to the passive behavior of the human. The task is, roughly speaking, holding a handle which is perturbed by 
random disturbances and trying to keep the handle still at a predefined position on the 2D plane. Hence, the task is simply to mimic 
a passive system. Had it been the case that measurements on the human arm would exhibit a non-symmetric stiffness matrix in the arm 
model, it would simply be a failure of the test subject (regardless of the physical limitations of the human arm in general). Note 
that this is a plausible situation for rehabilitation tasks. The other possibility would then be that the test subject was unable 
to keep up with the changes, or using the control theory jargon, the bandwidth of the subject was lower than the required agility to 
perform the test adequately. The well-known phenomenon due to such human input is the \enquote{pilot induced oscillations} in which the 
pilot of an aircraft, while trying to stabilize the aircraft, via overcorrecting inputs, destabilizes the system due to many distinct 
reasons (the phase lag of the pilot, response time of the aircraft etc.). We refer to the interesting report \cite{mcruer} for a more 
detailed exposition. Also, if for some reason, the task at hand is to prevent the system to reach a steady state at a certain position 
and the perturbations are applied accordingly i.e., to create a virtual negative potential, the results obtained from the experiments 
would most probably differ from that of \cite{mussa85}. Thus, it's emphasized here that the passivity of the human is closely linked 
to the requirements of the tasks. 


\begin{rem}\label{rem:filament}
A particular detail should be clarified about the measurements taken in \cite{hogan89}. It is stated that: 
\begin{displayquote}[{\cite{hogan89}}][.]
While normal human subjects held the handle of the manipulandum at a stable position
in the workspace, small perturbations were applied. Measurements
of the human's restoring force were made after the system
had returned to steady state following the perturbation but before
the onset of voluntary intervention by the subjects.
\end{displayquote}
Therefore, it is emphasized that only the involuntary response is taken into account during the measurements in order to
capture the natural properties of the human arm before the human correction intervenes. In fact, due to this crucial distinction
the results such as \cite{dyck2013} do not disprove Hogan's results since the voluntary input is included in the model hence the passive 
dynamics, if any, is mixed with the independent force input. It is the voluntary input of the human that should not be included in the 
analysis, since the human necessarily puts in energy, at the very least, to move the local device violating passivity condition trivially.
\end{rem}

We believe that it is unavoidable to introduce some concepts from the muscle physiology in order to put Hogan's 
argument into some mechanical engineering perspective. It is simply impossible for us to give a detailed analysis, 
however, mentioning the involved process via some mechanical analogies in order to relate the results of 
Hogan and Mussa-Ivaldi (\cite{mussa85}) seems feasible. This would emphasize the reason why we think that the common inference 
from their experiments in the literature is not inline with the conclusions of these studies. We refer the reader 
to the physiology literature e.g., \cite{spudich,millman,offer,horowits,yildiz,geeves} and references therein for a full 
treatment. Hence, we will only give a rough picture about the apparent behavior. Nevertheless the point that we want 
to emphasize is, fortunately, not related to the inner workings of the human muscles.

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mark options={fill=white},
mark indices={1,...,4,13,14,...,30,36,37,...,40,42,43,...,49}] plot[parametric] function{3*sin(t),0.1*sin(2*t)};
\filldraw[fill=white] (tpt-49) ++(10:2mm) arc (110:40:2mm) arc (-40:156:2mm) arc (65:5:2mm) ;
\filldraw[fill=white] (tpt-42) ++(150:2mm) arc (150:210:2mm) arc (330:30:2mm);
\end{scope}
\node[draw,anchor=west,shape=ellipse,ultra thick,minimum width=1cm] (a) at ([shift={(7mm,-3.5mm)}]tpt-42) (myo) {};
\node[draw,minimum height=5cm,minimum width=7mm] (actin) at ([shift={(2.5cm,-2mm)}]tpt-46) {};
\draw (myo.east) -- ($(myo.east)!0.5!(actin.north west)$) coordinate (a);
\draw[decorate] (a) -- ($(myo.east)!0.80!(actin.north west)$)coordinate (b);
\draw (b) --  (actin.north west);
\end{scope}
\begin{scope}[yshift=-7cm]
\begin{scope}[rotate=90]
\draw[domain=0.6*pi/3:6.5*pi/3,
samples=50,draw=none,
mark=o*,
mark size=2mm,
mark options={fill=white},
mark indices={1,...,4,13,14,...,30,36,37,...,40,42,43,...,49}] plot[parametric] function{3*sin(t),0.1*sin(2*t)};
\filldraw[fill=white] (tpt-49) ++(10:2mm) arc (110:40:2mm) arc (-40:156:2mm) arc (65:5:2mm) ;
\filldraw[fill=white] (tpt-42) ++(150:2mm) arc (150:210:2mm) arc (330:30:2mm);
\end{scope}
\node[draw,rotate=-20,anchor=west,shape=ellipse,ultra thick,minimum width=1cm] (a) at ([shift={(7mm,-3.5mm)}]tpt-42) (myo) {};
\node[draw,minimum height=5cm,minimum width=7mm] (actin) at ([shift={(2.5cm,-2mm)}]tpt-46) {};
\draw (myo.east) -- ($(myo.east)!0.5!(actin.north west)$) coordinate (a);
\draw[decorate] (a) -- ($(myo.east)!0.80!(actin.north west)$)coordinate (b);
\draw (b) --  (actin.north west);
\end{scope}
\end{scope}
\end{tikzpicture}
\caption[Muscle activation mechanism]{Muscle activation mechanism. The sequence is given in a clockwise (or anti-clockwise) fashion (adapted from \cite{geeves}).}%
\label{fig:lit:muscle}%
\end{figure}



The skeletal muscle activation takes place via a process described by the \emph{sliding filaments model} (\Cref{fig:lit:filament}). 
The muscles consist of muscle fibers and muscle fibers are made up of \emph{myofibrils}. The myofibrils involve different types of thin 
and thick filaments, mainly of the type \emph{actin}, \emph{myosin} and \emph{titin} filaments. The myosin filaments involve 
extensions that can bind to the thin actin filaments. The relative motion of these filaments are produced due to these 
extensions via ATP hydrolysis. Moreover, these extensions stay connected or disconnected to actin filament unless more 
ATP is utilized. Hence, the muscle needs extra energy to relax which is the reason behind \enquote{\emph{rigor mortis}} 
and some types of the muscle spasms when this additional energy source can not be provided or ATP can not bind to the 
actin filament for some reason. Moreover, binding or detachment of these extensions is also regulated by the 
$\mathrm{Ca}^{++},\mathrm{Mg}^{++}$ concentration in the muscle cell and controlled, eventually, by motor neurons using yet another 
chemical trigger. 

\begin{figure}%
\centering
\includegraphics[height=7cm]{\impath/literature/filament}%
\caption[The simplified illustration of the sliding filaments model for a single sarcomere.]%
{The simplified illustration of the sliding filaments model for a single sarcomere. (Source: \cite{ottenheijm})}%
\label{fig:lit:filament}%
\end{figure}


Therefore, the individual muscle activation at the basic level is analogous to a clock escapement mechanism \cite{headrick} 
i.e. each time the lever arm pulls back a thread of the gear and the cycle repeats. The resulting relative motion is very much 
like a graphite coming out of a mechanical pencil when pressed from the eraser cap. Obviously, muscle behavior is not rachet-like 
but smooth. This is because each of these mechanisms operates independently. Thus, at each time instant, different myosin extensions 
can be found at different phases of the cycle very much like a helical gear pair that are always in contact at one point. 
Using this analogy, ATP molecules are used to open and to close the pencil clutch made up of myosin extensions and actin filament 
is pushed forwards. Moreover, titin filaments can be thought of as the connecting rod of the pencil from the cap to the clutch which 
is mostly responsible for the passive elasticity of the muscles. 

In summary, the muscles have varying non/backdrivable configurations. Moreover, we can lock our muscles in place
e.g., we can try to keep our arm at one position during drilling etc. to increase the precision. Then, the arm becomes
a stiff object with inherent stiffness of the connection rod (titin), pencil clutch (myosin actin), muscle tendons and
 various other involved processes resisting to the applied strain. 

Coming back to our original discussion, the stiffness of the arm that has been the subject of the experiments mentioned 
above is, again invoking the analogy, based on the closed clutch response of the arm. In other words, what is measured 
is a cumulative spring-like behavior of the arm that is actively controlled to be kept at a certain configuration. This 
is related to the \enquote{human's restoring force} given in \Cref{rem:filament}. 

