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\dochead{Research}
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\title{Development of a Forceps Force Limiter Using Leaf Spring Buckling}
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\author[
addressref={sn}, % id's of addresses, e.g. {aff1,aff2}
corref={sn}, % id of corresponding address, if any
noteref={n1}, % id's of article notes, if any
email={snoda@fukushima-nct.ac.jp} % email address
]{\inits{SN}\fnm{Satsuya} \snm{Noda}}
\begin{comment}
\author[
addressref={aff1,aff2},
email={john.RS.Smith@cambridge.co.uk}
]{\inits{JRS}\fnm{John RS} \snm{Smith}}
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\address[id=sn]{% % unique id
\orgname{Department of Mechanical System Engineering, National Institute of Technology, Fukushima College}, % university, etc
\street{30 Tairakamiarakawa-aza-Nagao, Iwaki}, %
\postcode{970-8034} % post or zip code
\city{Fukushima}, % city
\cny{Japan} % country
}
\begin{comment}
\address[id=aff2]{%
\orgname{Department of Mechanical System Engineering, National Institute of Technology, Fukushima College},
\street{D\"{u}sternbrooker Weg 20},
\postcode{24105}
\city{Kiel},
\cny{Germany}
}
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\begin{abstractbox}
\begin{abstract} % abstract
To prevent accidents in minimally invasive surgeries, force limiters have been developed for forceps grippers.
When a force limiter is in use, if the absolute value of its spring constant is reduced, the risk of damage to the organs decreases.
This paper proposes the use of a leaf spring buckling mechanism as a force limiter for forceps.
The results obtained indicate that the spring constant of a buckled leaf spring is lower than that of a normal coil spring.
Furthermore, the use of a leaf spring allows the independent adjustment of its thickness and width, based on the stress and force values.
This enables an easy calibration of the threshold value.
In the experiments, the spring constant of the buckled leaf spring was $1.5 \times 10^{-1}$~\si{N/mm}, which is half of that of a normal coil spring.
After calibrating the gripping force, it was confirmed that the force limiter reduced the extent of damage to the dummy organs in the ex vivo experiments.
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\begin{keyword}
\kwd{\addpart{Buckling}}
\kwd{\addpart{Forceps}}
\kwd{\addpart{Force limiter}}
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%% Background %%
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\section*{Introduction}
An advantage of minimally invasive surgery (MIS) is the reduction of damage to the patient after the surgery; however, this type of surgery requires difficult techniques by surgeons as compared to conventional surgery.
The degree of freedom in general surgical instruments decreases from six to four because these instruments are inserted into trocars.
Hence, surgical robots~\cite{bodner2004first} or manipulators~\cite{frede2007radius} have been developed.
One problem with these surgical instruments is the transmission of force information.
Some surgical robots~\cite{bodner2004first} lose the force information to the surgeons.
Even though surgical manipulators with force feedback are used, surgeons with little experience in MIS have difficulty controlling the force acting on the instruments.
Applying excess force in gripping or exclusion might cause serious damage to the organs.
To prevent such accidents, it is generally effective to measure the force acting on the forceps; thus, many force sensors have been developed for forceps~\cite{peirs2004micro, puangmali2012miniature, 998943}.
However, equipping forceps only with force sensors does not eliminate accidents owing to the use of excess force.
To prevent such accidents, force limiters for forceps~\cite{heimberger1996forceps, sakaguchi2018development, michelini2017grasper} have been developed.
If the gripping force exceeds the threshold value, the coil spring begins to deform, as shown in Fig. \ref{fig:buckling_force_009}.
Because a normal force limiter employs a linear coil spring, the gripping force increases proportionally to the handle displacement, which might cause damage to the gripped organ.
On the other hand, if the spring force decreases with the displacement, the gripping force decreases, which might cause the organ to slip from the gripper.
Therefore, to decrease the change in force with respect to the displacement, the absolute value of the spring constant of a force limiter should be low.
\delpart{If the number of windings of a coil spring increases, the spring constant will decrease. }
\addpart{To reduce the spring constant without changing the threshold value of the force limiter, the number of windings of a coil spring should be increased. }
However, such springs will require more length and installing them for the gripper will become difficult.
Although a constant load spring or a coil spring with a noncircular pulley~\cite{gendo2010pulley} can be employed to reduce the spring constant, the size of this spring will pose difficulties in attaching it to the gripper.
This paper proposes the use of a leaf spring buckling mechanism as a force limiter for forceps.
The spring constant of the buckled leaf spring is lower than that of a normal coil spring.
%Moreover, the force limiter employs a linkage mechanism such that the spring constant will not increase.
Previous studies have examined mechanisms that use buckling; for example, some personal computer keyboards employ buckled coil springs~\cite{harris1978buckling}, leaf springs are used in constant load springs~\cite{lan2010compliant}, and a force limiter for a toothbrush employs a leaf spring~\cite{slocum2019buckling}.
Compared to previous studies, the advantage of this study is the ease of adjusting the design parameters.
%The use of a leaf spring allows the thickness and width to be adjusted to the stress and force independently.
The use of a leaf spring allows the independent adjustment of its thickness and width, based on the stress and force values. %\cite{koon,oreg,khar,zvai,xjon,schn,pond,smith,marg,hunn,advi,koha,mouse}
\section*{Force limiter mechanism}
Figure \ref{fig:fs_ref_001} shows the mechanism of the proposed force limiter.
The handle of the force limiter consists of base and movable handles.
Generally, the movable handle is used for gripping, whereas the base handle does not move.
If the gripper of the distal side grasps an organ and the movable handle rotates in the gripping direction, the gripping force increases.
Hence, the gripping force will not increase if the gripper of the distal side does not rotate.
\addpart{In the prototype model of the force limiter, the ease of a leaf spring attachment is significant.
Thus, }this study proposes the base handle mechanism that moves only when the gripping force exceeds the threshold value.
If the base handle can be driven by a constant force, the gripping force will not increase (see Fig. \ref{fig:fs_ref_001} (b)).
\addpart{The use of the leaf spring buckling enables to prevent the gripping force increase.
Furthermore, the linkage mechanism of the base handle also assists in preventing the gripping force increase.}
\subsection*{Leaf spring buckling}
Figure \ref{fig:buckling_cordinate_001} shows the coordinate system of the leaf spring.
The bending formula is given by the following expression:
\begin{eqnarray}
\kappa(s) =\theta' (s) = - \frac{M(s)}{EI} ,
\label{eq:bend_eq}
\end{eqnarray}
where $s$ is the arc length from the edge, $x(s)$ and $y(s)$ indicate the position of the leaf spring at $s$, $\theta(s)$ is the angle of the tangential line from the $x$-axis at $s$, $\ '$ denotes the derivative with respect to $s$, $\kappa(s)$ is the curvature at $s$, $M(s)$ is the bending moment at $s$, $E$ is the Young's modulus of the leaf spring, and $I$ is the moment of inertia of the cross-sectional area of the leaf spring. \\
Because the cross section of the leaf spring is a rectangle, $I$ is given by
\begin{eqnarray}
I=\frac{bt^3}{12},
\label{eq:I}
\end{eqnarray}
where $b$ and $t$ denote the width and thickness of the leaf spring, respectively.
$M'(s)$ denotes the shearing force at $s$.
By differentiating (\ref{eq:bend_eq}), the following equation is obtained:
\begin{eqnarray}
\frac{d^2 \theta(s)}{ds^2} &=& - A \sin \theta
\label{eq:swang_eq}
\\
A &=& \frac{F_b}{EI},
\label{eq:deg_a}
\end{eqnarray}
where $F_b$ denotes the buckling load.
Equation (\ref{eq:swang_eq}) is analogous to the equation of motion of a nonlinear pendulum, whereby the buckling load corresponds to the period of the pendulum.
Let $\theta_{\rm max}$ be ${\rm Max} |\theta(s)|$.
It is known that the period of a nonlinear pendulum is given by the complete elliptic integral of the first kind and depends on $\theta_{\rm max}$~\cite{belendez2007exact}.
This study assumes $\theta_{\rm max} \leq 45\ ^{\circ}$, as described in the Simulation section.
In this range, the period error of a nonlinear pendulum is less than 5 $\%$ of that of a linear pendulum.
Similarly, the buckling load error of a nonlinear deformation is less than 10 $\%$ of that of a linear deformation.
The buckling load of a linear deformation is given by the following equation:
\begin{eqnarray}
F_b=c \pi^2 \frac{EI}{l^2},
\label{eq:buckle_begin}
\end{eqnarray}
where $l$ denotes the length of the leaf spring and $c$ denotes the coefficient of fixity.