Having a spring-like property naturally implies that the arm is passive. But the problem with this argument, as far as 
we understand, is that there is no reason to assume that the human arm model involves a symmetric positive definite 
stiffness matrix at each time instant for an arbitrary trajectory. In fact, as shown clearly in \cite{mussa85}, the major eigenvector 
of the stiffness matrix varies both in terms of direction and magnitude. Therefore, it might be possible to extract energy 
from the human arm with some particular pathological trajectory. Just in the case of a frozen time analysis of a time-varying 
operator does not imply stability, the conclusion is only valid for postural analysis at a fixed configuration of the human arm 
but not for an arm trajectory. 

\begin{quote}
Therefore, it is our belief that the assumption of human arm being a passive system is incorrect for most teleoperation tasks.
\end{quote}
 

Since it is customary in the literature to include the passivity hypothesis, let us invoke it here too for the sake of the argument. 
The question of how, then, a human can possibly move anything while remaining passive is one that makes the whole story even more 
complicated. The voluntary input of the human is taken as an exogenous and state independent input to the system. Hence, the human 
cognitive input is an additional but independent force source acting on the handle together with the passive human arm immitance 
as depicted in \Cref{fig:lit:stateindep}. In other words, the contribution of the exogenous force and the arm dynamics are assumed 
to remain mutually exclusive. 

\begin{figure}%
\centering
\begin{tikzpicture}
\tikzstyle{spring}=[thick,decorate,decoration={coil,pre length=0.3cm,post length=0.3cm,segment length=1mm,amplitude=2mm}]
\tikzstyle{damper}=[thick,decoration={markings,  
  mark connection node=dmp,
  mark=at position 0.5 with 
  {
    \node (dmp) [thick,inner sep=0pt,transform shape,rotate=-90,minimum width=15pt,minimum height=3pt,draw=none] {};
    \draw [thick] ($(dmp.north east)+(2pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(2pt,0)$);
    \draw [thick] ($(dmp.north)+(0,-5pt)$) -- ($(dmp.north)+(0,5pt)$);
  }
}, decorate]
\draw (-5mm,10mm) arc (120:240:1.7mm) 
                  arc (140:240:2mm) 
                  arc(140:230:2mm) 
                  arc(150:230:1.7mm) 
                  to[out=-20,in=190] ++(1.3cm,1mm) 
                  to[out=-5,in=200] ++(3cm,-1mm) 
                  coordinate (kose);
\coordinate (ustkose) at ([yshift=10mm]kose);
\draw (kose) .. controls +(60:2mm)  and +(-60:2mm) .. ++(0,5mm) coordinate (orta); 
\filldraw[pattern=north east lines]  
(orta) .. controls +(120:3mm) and +(190:2mm) .. (ustkose) 
       .. controls +(-10:2mm) and +(60:3mm) .. (orta);
\draw (ustkose) to[in=-15,out=160] ++(-3cm,1mm) 
                to[in=20,out=170] 
                coordinate [pos=0.6](b) 
                coordinate [pos=0.8](a) (-5mm,10mm);
\fill[gray] (a) |- ++(2mm,15mm) -|(b);
\draw[-DIN,ultra thick] ([shift={(2mm,10mm)}]a) -- ++(-2cm,0) node[above,midway] {$F_{exo}$};
\coordinate  (yay) at ([shift={(5mm,5mm)}]orta);
\coordinate  (damp) at ([shift={(5mm,-5mm)}]orta);
\coordinate (duvar) at ([shift={(2cm,0)}]orta);
\draw[thick] (orta) -| (yay) (orta) -| (damp) ([yshift=7mm]duvar) |-(-1cm,-5mm);
\fill[pattern=north east lines] ([yshift=7mm]duvar) |-(-1cm,-5mm) --++(0,-2mm) -| ([shift={(2mm,7mm)}]duvar);
\draw[spring] (yay) -- ++(15mm,0);
\draw[damper] (damp) -- ++(15mm,0);
\end{tikzpicture}
\caption[A pictorial representation of the state independent human force input]%
{A pictorial representation of the state independent human force input. By state dependence, we mean that the
vector $F_{exo}$ is completely decoupled from the arm characteristics and can be taken as a genuine exogenous input.}%
\label{fig:lit:stateindep}%
\end{figure}
The reader would spot that this is not in line with Hogan's remark since the activation state in the muscles should be altered in 
order to apply some force. In other words, we need to alter the stiffness matrix of the arm to be able to move but, to the best of our 
knowledge, this is not given in any teleoperation study. In other words, the 
answer to the question of whether the human's restoring force would exhibit the same properties during the force exertion phase is 
not known to us. It has been also shown that the applied force is also related to the muscle tension-length properties \cite{eltit} 
which we have not considered in our analogy.

Therefore, we have to further separate the human force into two parts, namely, the active neuromuscular feedback force that keeps the 
human arm passive and the voluntary and cognitive force input applied to the system. How this is usually performed is not clear in the 
literature. To the best of our knowledge, this issue is considered (but still briefly) only in \cite[Sec. II.B]{kazeroonitsay} and 
references therein.


This ambiguity becomes much more important since the control oriented focus of this thesis necessitates that we concentrate on
worst cases rather than the experiments performed within the cognitive range of human operators. In other words, we are interested in
the cases where things go wrong due to various reasons, such as sampling disturbances, measurement noises, directionality 
etc. Therefore it cannot be a satisfactory argument if stability depends on the user's neuromuscular feedback or, simply, the user's 
stabilization capabilities. In order to use the bilateral teleoperation devices in real-life, stability should be addressed 
regardless of the set of prescribed human actions. Hogan's findings are not sufficient for supporting the passivity assumption 
often found in the literature. 


In summary, the passivity of the human and the environment (or the virtual environment in haptics/virtual reality applications), is 
only plausible in certain cases; then it should be verified regardless, in order to assume that the corresponding mathematical models 
are passive. Still, analysis and synthesis methods that invoke this assumption lead to many real-world implementations 
with varying degrees of realism. An obvious follow-up question is \enquote{how can then the reported analysis and synthesis results
based on this passivity assumption lead to successful implementations which would imply the validity of the assumption?}. We need to 
focus on the \enquote{successful} part in order to answer this question. If we again scan through the literature these successful  
implementations are only successful under strict assumptions about the user behavior together with any combination of the following
device properties
\begin{itemize}
	\item The controller stabilizes the loop via excessive apparent damping hence either or both free air or hard contact realism is lost.
    \item The bandwidth of the force and position tracking are exceedingly small.
    \item The remote and local devices exhibit position drifts over time.
    \item The performance of the device is time-varying and not uniform.
\end{itemize}
 
We argue that the success of these methods is due to the conservatism of the analysis/synthesis tools and does not validate the 
passivity hypotheses on the respective models. Many frequency-domain methods, which assume LTI terminations at both ends of the 
teleoperation network, are reported to achieve stable bilateral interaction in real setups. While impressive, this shouldn't have 
been possible had the tests not been exceedingly conservative since the real setups involve sudden contacts i.e. they are essentially 
time-varying systems. We recall the well-known fact that the behavior set of the time-varying systems are significantly richer than 
LTI systems. Once again, we remind the reader that the proper metrics to gauge the success of teleoperation systems are not known and 
we use the typical error-norm based control design rationale. 

A compact version of the argument above is given by Yokokohji and Yoshikawa in \cite{yokokohjiyoshikawa}: 
\begin{quote}
Passivity of the system can be a
sufficient condition of stability only when the system interacts
passive environments. In the case of master-slave systems, if
we could assume that the operator and the environment are
passive systems, then the sufficient condition of stability is
that the master-slave system itself must be passive. Strictly
speaking, however, the operator is not passive because he/she
has muscles as the power source. Colgate et al. [21]\footnote{Reference 
\cite{colgatehogan88} of this thesis.} mentioned
that even if the system has an active term, the system stability
is guaranteed unless the active term is in some way state dependent.
Obviously, the operator is passive when $\tau_{op}= 0$.
Therefore, we will give the following assumption about $\tau_{op}$:
\enquote{\emph{The operators input $\tau_{op}$ independent to the state of the
master-slave system. In other words, the operator does not
generate $\tau_{op}$ that will cause the system to be unstable.}}
Dudragne et al. [3]\footnote{Reference \cite{dudragne} of this thesis.} gave a similar assumption in order to use
the concept of passivity for stability distinction. The above
assumption seems tricky in a sense, but it is necessary to ensure
the system stability by the passivity.
\end{quote}

\noindent Finally, a supplementary remark is also given by Buerger and Hogan in \cite{buergerhogan1}: 
\begin{quote}
When passivity is used as a stability objective, the only assumption
made about the environment is that it, too, is passive.
This is likely sufficient to guarantee coupled stability with humans
(though, to date, it has not been conclusively proven that
human limbs are passive; see [29]\footnote{Reference \cite{hogan89} 
of this thesis.} for an argument for treating them as such). However, 
given the properties of human arms 
described above, passivity is unnecessarily restrictive. Our experience
has shown that some controllers that are known to be
nonpassive are adequately stable in clinical rehabilitation tasks [26]%
\footnote{Reference \cite{buergerconf} of this thesis.}.
\end{quote}


We should mention here that Hogan's paper together with other identification experiments are extremely important for many
fields and needs no motivation. The discussion above only points out that the frequently reported inference that follows from 
his results is not in line with the results. 

The idea of modeling the teleoperation as a two-port network seems to have multiple origins and we have no reference to 
point out a common source. However, in general, the popularity of two-ports can be attributed to \cite{andersonspong,nieslotine,
rajuphd,hannaford89,yokokohjiyoshikawa}. 


\subsection{Uncertain Models of Bilateral Teleoperation for Robustness Tests}\label{sec:lit:uncmodel}

Another possibility of modeling the human arm and its cognitive input is to define a reference position signal \enquote{filtered} by the 
human arm impedance\footnote{The term \emph{impedance} is used in a more general sense than its common usage to denote
LTI transfer functions such that no distinction is made between linear and nonlinear or time-invariant and time-varying 
operators.} e.g., \cite{leelee,kazeroonitsay}. Various studies pointed out that the identification experiments suggest a 
mass-spring-damper system pattern is evident in the frequency response data of the human arm recorded under various task 
performance similar to the one given in \cite{hogan89}. The general method is to instruct the human to perform a specific 
task and then perturb the hardware with certain predesigned disturbance signals such that the output can be evaluated to obtain 
a mathematical model. In the literature, the model structure is often set a priori to be a second order transfer function and 
the parameters are optimized to minimize the mismatch between the experimental and predicted response. It is also well-known 
that the human can change the inherent impedance of the arm during the task execution (see, e.g., \cite{tsujimorasso}). 
Therefore, the studies are performed in the ranges where it is safe to assume that the human arm characteristics are constant 
or constant up to negligible changes. 


The modeling is straightforward via uncertain mass-spring-damper system differential equation manipulations. Suppose the human arm
admits the second order model
\[
M(\Delta_1)\ddot{x} + B(\Delta_2)\dot{x} + K(\Delta_3)x = F_h -F_m
\] 
where $F_h,F_m$ denote the human force and the force feedback inputs, respectively. Then choosing a multiplicative or additive uncertainty 
structure and via basic linear fractional transformations, the signal relations are converted to the interconnection shown in 
\Cref{fig:lit:figunc}. The arrows are deliberately left out as it is up to the designer to get different immitance models.
\begin{figure}%
\centering
\begin{tikzpicture}
\matrix[draw,matrix of math nodes] (m) {\Delta_1 &&\\&\Delta_2 &\\&&\Delta_3\\};
\node[below=1cm of m,draw,minimum size=2.0cm] (g) {$G_{nom}$};
\draw (g.155) -| ++(-1cm,1cm) |- (m.west);
\draw (g.25) -| ++(1cm,1cm) |- (m.east);
\draw (g.-155) -- ++(-1cm,0cm) (g.-25) -- ++ (1cm,0cm);
\end{tikzpicture}
\caption{Uncertain model representation by taking out the uncertainty blocks}%
\label{fig:lit:figunc}%
\end{figure}


Many studies have appeared in the literature regarding such modeling and the majority of these assume a 
mechanical model of order from two to five. Note that this is an assumption made a priori and only applies 
to the specific task performed by the human in the experiment from which the frequency response data is 
collected. The commonly utilized models, unfortunately most given for the fixed postural arm 
configuration can be found in \cite{kazeroonitsay,tsujigoto,kosuge,colgate1,speich,fucavus,buergerhogan1,leungfa,
husalculoewen,andrifour,laroche}. Obtaining these measurements are time-consuming and difficult to parameterize. For this reason, although
the results along this direction are scarce, they are, as in the passivity case, very valuable. 

The disadvantage of such parametrization of the human arm is contrasted with the passivity approach methods
invoking the argument of the time-varying nature of the arm parameters. It is often rightfully argued that the 
uncertainty ranges in which the stiffness and damping (and partially inertial) coefficients change, are too 
large to be considered in the structured singular value based robustness tools. Moreover, many auxiliary 
effects such as the visual feedback, cognitive lag of the brain etc. are not considered in the identification 
experiments; in the passivity approach all of these are lumped into a single port condition. Obviously, 
the main difficulty is to get a model (out of hypothetical insights, identification experiments etc.) which is
not required in the passivity approach.

The papers \cite{leelee,prekopiou} offer interesting alternatives for human modeling as they attempt to incorporate 
many of the aforementioned effects, but the results are prospective and yet to be utilized. 



\section{Analysis}\label{sec:litchapanalysis}

The stability analysis is one of the major problems in designing stable yet high-performance teleoperation 
systems. It's often not feasible to manually tune some local controllers and make test subjects use it in order to verify the
design specifications. Moreover, by relying only on the experiments, one can miss an important destabilizing scenario if the field 
experiments do not cover that particular case. Hence, an a priori certificate of stability is much sought after. The stability 
analysis can also give some guidelines about the parameter selection in the hardware design phase and can lead to minimized 
design iterations. Therefore, having a realistic stability test is essential in building these systems. 

Similar to the modeling section, the analysis in the literature extensively relies on network theory based results. In fact 
this is where the network theory stands out as a complete tool for analysis and synthesis of bilateral teleoperation systems
via the hypothesis that human and the environment models are passive.  

The common terminology for stability is somewhat different than that of the contemporary control theory as \emph{nominal stability} 
is used for the stability properties of isolated two disjoint media; when the interaction is set up between these two media the closed-
loop stability problem is called \emph{coupled stability}. To the best of our knowledge, this terminology is introduced in 
\cite{colgatehogan88} hence we refer to this paper (or Colgate's thesis \cite{colgatephd}) for more details. 


It's also worth mentioning that the passivity and stability is used often interchangeably and also usually referred 
to the classical texts \cite{haykin,mitra69,chen91} for the precise definitions. Hence, there is a little guesswork 
required to classify the stability definitions given in the literature in order to locate which version is meant. 
The important distinguishing point is that marginal stability is often accepted in the definition of stability results 
since it arises frequently in lossless (hence passive) models where energy conservation is assumed. However, the analysis 
results that rely on such assumptions do not guarantee asymptotic interconnection stability, but only lead to certain 
passivity properties of the interconnection (see \cite[Thm. 6.1]{khalil} and \cite[Sec. V]{lawrence} for the discussion 
on strict passivity).  

As given in \Cref{sec:litchapmodeling}, the passivity property is crucial to many studies in the literature. The direct 
physical interpretation of the abstract concepts gives even more appeal to such energy book-keeping methods. Another 
advantage of passivity methods is that the nonlinear counterparts of the results are also available in the literature 
and relatively easy to utilize. However, this convenience misses out many relevant details that are specific to 
teleoperation systems and result with too general conditions. Let us recall a general version of the passivity theorem:

\begin{thm} The negative feedback connection of two passive systems is passive. The negative feedback connection of a passive system with
a strictly passive system is asymptotically stable.\end{thm}

Note that, this result is valid for both nonlinear and linear systems. Invoking the theorem twice on the teleoperation 
system allows us to conclude that, under the passivity assumption of the human and the environment, if the two-port is 
passive then the interconnection is passive. Moreover, if any of the involved operators is strictly passive the teleoperation 
system is asymptotically stable. 

\begin{figure}%
\centering
\includegraphics[width=0.6\columnwidth]{\impath/literature/train}%
\caption[A transparent two-port network with passive terminations]{A transparent two-port network with passive 
terminations. The actuation of the railcar is taken as a state-independent input to the system. (Source: \cite{train})}%
\label{fig:lit:train}%
\end{figure}


The passivity theorem is in general not necessary for stability but only sufficient. Because there do exist stable 
interconnections that involves nonpassive subsystems. In particular, the conservatism brought in by passivity 
assumption is arbitrarily high (especially in the nonlinear case). Facetious as it may seem, the test also takes 
into account the port terminations shown in \Cref{fig:lit:train} for a table-top joystick. We have to emphasize 
that the three-carriage railcar with two cabs, is a valid, almost perfectly transparent two-port network with 
passive end terminations. Hence, a regular passivity-based stability test includes those end terminations for 
a simple hand-held device. In other words, if we only consider the passivity property of the involved subcomponents and do not 
distinguish further, there is no possible way to distinguish a table-top teleoperation system from a train since
both are passive. It's that conservative.


The major disadvantage of the passivity methods is that the procedure is focused almost only on the 
energy exchange. The performance specifications are very difficult to formulate and also difficult 
to integrate into the analysis and synthesis steps using only the inner product structure. As an 
example, the signals that are not port variables such as position errors, nonlinear effects etc.,
that are functions of these signals, can't be utilized easily in the performance specifications. 
Same difficulty arises in the normed space structures though much more can be achieved. Similarly,
peak-to-peak gain minimization methods are not mature enough to handle any practical system without 
excessive conservatism. 


Another disadvantage is that the power- or the energy-based analysis, due to the inner-product structure, 
can not distinguish the individual signals. Consider the ideal case where the human and the local 
device is pushing each other and cancelling each other's contribution. In this case the external or observable 
energy exchange based on the port variables is zero (negligible) which can not be distinguished from the case of 
not touching at all (a small motion on the device). 


When the network, human, and the environment models are assumed to be Linear Time Invariant (LTI), the frequency domain methods
allow us the analyze the teleoperation systems for stability and performance. The most common stabiliy analysis tool for such 
models is the Llewellyn's stability criteria (also often called absolute stability theorem or unconditional stability theorem). 
For linear networks, the following definitions seem to be used quite widely (modified from \cite{chen91}): 

\begin{define}[Potential Instability] A two-port network is said to be potentially unstable if there exist two passive one-port 
immitances that, when terminated at the ports, produce a persisting natural frequency.
\end{define}

Notice that this definition is equivalent to the common \emph{robust stability} definition which states that if a two port 
network is robustly stable there exists no pair of one-port immitances that makes the network exhibit a persisting natural 
frequency. 

\begin{define}[Absolute Stability at $\iw_0$] A two-port network is said to be absolutely stable if it is not potentially unstable.
\end{define}


\subsection{Llewellyn Stability Criteria}\label{sec:llewellyn}
The well known conditions for stability of a two-port network, formulated in \cite{llewellyn,bolinder,rollett}, 
are recalled in \Cref{chap:apdxnetwork}. As shown in \cite{rollett}, the conditions stated in \Cref{thm:apdx:llw} 
are invariant under immitance substitution. This result forms the basis for almost all passivity-based frequency 
domain bilateral teleoperation stability analysis approaches in the literature. We also derive this theorem 
from an IQC perspective and show that it is actually the passivity counterpart of the of the $D$-scalings in the 
$\mu$-tools. This has also been derived in a scaled teleoperation context in \cite{poortenyokokohji} using only 
Structured Singular Value (SSV) arguments. 