From (\ref{eq:I}) and (\ref{eq:buckle_begin}), it is obvious that $b$ can be used to adjust $F_b$ because $F_b \propto I \propto b$.
The coefficient $c$ is determined as follows: \\
\noindent (a) $c=1$, if both edges cannot support momentum loads (see Fig. \ref{fig:spring_edge} (A)). \\
\noindent (b) $c=4$, if both edges can support momentum loads (see Fig. \ref{fig:spring_edge} (B)). \\
Because the proposed force limiter can support the moment of the buckling direction, as shown in Fig. \ref{fig:fs_ref_001}, the procedure for this study sets $c=4$.
\section*{Design procedure for the force limiter mechanism }
Because the spring constant of the buckled leaf spring is lower than that of a normal coil spring, it is difficult to analyze the stress produced by the load. The analysis of this stress requires the calculation of the leaf spring deformation.
Therefore, the parameters are determined in the order of $l$, $t$, and $b$.
Considering the handle size, the parameters were set so as to minimize $l$.
A detailed description of each step in the design of the leaf spring is given below:
\\
\noindent Step 1: Determine the range of $L_3$ and $L_4$.
Let $F_{\rm in}$ be the input force to the linkage and $\Delta x_{\rm in}$ be the base handle displacement in the $x$ direction.
This step provides $F_{\rm in}$ and $\Delta x_{\rm in}$.
The designed buckling load $F_i$ and displacement of the leaf spring $\Delta x_i$ are calculated using
\begin{eqnarray}
F_i&=&\frac{L_{3i}}{L_{4i}}F_{\rm in}\\
\Delta x_i&=&\frac{L_{4i}}{L_{3i}} \Delta x_{\rm in}.
\label{eq:f_fin}
\end{eqnarray}
$L_{3i}$ and $L_{4i}$ are determined using
\begin{eqnarray}
L_{3i} &=& L_{\rm 3min} + \frac{L_{\rm 3max} - L_{\rm 3min}}{m_1} i\\
L_{4i} &=& L_2 - L_{3i},
\end{eqnarray}
where $i=0,\ 1 \ldots m_1$.
In this range, the value of $L_3$ that minimized $l$ is determined. \\
\noindent Step 2: Determine the range of $l$.
Let $l_i$ be the value of $l$ that corresponds to $L_{3i}$ and $L_{4i}$.
From the leaf spring length limitation, $l_i$ must satisfy $l_i > \Delta x_i$.
$l_{\rm max}$ denotes the maximum length, which is limited by the handle size. Thus, $l_i \leq l_{\rm max}$.
Hence, $l$ must be sampled in the range $\Delta x_i < l \leq l_{\rm max}$.
Let $l_{ij}$ be the $j$th sample of $l_i$. Then,
$l_{ij} = \Delta x_i + \frac{l_{\rm max} - \Delta x_i}{m_2}j$, where $j=0,1 \ldots m_2$.
In Step 2, $j=0$. \\
\noindent Step 3: Simulate the deformed shape.
The linear constitutive relationship is given by
\begin{eqnarray}
\left[
\begin{array}{c}
x'(s) \\
y'(s) \\
\end{array}
\right]
=
\left[
\begin{array}{c}
\cos \theta (s) \\
\sin \theta (s) \\
\end{array}
\right],
\label{eq:frennet}
\end{eqnarray}
where the initial conditions are as follows:
\begin{eqnarray}
(x(0),\ y(0))&=&(0,\ 0)\\
\theta (0) &=& {\rm atan2}(y'(0), x'(0)) \\
(x'(0),\ y'(0))&=&(1,\ 0).
\end{eqnarray}
The boundary conditions are as follows:
\begin{eqnarray}
(x(l_{ij}),\ y(l_{ij}))&=&(l_{ij}-\Delta x_i,\ 0)
\end{eqnarray}
Equations (\ref{eq:swang_eq}) and (\ref{eq:frennet}) are solved by a numerical method, such as the Runge--Kutta method.
Let $A_{ij}$ be $A$ with respect to $l_{ij}$.
To satisfy the boundary conditions, these calculations are nested within the adjustment procedure loop of $A_{ij}$ and $\kappa (0)$.