Thanks to the frequency domain formulation, it is possible  to rewrite the condition \eqref{eq:lit:llew3} as a fraction 
and see the problematic regions in which the fraction gets close to or crosses to the instability, together with one of the 
conditions given in \eqref{eq:lit:llew2}. 


\begin{figure}%
\centering
\begin{tikzpicture}[>=stealth]
\node[draw,minimum size=1cm] (h2) {$H_2$};
\node[draw,minimum size=1cm, above=1cm of h2] (h1) {$H_1$};
\node[draw,circle,inner sep=1mm,right= 1cm of h2]   (a1) {};
\node[draw,circle,inner sep=1mm,left= 1cm of h1,label={[inner sep=0]-60:$-$}] (a2) {};
\draw[->] (h1) -| (a1) node[right,midway] {$y_1$};
\draw[->] (h2) -| (a2) node[left,midway] {$y_2$};
\draw[->] (a1) -- (h2) node[above,midway] {$e_2$};
\draw[->] (a2) -- (h1) node[above,midway] {$e_1$};
\draw[<-] (a1) -- +(1cm,0) node[right] {$u_2$};
\draw[<-] (a2) -- +(-1cm,0) node[left] {$u_1$};
\end{tikzpicture}
\caption{Negative feedback interconnection}%
\label{fig:lit:passint}%
\end{figure}

\subsection{\texorpdfstring{$\mu$}{mu}-analysis}
As given in \Cref{sec:lit:uncmodel}, stability in the face of uncertainties can also be analyzed in the generalized plant framework 
of robust control. After rewriting the signal relations, the teleoperation system can be written as an uncertain interconnection as 
shown in \Cref{fig:lit:uncincgeneral}. In this setting, $G$ is the model of the nominal bilateral teleoperation system and $\Delta$ 
is a block diagonal collection of uncertainties, such as the human, the environment, delays, etc. Stability tests are based on structural 
hypotheses on the diagonal blocks of the operator $\Delta$ such as gain bounds or passivity. These properties should allow us to develop 
numerically verifiable conditions for the system $G$ that guarantee interconnection stability. This is intuitive because we have no access 
to the actual $\Delta$ and we can only describe its components by means of indirect properties. 

If the interconnection subsystems are represented in the scattering parameters, the $\mu$ test is precisely equivalent to the test of
Llewellyn's theorem and often called as Rollett's stability parameter. This is due to the well-known equivalence between the small-gain 
and passivity theorem \cite{desvid}. We have to note that the equivalence is stated in terms of the stability characterization. Otherwise, 
the small-gain theorem requires a normed space structure whereas passivity theorem requires an inner product space structure, hence the 
applicability is relatively limited. This is also related to the fact that we need to work with power variables exclusively in the 
passivity framework and this is not always convenient if the performance specifications are related to other variables.

In the literature, this analysis method often follows the scattering transformations such that, the passivity assumption avoids the
explicit modeling and then small-gain theorem is utilized mainly to handle the delay problem via with the involved norm bounded operators.
A refinement can be found  in \cite{poortenyokokohji} where the authors utilize a direct $\mu$-analysis to reduce the conservatism, 
however rather remarkably, it's not picked up by other studies and the analysis is mainly limited to small-gain conditions even in 
the linear systems. 


We could have also directly choosen to utilize the uncertain modeling of the human and the environment and utilize $\mu$-analysis for the 
teleoperation system had the specific models been available.  


\begin{figure}%
\centering%
\begin{tikzpicture}[>=stealth]
\node[draw,minimum size=0.75cm] (plant) at (0,0) {$G$};
\node[draw,minimum size=0.75cm] (unc) at (0,1.5) {$\Delta$};
\draw[->] (plant.west) -| ++(-5mm,10mm) node[left] {$v$} |- (unc.west);
\draw[->] (unc.east)   -| ++(5mm,-5mm) node[right] {$u$} |- (plant.east);
\end{tikzpicture}
\caption{Uncertain Interconnection}%
\label{fig:lit:uncincgeneral}%
\end{figure}



\subsection{Modeling the Communication Delay}

Over the past two decades, it has been confirmed in various studies that, if present, communication delays are a major 
source of instability (reports date back to 60's, e.g., \cite{sheridanferrell} and the references in \cite{andersonspong}).
Even when the delay duration $t$ is known and constant, the delay operator can be shown to be nonpassive since $e^{-st}$ is 
not positive real. Hence, when combined with the passivity framework, it violates the assumptions on the uncertain operators. 


At end of the 80's and early 90's, two prominent studies (\cite{andersonspong,nieslotine}) proposed to handle the delay 
robustness problem using scattering transformations. This notion is best explained, in our humble opinion, by loop transformations 
since the original articles refer to microwave and transmission line theories which use quite specialized terminology. One can also
find a slightly different system theoretical view of these transformations in \cite{colgate3}. If we restrict the discussion to LTI
operators\footnote{In the nonlinear case, it's a \emph{completion of square} argument to switch from the inner product structure to a 
norm structure provided that the signal space is suitable for such operation.}, the key concept of the scattering transformation
or the wave variables methods is to map the closed right half plane to the closed unit disk via a special case of bijective M\"{o}bius 
(or linear fractional or bilinear) transformation: 
\begin{equation}
W: \Complex_+\cup \Complex_0 \mapsto \left\{ z\in\Complex \mid \abs{z} \leq 1 \right\}\, ,\, W(z) = \frac{z-1}{z+1}
\label{eq:lit:smith}
\end{equation}

One can directly verify that $1\mapsto 0,\infty\mapsto 1$ and $0\mapsto -1$ under $W$. Pictorially, the mapping is given in 
\Cref{fig:lit:smith} using a Smith chart which is located at the origin. Hence, positive real transfer matrices
become norm bounded by $1$ such that we can analyze the interconnection using the small-gain theorem.


Let us demonstrate a few properties of this transformation. First, this mapping can be shown with a block diagram. 
Assume that $G$ is a proper positive real LTI SISO system and let the input/output relation be given by $y=Gu$. Then, 
with a standard manipulation, we obtain a feedback interconnection that leads to the mapping
\begin{equation}
W(G(s)) = \frac{G(s)-1}{G(s)+1} = -1 + \frac{2G(s)}{G(s)+1} \Longrightarrow 
\begin{tikzpicture}[baseline=(g.center),scale=0.5,transform shape,every node/.style={draw,minimum size=1cm},every path/.style={->},>=stealth]
\node[draw] (g) at (0,0) {$G$};
\node[draw,left=1cm of g] (s1) {$\sqrt{2}$};
\node[draw,right=1cm of g] (s2) {$\sqrt{2}$};
\node[draw,circle, right=1cm of s2,minimum size=0,inner sep=3pt] (j) {};
\draw (s1) -- (g) node[midway,circle,fill=white,minimum size=2mm,inner sep=0] (f) {};
\draw ($(g)!0.5!(s2)$) |- ++(-1cm,-1cm) -| (f);
\draw (g) -- (s2);\draw (s2)--(j);\draw (j)--+(1cm,0);
\draw (s1) ++(-25mm,0) -- (s1);
\draw (s1) ++(-15mm,0) |- ([shift={(-1cm,1cm)}]j.center) -| (j);
\path (j)++(50:3mm) node[draw=none,minimum size=0,inner sep=3pt] {$-$};
\node[draw=none] at ([shift={(2mm,2mm)}]f.north) {$u$};
\node[draw=none] at ([shift={(-23mm,3.5mm)}]s1) {$\hat{u}$};
\node[draw=none] at ($(g)!0.5!(s2)+(0,2.7mm)$) {$y$};
\node[draw=none] at ($(j)+(0.8cm,3.2mm)$) {$\hat{y}$};
\end{tikzpicture}
\label{eq:wavetrafo}
\end{equation}
Simply following the signal paths, we also see that the input/output relation becomes
\[
\tilde{y}\coloneqq W(G(s))\tilde{u}.
\]
where 
\begin{equation}
\pmatr{\tilde{y}\\\tilde{u}} = \sqrt{2}\pmatr{-1 &1\\1 &1}\pmatr{u\\y}
\label{eq:scatrafo}
\end{equation}

Note that, physical realization of this transformation requires a feedforward and 
a feedback control action to which typically referred with \emph{\enquote{Wave encoding}}.  
Often these variables are normalized with $\sqrt{2}$ at the outset. The seperation of $\sqrt{2}$ 
blocks is a matter of convention and provides symmetry in the block diagrams. Also we have, 
\begin{align*}
(W\circ W)(G) &= \frac{-1}{G},\\
(W\circ W \circ W) (G) &= \frac{-1}{W(G)},\\
(W\circ W \circ W \circ W) (G) &= G\\
\end{align*}
which shows the effect of the $90^\circ$ clock-wise rotations of the Riemann sphere about the axis 
parallel to the imaginary axis ($W$ is an element of M\"{o}bius group with $\circ$ operation). This 
stereographic projection idea is also the main idea behind the derivation of the stability parameter 
of Edwards and Sinsky (\cite{edsin}). Moreover, from \Cref{fig:lit:smith}, we can 
see a visual proof of Theorem 3.1 in \cite{andersonspong} which states that; 
\[
\Re{G(\iw)}\succeq 0 \iff \norm{W(G(s))}_\infty\leq 1
\]  



\begin{figure}%
\centering
\begin{tikzpicture}[scale=0.6]
\begin{smithchart}[show origin,
axis background/.style={shading=mysphere},
no marks,
grid=major,
axis equal,samples=150,
]
\foreach \t in {0,0.2,0.5,1}{
\addplot+[ultra thick,domain=-10:10] (\t,\x);
}

\end{smithchart}
\begin{scope}[shift={(8cm,0)}]
\begin{axis}[axis lines=middle,
grid=major,
no marks,
axis background/.style={right color=black!50,left color=white},
xmax=1.5,
%ytick=\empty,
every axis/.append style={
extra description/.code={
\node[xshift=-5mm] at (0,1) {Im$(z)$};
\node[xshift=-5mm] at (1,0.55) {Re$(z)$};
}}]
\foreach \x in {0,0.2,0.5,1}{
\addplot+[ultra thick] coordinates {(\x,-1.5) (\x,1.5)};
}
\end{axis}
\end{scope}
\end{tikzpicture}
\caption[Mapping the closed right half plane onto the unit disc.]%
{Mapping the closed right half plane onto the unit disc.}%
\label{fig:lit:smith}%
\end{figure}
%
%\begin{figure}[b]%
%\centering
%\begin{tikzpicture}
%\begin{axis}[unit vector ratio=1 1 1,view={65}{20},hide axis,colormap/invgray,ymax=4]
%\begin{scope}[/pgfplots/restrict z to domain=-1:0];
%\addplot3[surf,shader=interp,z buffer=sort,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=0:180]({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\addplot3[opacity=0.5,mesh,draw=black,ultra thin,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=180:360] ({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\end{scope}
%\shade[right color=black!50,left color=white](axis cs:-2,0,0) -- (axis cs:2,0,0) -- (axis cs:2,2,0) -- (axis cs:-2,2,0)-- cycle;
%\draw (axis cs:0,-1.2,0) -- (axis cs:0,2,0);
%\draw[white,thick] (axis cs:-2,0.3,0) -- (axis cs:2,0.3,0);
%\draw[white,thick] (axis cs:-2,0.70,0) -- (axis cs:2,0.70,0);
%\begin{scope}[/pgfplots/restrict z to domain=-0.01:1]
%\addplot3[surf,shader=interp,z buffer=sort,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=0:180]({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\addplot3[opacity=0.5,mesh,draw=black,ultra thin,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=180:360] ({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\draw (axis cs:-2,0,0) -- (axis cs:2,0,0);
%\end{scope}
%\addplot3+[domain=0:18,samples=100,no marks,white,samples y=0] ({cos(x)*sin(5*x)},{sin(x)*sin(5*x)},{cos(5*x)});
%\addplot3+[solid,domain=0:45,samples=100,no marks,white,samples y=0] ({cos(x)*sin(2*x)},{sin(x)*sin(2*x)},{cos(2*x)});
%\end{axis}
%\begin{scope}[xshift=4cm,yshift=4.1cm,rotate=-80]
%\begin{axis}[unit vector ratio=1 1 1,view={65}{-25},hide axis,colormap/invgray,ymax=4]
%\begin{scope}[/pgfplots/restrict z to domain=0:1]
%\addplot3[surf,shader=interp,z buffer=sort,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=0:180]({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\addplot3[opacity=0.5,mesh,draw=black,ultra thin,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=180:360] ({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\end{scope}
%\begin{scope}[/pgfplots/restrict z to domain=-1:0];
%\addplot3[surf,shader=interp,z buffer=sort,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=0:180]({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\addplot3[opacity=0.5,mesh,draw=black,ultra thin,samples=19,variable=\u, variable y=\v,domain=0:180, y domain=180:360] ({cos(u)*sin(v)}, {sin(u)*sin(v)}, {cos(v)});
%\end{scope}
%\addplot3+[solid,domain=0:18,samples=100,no marks,white,samples y=0] ({cos(x)*sin(5*x)},{sin(x)*sin(5*x)},{cos(5*x)});
%\addplot3+[solid,domain=0:45,samples=100,no marks,white,samples y=0] ({cos(x)*sin(2*x)},{sin(x)*sin(2*x)},{cos(2*x)});
%\draw (axis cs:0,0,-1.2) -- (axis cs:0,0,2);
%\draw (axis cs:-2,0,0) -- (axis cs:2,0,0);
%\end{axis}
%\end{scope}
%\end{tikzpicture}
%\caption[Mapping the closed right half plane onto the unit disc in 3D.]%
%{Same mapping shown in \Cref{fig:lit:smith} but in 3D coordinates. If we trace the horizontal lines with rays originating from the north 
%pole of the sphere. The intersection of the ray with the sphere uniquely maps the line onto the 
%eastern hemisphere. The rotation of the sphere gives the left figure of \Cref{fig:lit:smith}.}%
%\label{fig:lit:smithmobius}%
%\end{figure}





Obviously once this transformation is introduced, there is a need for the \enquote{inverse} of $W$ on the operator that is seen
by the operator $G$ such that the loop equations remain unchanged, that is to say we have to introduce another transformation that undoes
$W$. The simplest way to obtain a mapping $\hat{W}$ is to follow the block diagram backwards as shown in 
\Cref{lit:syn:invscat}. A block diagram reduction step(or rotating three more times as shown above) shows that 
\[
\hat{W}(z) = -\frac{z+1}{z-1} = -\frac{1}{W(z)}
\]
which is nothing but the inverse of the relation \eqref{eq:scatrafo}. The negative sign usually does not show up in 
the formulations in the literature because the passive interconnections 
require a sign change in the loop to indicate the \enquote{from} and \enquote{to} ports. A more detailed derivation
is given in \cite{colgate3}. Also note that a positive real transfer matrix, when negated, has its Nyquist curve 
confined to the close left half plane (anti-positive real) which is equivalent to a $180^\circ$ rotation or application
of $W$ to the closed left half plane twice. Thus, some care is needed for the book-keeping of the negative signs and 
seemingly the best practice is to absorb the negative sign into $H$ at the outset and work with $-H$ afterwards. This 
makes the required forward and backward transformations in the loop identical. One can see that there are many variants 
of this mapping, especially in wave variables context, e.g., $\frac{z-b}{z+b}$ but for simplicity we take $b=1$ as it 
doesn't play any role in our presentation of the method. 


\begin{figure}%
\centering
\begin{tikzpicture}[scale=0.7,transform shape,every node/.style={draw,minimum size=1cm},every path/.style={->},>=stealth]
\node[draw] (g) at (0,0) {$G$};
\node[draw,left=1cm of g] (s1) {$\sqrt{2}$};
\node[draw,right=1cm of g] (s2) {$\sqrt{2}$};
\node[draw,circle, right=1cm of s2,minimum size=0,inner sep=3pt] (j) {};
\draw (s1) -- (g) node[midway,circle,fill=white,minimum size=2mm,inner sep=0] (f) {};
\draw ($(g)!0.5!(s2)$) |- ++(-1cm,-1cm) -| (f);
\draw (g) -- (s2);\draw (s2)--(j);
\draw (s1) ++(-15mm,0) |- ([shift={(-1cm,1.5cm)}]j.center) -| (j);
\path (j)++(50:3mm) node[draw=none,minimum size=0,inner sep=3pt] {$-$};
\node[draw] (h) at (0,-2cm) {$H$};
\node[draw,left=1cm of h] (is1) {$\frac{1}{\sqrt{2}}$};
\node[draw,right=1cm of h] (is2) {$\frac{1}{\sqrt{2}}$};
\node[draw,circle, right=1cm of is2,minimum size=0,inner sep=3pt] (j2) {};
\draw (j)-|++(1cm,-1cm)|-(j2);
\draw (h) -- (is1) node[midway,circle,fill=white,minimum size=2mm,inner sep=0] (f2) {};
\draw (is2) -- (h);\draw (j2)--(is2);
\draw ($(h)!0.5!(is2)$) |- ++(-1cm,-1cm) -| (f2);
\draw (is1) --++(-25mm,0) |- (s1);
\draw (is1) ++(-15mm,0) |- ([shift={(-1cm,-1.5cm)}]j2.center) -| (j2);
\path (f) ++(-70:5mm) node[draw=none] {$-$};
%\path (f2) ++(-70:5mm) node[draw=none] {$-$};
\end{tikzpicture}
\caption[Scattering transformation and its inverse.]{Scattering transformation and its inverse.}%
\label{lit:syn:invscat}%
\end{figure}


Under the mapping $W$, the Nyquist curve of $e^{-sT}$ (the unit circle) is mapped onto the imaginary axis, 
and hence unbounded, i.e.
\[
\frac{e^{-\iw T}-1}{e^{-\iw T}+1} = i\tan{\frac{\omega T}{2}}
\]
One can also obtain a similar qualitative result by seeing the unit circle $\mathbb{T}$ as the image of 
the imaginary axis under $W$: 
\[
W(\mathbb{T}) = (W\circ W)(i\widetilde{\omega} T) = \frac{1}{i\widetilde{\omega} T}
\]
We don't need to track the points individually as we are only interested in the domain and its image
under these transformations. 


Therefore when we connect a passive operator to a small-gain operator, neither small-gain nor passivity theorem
can be applied directly since each operator is not in the class of the other operator. In other words, delay 
uncertainty does not satisfy the norm constraint $\|W(e^{-\iw t})\|_\infty \leq 1$ to invoke the small-gain 
theorem in the transformed coordinates. Had it been the case that the uncertainty was bounded by one, then it 
would have been possible to conclude stability directly in the passivity theorem anyhow. Thus, these transformations 
are not directly beneficial for analysis, however, following the cue from the previous mapping results, studies 
\cite{andersonspong} and \cite{nieslotine} made it possible to design controllers that renders the passive subsystem 
a norm bounded one and hence allowing delay robustness to be inspected via small-gain theorem. The resulting 
loop is stable regardless of delay period, hence they belong to the class of methods often distinguished
as \enquote{delay-independent} methods. According to the literature, these are the most common methods
applied in the face of delay uncertainty.

\subsubsection{Delay is Small-Gain}
Another possibility is to utilize the simple fact that the delay operator is gain-bounded and obtain the 
generalized plant by pulling out the uncertainty out of the loop. Obviously, this would be a very crude
characterization of the unit circle since a unit disk is used as the uncertainty instead. However, as 
we show later, wave variables/scattering transformations directly use this conservative formulation to 
model the delay in the stabilization of the loop.

A similar approach is reported in \cite{leungfa} using $\mu$-synthesis. By exploiting the
low frequency property of the operator $e^{-sT}-1$ and covering with a dynamic filter, the conservatism
is reduced. But the authors have omitted the uncertainty of the human and the environment. Therefore, 
their analysis is only valid for nominal teleoperation systems. Though, this can be extended to more 
general cases, we have to note that, they don't consider the human as \enquote{some impedance$+$state-independent 
force input} but as an finite-energy force input signal filtered through the human characteristics. 
Technically, this amounts to the common disturbance input-filtering often used in the $H_\infty$ design problems. 

 




\section{Synthesis}

Complementary to the analysis section, we cover a few popular controller structures among many others except the 
art of control engineering; manual PID tuning. 

\subsection{Two-, Three-, and Four-Channel Control Architectures}
In the teleoperation literature, the control laws are categorized in terms of how many measurement signals 
are sent over to the opposite medium during the teleoperation for control. The actual controller synthesis 
method is often not considered in this classification.  Hence the naming \enquote{$n$-channel control}. The
naming scheme can be better visualized as shown in \Cref{lit:syn:ela}. 


\subsubsection{Position-Postion and Position-Force Controllers}
The most basic control architecture among all is probably the PERR (position error) control in which the 
position of both the local and the remote site devices are collected by the controller and control commands
are applied to both devices to minimize the position difference regardless of which device is falling behind
in terms of tracking.

Assume that the local and the remote device are at rest at position $x=0$ in the respective world coordinates. 
Also assume that the user moves the local device to position $x=\SI{10}{\centi\metre}$. What the control algorithm
should do is to measure the position difference and force each device accordingly to minimize the error. Hence, the
control law is of the form
\[
\pmatr{F_l\\F_r} = \pmatr{-1\\1}K_p(s)\pmatr{1 &-1}\pmatr{x_l\\x_r}.
\]
where $F_r,F_l$ denote the control action at the local and the remote sites. $K_p(s)$ can be a constant or a SISO dynamical 
system or any other esoteric control law. One can perform the same with velocity signals (to comply with the passivity 
analysis) if available in noise-free measurements. Otherwise, a position drift is unaviodable even with integral action. 

Typically, this control architecture would give a sluggish performance since there is no preference or priority in 
correcting the error signal on each side. Therefore, while the remote site is pulled forward to track the local device,
simultaneously, the local device is pushed back with the same force. This results in a feel similar to extending a 
damper, only in this case, it softens up according to the position error instead of the travel velocity.

Another widely used control architecture is the so-called Position-Force Controller. In this method the first channel 
in the PERR control structure is replaced with the remote site force input. Hence, the local site device tracks the remote site 
encountered force while the remote site device tracks the position of the local site device which can be represented by the 
following description 
\[
\pmatr{F_l\\F_r} = \pmatr{K_f(s)F_{env} \\ K(s) (x_l-x_r)}.
\]
Clearly, the side-effect of PERR type control is avoided since the position and force errors are tracked independently 
in two seperate channels. But this brings in another tuning problem: If the position control gain dominates, the force tracking 
behaves aggresively in the hard contact case due to the overuse of control action to drive the remote device into the obstacle
and generally results with a kickback of the local device. Conversely, a domination of the force gain results in chattering of 
the remote device on the obstacle due to the discontinuous nature of the force reference signal if the user touches the handle 
just softly enough to sustain an oscillation. Therefore, not only the gains of the individual channels are hard to tune, but
also the relative magnitudes of the gains makes the tuning more tedious.

\subsubsection{Force-Force+PERR}

To increase the bandwidth and to reduce the side-effects of the aforementioned methods, a feedforward controller is added to the 
position-force control architecture. 
\[
\pmatr{F_l\\F_r} = \pmatr{K_{f1}(s)F_{env} \\ K(s) (x_l-x_r)+K_{f2}(s)F_{hum}}.
\]
Hence, this scheme is called a three-channel controller. This has been introduced in \cite{hashzaad1999} and also analyzed in \cite{naerumhannaford}.

\subsubsection{Lawrence Control Architecture}

In \cite{lawrence}, a general control scheme was proposed (later extended by \cite{salcudeanzhu,hzaadsalcu2}). In this architecture, 
also the remaining channel of remote position is sent over to the local site, completing the number of measurement channels to four.
The individual controller blocks and the resulting overall block diagram is shown in \Cref{lit:syn:ela}.  

\begin{figure}%
\centering
\begin{tikzpicture}[
imp/.style={draw,inner sep=4pt},
impc/.style={draw,inner sep=4pt,ultra thick},
junc/.style={draw,circle,inner sep=3pt},>=stealth]
\begin{scope}[xshift=-3cm]
\node[impc] (c4) at (0,0) {$C_4$};
\node[impc] (c3) at (2cm,0cm) {$C_3$};
\node[impc] (c2) at (4cm,0cm) {$C_2$};
\node[impc] (c1) at (6cm,0cm) {$C_1$};

\node[imp] (zinvs) at (1cm,2cm) {$Z_s^{-1}$};
\node[impc] (cs) at (1cm,3cm) {$C_s$};
\node[impc] (c5) at (5cm,3cm) {$C_5$};
\node[imp] (ze) at (3cm,5cm) {$Z_e$};

\node[imp] (zinvm) at (5cm,-2cm) {$Z_m^{-1}$};
\node[impc] (cm) at (5cm,-3cm) {$C_m$};
\node[impc] (c6) at (1cm,-3cm) {$C_6$};
\node[imp] (zh) at (3cm,-5cm) {$Z_h$};
\end{scope}
\foreach \x in {1,4}{
\draw[dotted] (-3.5cm,\x) --(3.5cm,\x) (-3.5cm,-\x) --(3.5cm,-\x);
}
\foreach \x[count=\xi] in {(0.25,2),(3,5),(0,-2),(-3,-5)}{
\node[junc] (j\xi) at \x {};
}
\draw[->] (zinvs.west) -| (c4);
\draw[->] (zinvs.east -| c4) |- (cs.west);
\draw[->] (cs-|c4) |- (ze);
\draw[->] (ze) -- (j2);
\draw[->] (j2) |- (c5);
\draw[->] (j2 |- c5) |- (j1);
\draw[->] ({{j2 |- j1}}-| c2) -- (c2.north);
\draw[->] (c1.north) -- ++(0,2mm) --(j1.-45);
\draw[->] (cs.east) -- ++(5mm,0) --(j1);
\draw[->] (j1) -- (zinvs);
\draw[->] (c5.west) -- ++(-5mm,0) --(j1);
\draw[->] (c3.north) -- ++(0,5mm) --(j1);

\draw[->] (zinvm.east) -| (c1);
\draw[->] (zinvm.east -| c1) |- (cm.east);
\draw[->] (cm-|c1) |- (zh);
\draw[->] (zh) -- (j4);
\draw[->] (j4) |- (c6);
\draw[->] (j4 |- c6) |- (j3);
\draw[->] ({{j4 |- j3}}-| c3) -- (c3.south);

\draw[->] (c4.south) -- ++(0,-2mm) --(j3.135);
\draw[->] (cm.west) -- ++(-5mm,0) --(j3);
\draw[->] (j3) -- (zinvm);
\draw[->] (c6.east) -- ++(5mm,0) -- (j3);
\draw[->] (c2.south) -- ++(0,-5mm) --(j3);

\draw[<-] (j2) -- ++(8mm,0) node[midway,above] {$f^*_e$}; 
\draw[<-] (j4) -- ++(-8mm,0) node[midway,above] {$f^*_h$};

\path (j2) ++(200:4mm) node {$\scriptstyle-$};
\foreach \x/\y in {20/+,60/-,120/-,240/+,-20/-}{
\draw (j1) ++(\x:5mm)node   {$\scriptstyle\y$};
}
\foreach \x/\y in {160/+,120/-,60/-,-60/-,200/+}{
\draw (j3) ++(\x:5mm)node   {$\scriptstyle\y$};
}
\end{tikzpicture}
\caption[Extended Lawrence Architecture]{Extended Lawrence Architecture, adapted from \cite{hashzaad1999}. 
Starred signals are the exogenous force inputs by the human/environment}%
\label{lit:syn:ela}%
\end{figure}


Instead of a such classification, one can directly start with a MIMO control structure while keeping the 
control problem formulation fixed. For this purpose, let $Y_1\subseteq \Real^{f_1}\times\Real^{f_2}$ denote the force 
measurement space, $Y_2\subseteq \Real^{p_1}\times\Real^{p_2}$ denote the position measurement space harvested from 
arbitrary number of sensors and consider the control mapping $K:Y_1\times Y_2\to \Real^{m_1}\times\Real^{m_2}$ from the 
measurements to the local and remote control actions via 
\begin{equation}
\pmatr{F_l\\F_r} = K \pmatr{x_l\\x_r\\F_{hum}\\F_{env}} 
\vcentcolon = 
\tikz[baseline=(m.center),
every left delimiter/.style={xshift=1ex},%tighter delimiter spacing
every right delimiter/.style={xshift=-1ex}]
{\matrix[matrix of math nodes,
ampersand replacement=\&,
left delimiter={[},
right delimiter={]},
nodes={minimum width=1cm}
](m){
-C_m \& |[fill=gray]| C_4 \& C_5 \& |[fill=gray]|-C_2\\ 
|[fill=gray]|C_1  \& C_s \& |[fill=gray]|C_3 \& -C_6\\
};
}
\pmatr{x_l\\x_r\\F_{hum}\\F_{env}}
\label{eq:lit:MIMOcont}
\end{equation}
where $m_1,m_2$ denote the actuation inputs with overactuated robotic manipulators in mind. The grayed entries are 
the control subcomponents that work on the variables sent over the network. If we recap the architectures above 
with this notation, they can be represented as
\[
K_{PERR} = \bmatr{
-k &k&0&0\\
k&-k&0&0
}\pmatr{x_l\\x_r\\F_{hum}\\F_{env}},
\]
\[
K_{PF} = \bmatr{
0&0&0&k_f\\
k&-k&0&0
}\pmatr{x_l\\x_r\\F_{hum}\\F_{env}},
\]
\[
K_{3\text{-channel}} = \bmatr{
0&0&0&k_{f1}\\
k&-k&0&k_{f2}
}\pmatr{x_l\\x_r\\F_{hum}\\F_{env}}
\]
respectively. Each $k_i$ represents some constant or dynamic controller. Obviously, one can 
generate many more architectures by populating different entries and the zero blocks. 

In \cite{hashzaad2002} (see also \cite{rajuverg} for a specialized treatment), the perfect transparency conditions are given for the 
undelayed case as the following; If $C_1,\cdots,C_6$ are not functions of $Z_h$ and $Z_e$, transparency is achieved if and only if the 
transparency-optimized control law

\begin{itemize}
	\item $C_1=Z_{cs}$
    \item $C_2 = 1 + C_6$
    \item $C_3 = 1 + C_5$
    \item $C_4 = -Z_{cm}$
\end{itemize}
where $Z_{cs} \coloneqq Z_e + C_m,Z_{cm} \coloneqq Z_h + C_s$ hold for nonzero $C_2,C_3$.

\subsection{Wave Variable-Scattering Transformation Control for delays}

We have discussed the transformation from a passive interconnection to small-gain interconnection via
the M\"{o}bius transformation $W$. However we have assumed that the interconnection did not involve any 
communication delays. When the delays are introduced in the loop as depicted in \Cref{lit:syn:delinc}, 
the transformations actually shift the stability problem from one domain to another. In other words, 
the transformations make the delay operators unbounded-gain as we have showed previously. Therefore, 
interconnection of passive and small-gain operators avoids to be handled by neither small-gain nor 
passivity theorems. Hence, we are left with the only option to modify the system. This technique has 
dominated the literature thanks to \cite{andersonspong,nieslotine, nieslotine2}. 

%
%\begin{figure}%
%\centering
%\begin{tikzpicture}[%scale=0.8,
%transform shape,
%every node/.style={draw,minimum size=1cm},
%every path/.style={->},>=stealth]
%\node[draw] (g) at (0,0) {$G$};
%\node[draw] (h) at (0,-2cm) {$H$};
%\node[draw] (d1) at (2cm,-1cm) {$e^{-sT}$};
%\node[draw] (d2) at (-2cm,-1cm) {$e^{-sT}$};
%\draw (h) -| (d2);\draw (d2) |- (g);
%\draw (g) -| (d1);\draw (d1) |- (h);
%\end{tikzpicture}
%\caption[A delayed interconnection]{A delayed interconnection (Negative feedback sign is included in $H$)}%
%\label{lit:syn:delinc}%
%\end{figure}

\begin{figure}%
\centering
\begin{tikzpicture}[%scale=0.7,
>=stealth,
]
\begin{scope}[transform shape,
every node/.style={draw,minimum size=1cm},
every path/.style={->},
shift={(-7cm,0cm)}
]
\node[draw] (g) at (0,0) {$G$};
\node[draw] (h) at (0,-2cm) {$-H$};
\node[draw] (d1) at (2cm,-1cm) {$e^{-sT}$};
\node[draw] (d2) at (-2cm,-1cm) {$e^{-sT}$};
\draw (h) -| (d2);\draw (d2) |- (g);
\draw (g) -| (d1);\draw (d1) |- (h);
\end{scope}

\matrix (p) at (0,-2cm) [matrix of math nodes,nodes in empty cells,draw,nodes={minimum width=8mm}] {
G &0\\
0 &\!\!\!\!\!-H\\
};
\matrix (d) at (0cm,0cm) [matrix of math nodes,nodes in empty cells,draw] {
0 &e^{-sT}\\
e^{-sT} &0\\
};
\draw[->] (d) -| ++ (2cm,-1cm) |- (p);
\draw[->] (p) -| ++ (-2cm,1cm) |- (d);
\end{tikzpicture}
\caption[A delayed interconnection]{A delayed interconnection in the input/output setting and 
block diagram as a two block interconnection.}%
\label{lit:syn:delinc}%
\end{figure}

Suppose we are given strictly passive LTI systems $G,H$ interconnected as shown in \Cref{lit:syn:delinc} on the 
left with communication delays. We, then, rewrite the interconnection as a two block interconnection as shown on 
the right for which we will use the shorthand $P-\Delta$ interconnection (delay block being the $\Delta$). Now if 
the system $P$ was strictly small-gain i.e. \(\|P\|_\infty< 1\),  thanks to the small-gain theorem, we would have 
directly conclude with stability since $\norm{\Delta}_\infty = 1$, i.e.,
\[
\bmatr{0 &e^{\iw T}\\e^{\iw T}&0}\bmatr{0 &e^{-\iw T}\\e^{-\iw T}&0}= I \quad \ \forall\omega,T 
\]
This fact also justifies why this methodology works regardless of the delays involved. However, $P$ is strictly 
passive and thus not necessarily a unity gain-bounded operator. But we have showed how to transform 
such operators into norm bounded ones. This is actually the key point of the wave variable transformations. We 
simply use the mapping $W(P)$ and obtain 
\begin{align}
W(P) = (P-I)\inv{(P+1)} &= \bmatr{G-I & 0\\0 &-H-I}\inv{\bmatr{G+I & 0\\0 &-H+I}}\\ &= \bmatr{W(G) &\\&W(-H)}
\end{align}

This constitutes as a simple justification of the common \enquote{left} and \enquote{right} scattering transformation 
of the port terminations, leaving the delay operators in the \enquote{hybrid} structure untouched in the two port network 
terminology. Note that, we have only applied the mapping $W$ to $P$ and there is no inverse mapping to \enquote{undo} 
this in the loop. Technically speaking, this is not precisely a complete loop transformation since we have applied $W$
to system $P$ but not to the delay block $\Delta$. Thus, the additional feedback and feedforward branches together with 
the $\sqrt{2}$ gains shown in \eqref{eq:wavetrafo} constitutes a genuine controller. In other words, the transformation 
block diagram is precisely achieved by the use of a feedforward/feedback control law, which can be represented by 
\Cref{eq:lit:MIMOcont}. The explicit derivation of the wave variable controller entries, in terms of a MIMO controller, 
is given in \cite{christiansson2008}. One can also verify that the control law given in \cite{andersonspong} can be 
obtained via this formulation. 

It is this very reason that the motivation often found in the literature is slightly misleading. Because, we did not 
and also could not do any modification on the delays. Quite the contrary, we have modified our system such that we can use 
the small-gain theorem to conclude stability in the face of $\Delta$. Therefore, we refrain from seeing 
this method as a passification of the communication channel. It is certainly possible to reflect the transformation 
on $\Delta$ and invoke an impedance matching argument but we think that this 
only complicates the presentation. Because it's the change in the control action that stabilizes the loop, and not the 
change in the characteristics of the delay operator or making the communication line a lossless $LC$ line.
We have to emphasize that this transformation does not guarantee stability if the original $P$ is not strictly passive.
However if any other plant $\hat{P}$ from an arbitrary class of systems $X$ can be brought into a unity norm-bounded form 
with some other transformation/control law. Then we can infer another set of physical interpretations as we would have 
$X$-ified the communication channel. Thus, if we replace $X$ with \enquote{passive} the class of systems become the passive 
system class and we declare that we have passified the communication channel. However, the communication line stays untouched 
in both cases, though the control action on the system would be different. Therefore, it is, in our opinion, better to state 
the changes done on the system instead in terms of control laws.  


Clearly, we have introduced the same conservatism by treating $\Delta$ as a gain-bounded operator 
that $\mu$-analysis approaches also utilize. Moreover, we have directly transformed the strictly passive 
operator to a small-gain operator. In case of a passivity excess, i.e., the mapped operator is confined only 
in a subregion of the unit disk, we, yet again, introduce further conservatism by using encapsulating the smaller disk 
with the unit disk, for which one can shift the disk to the origin and use the scaled small-gain theorem to reduce the 
conservatism. We should note also that, this method works for any norm-bounded linear/nonlinear $\Delta$ operator as long as the 
passivity structure is preserved and certainly not limited to delays (as in the passivity case, it's that conservative). 

Furthermore, if one shuffles the loop equations and bring $\Delta$ to a block diagonal form while the system in turn admits 
an anti-diagonal structure, it's possible to formulate a $\mu$-synthesis problem. By doing so we can recover the setup of 
\cite{leungfa}. Then, we can easily see that a wave variable control law above is in the subset of all stabilizing controllers 
set (due to the particular zero blocks in the controller structure). Thus, in terms of the conservatism involved, $\mu$-synthesis 
with constant $D$-scalings ($D=I$) covers the wave variables controller design implicitly.

It has been noted that these control algorithms are prone to position drifts due to the velocity communication and 
different alternatives have been proposed to tackle this mismatch e.g., \cite{yokokohji,chopratro06}. 
Also there are generalizations of the scattering transformations available in the literature, e.g., 
\cite{hirchebuss} to exploit the degree of freedom on the mapping $W$ using different rotation-scaling 
combinations for the unitary transformation matrix and also \cite{stramigioli} for multidimensional systems. 
Delay problems are also addressed in this context e.g., \cite{chopraberes,munirbook,nieslotine97,uedayoshikawa} 
and references therein. 



\subsection{Time-Domain Passivity Control}
In \cite{hannafordryu}, the passivity approach is formulated in the time domain and the energy exchange is literally
monitored and regulated. An initial version of this idea can also be found in \cite{yokokohji}. The basic idea is 
to see whether at any port energy is generated, using a \enquote{passivity observer} (PO): for an $N$-port 
network the observed total energy is given by
\[
E_{obsv}(n) = \sum_{k=0}^n \Delta T_k(F(k)^TV(k))
\]
where $\Delta T_k$ is the sampling period at each step with nonuniform sampling in mind. If 
$E_{obsv}(n) $ is greater than or equal to zero then the energy is dissipated by the network, conversely if
it is negative at some $k$, then the network has generated energy equal to the amount of $-E_{obsv}(n)$. Note that
this is a cumulative term and it is not implied that the energy is generated in the last step analogous to the 
integral-action control. Also, it has been shown that observing the energy flow only at the open-ports is 
sufficient to monitor the total \enquote{net} energy flow which is analogous to the observability concept in the 
linear control theory. 

The \enquote{Passivity Control} (PC) is implemented on top of this observer architecture as a virtual dissipative 
element. It relies on the passivity observer and if the energy generation is detected a dissipative element is 
introduced. The practical implementation is very similar to a safety relay circuit, i.e., it's only active when 
some relay switch is triggered. Depending on the causality (following the our analogy, as a current sensing or a 
voltage sensing relay), the PC can be implemented in series or parallel to the port. 

\begin{figure}%
\centering
\begin{tikzpicture}[contact/.style={circle,draw,fill=white,inner sep=0.5pt}]
\node[draw,minimum size=1.5cm] (n) {$N$};
\draw (n.140) -- +(-2cm,0)  (n.-140) -- +(-2cm,0);
\draw ([xshift=-1cm]n.140) -- ++(0,-2mm) node[draw,anchor=north,inner sep=2pt] (a) {$\alpha$};
\draw (a) -- ++(0,-3mm) node [shift={(-1mm,0cm)},anchor=east,rotate=90,shape=var make contact IEC,minimum width=3mm,minimum height=2mm
,draw] (s) {};
\draw (s) -- (s|- n.-140);
\node[scale=0.5,minimum height=3mm,minimum width=7mm,draw] at ([xshift=-8mm]s) (r) {$PO$};
\draw[densely dashed] (r) -- ([xshift=2mm]s.south);

\begin{scope}[xshift=5cm,decoration={markings,
mark connection node=my node,
mark=at position .5 with
{\node [draw] (my node) {$\alpha$};}}]
\node[draw,minimum size=1.5cm] (n) {$N$};
\draw[decorate] (n.155) -- +(-1cm,0) node[contact](c2){};
\draw (n.140) -- +(-1cm,0) node[contact] (c1){};
\draw  (n.-140) -- +(-2cm,0);
\draw ([xshift=-1cm]$(c1)!0.5!(c2)$)  -- ++(7mm,0) -- (c1.south);
\node[scale=0.5,minimum height=3mm,minimum width=7mm,draw] at ([shift={(-1.5mm,-3mm)}]c2) (r) {$PO$};
\draw[densely dashed] (r) -- ++(0,8mm);
\end{scope}
\end{tikzpicture}
\caption{Passivity Controller (PC) implemetations.}%
\label{lit:syn:pc}%
\end{figure}

This concept is then generalized to two-ports in \cite{ryukwonhanna,ryukwonhanna2}; also results regarding
the delay problem in the Time Domain Passivity Control context can be found in \cite{ryu}. Since the essential 
architecture is a PI controller, it also suffers from the same problems that integral-action controllers suffer 
such as, wind-up and integrator reset etc. Some of these problems are addressed in \cite{kimhannaford}. 



\subsection{Others}


We have shown that most of the proposed methods in the literature use passivity or small-gain theorems and 
the involved mathematical operators are often indistinguishable from any other physical setup that is not a 
teleoperation system. It's our belief that, without any further refinements, all the above methods can be shown 
to be, essentially, equivalent stability characterizations as far as practical implications are concerned.

There are other approaches such as Energy Bounding Algorithm (EBA), \cite{kimryu,seokim}, sliding mode control, 
\cite{parkcho,buttolo}, reset control, \cite{villaverdereset}, model predictive control \cite{bemporad,shengspong} 
and many more which we will omit here. These studies also involve model-based control techniques with varying degree 
of dependence on the model.


The proposed methods often avoid the discussion on how to tune the PID controllers for the local, remote,
and the communication line. It's as if those details are easy to handle and the attention is shifted to the $n$-channel
architecture. We have no evidence supporting this in the control design papers. Alternatively, very conservative 
upper and lower bounds are given on the parameters of individual controllers and, therefore, the underlying 
control architecture is extremely simple since otherwise the proposed methods suffer from exceeding complexity in 
the derivations. This is yet another reason for why we have chosen the IQC framework for robustness analysis and 
model-based control path in the first place. 

It's our humble opinion, however, that a \enquote{one-size-fits-for-all} design toward operator- and task-aware 
control laws without dedicated modeling and/or classification, seems very unlikely to produce a generally applicable 
results with high-performance guarantees. Nevertheless, robust control at least addresses the conservatism reduction 
in a systematic way, if there is no other way to model the teleoperation systems. Moreover, Linear Parameter Varying
controllers are to the best of our knowledge, not yet pursued to the extent the theory allows for.


We cannot claim that we have surveyed the literature in this brief and biased survey in a comprehensive manner.
The omission, as we have mentioned before, is a pragmatic choice. Though a body of research is left out (in particular
most of the nonlinear methods), the covered part also constitutes a significant volume and in the next chapter 
they are detailed further. However, we have to note that, most nonlinear methods are the counterparts of the linear 
passivity theorem methods given here and they do not introduce a different look on the modeling and analysis problem 
per se. Thus, the motivation holds for some of those nonlinear cases. 

Moreover, the number of studies reported in the literature is rapidly increasing and quite difficult to follow 
even under the guidance of at least two survey papers published recently \cite{hokayemspong,
passenberg}. We have to point out that there is still no consensus on the simplest conventions and the results are 
often very challenging to differentiate. Hence, there is a great need to have authoritative and comprehensive sources
since the survey papers above don't go into details. As a closing remark, we would like refer to the outstanding 
survey of \cite{klomp} which provides also a comprehensive outlook to the teleoperation literature in a relatively detailed 
fashion up to 2006. It should give the reader a good idea how involved and, as a subjective note, unnecessarily 
complicated the bilateral teleoperation literature is.