To increase the convergence precision, the adjustment procedure is described as follows: \\
\noindent Loop 1: Let $\varepsilon$ be the allowable error of the edge position.
The procedure adjusts $A_{ij}$, which satisfies $|y(l_{ij})|<\varepsilon$. An optimization process is used to determine $A_{ij}$. \\
\noindent Loop 2: The procedure adjusts $\kappa (0)$, which satisfies $|x(l_{ij})-(l_{ij} - \Delta x_i)|<\varepsilon$. If $\kappa(0)$ is updated, then Loop 1 is performed. \\
\noindent Step 4: Determine the range of $t$ at $l_{ij}$.
The constraints on $t$ are described as follows:
\noindent (a) The lower limit of $t$
Let $t_{\rm min}$ be the lower limit of $t$ and $t \geq t_{\rm min}$.
\noindent (b) The maximum stress
Let $\sigma(s)$ be the maximum stress at $s$, then $|\sigma(s)|$ is given by
\begin{equation}
|\sigma(s)| = \frac{|M(s)|}{Z},
\label{eq:sigma}
\end{equation}
where $Z$ denotes the modulus of the section, which is given by
\begin{eqnarray}
Z=\frac{bt^2}{6}.
\label{eq:z_lect}
\end{eqnarray}
From (\ref{eq:bend_eq}), (\ref{eq:I}), (\ref{eq:sigma}), and (\ref{eq:z_lect}), $\sigma(s)$ is calculated by the following equation that does not include $M(s)$:
\begin{equation}
|\sigma(s)| = \frac{|\kappa(s)|Et}{2}.
\label{eq:det_sigmas}
\end{equation}
According to (\ref{eq:det_sigmas}), $\sigma(s)$ does not depend on $b$ but depends on $\kappa(s)$, $E$, and $t$.
Let $\sigma_{\rm lim}$ be the proof stress.
Because $|\sigma(s)| \leq \sigma_{\rm lim}$, $t$ must satisfy
\begin{equation}
t \leq \frac{2\sigma_{\rm lim}}{ E \kappa_{\rm max}},
\label{eq:t_sigma}
\end{equation}
where
\begin{equation}
\kappa_{\rm max} = \mathrm{Max} | \kappa(s) |.
\label{eq:def_k_max}
\end{equation}
\noindent (c) The limitation of $b$
From (\ref{eq:I}) and (\ref{eq:buckle_begin}), if $t$ is determined, then $b$ can be calculated by
\begin{equation}
b=\frac{12l_{ij}^2 F_i}{c \pi^2 E t^3}.
\label{eq:det_b}
\end{equation}
Let $b_{\rm max}$ be the maximum $b$, which is determined by the gripper size.
Then $b$ must satisfy $b \leq b_{\rm max}$, thus
\begin{equation}
t \geq \sqrt[3]{\frac{12l_{ij}^2F_i}{c \pi^2 b_{\rm max} E}} .
\label{eq:t_b_lim}
\end{equation}
If $t > b$, then (\ref{eq:det_b}) cannot be used to determine $b$ because the buckling direction in this case is orthogonal to the direction shown in Fig. \ref{fig:fs_ref_001}.
Hence, to satisfy $t \leq b$, $t$ must satisfy
\begin{equation}
t \leq \sqrt[4]{\frac{12l_{ij}^2F_i}{c \pi^2 E}}.
\label{eq:t_b}
\end{equation}
Let $t_{ijk}$ be $t$ with respect to $l_{ij}$.
Hence, $t_{ijk}$ must satisfy
\begin{equation}
t_{ij\alpha}\leq t_{ijk} \leq t_{ij\beta},
\label{eq:det_t}
\end{equation}
where $k=0,1 \ldots n_{ij} \ (n_{ij} < (t_{ij \beta} - t_{ij \alpha}) / \Delta t)$,
\begin{eqnarray}
t_{ijk}& =& t_{ij \beta} - k \Delta t \\
t_{ij \alpha}&=&\mathrm{max} \left( t_{\rm min},\ \sqrt[3]{\frac{12l_{ij}^2F_i}{c \pi^2 b_{\rm max} E}} \right)
\label{eq:det_ta}\\
t_{ij \beta}&=&\mathrm{min} \left( \frac{2\sigma_{\rm lim}}{E \kappa_{\rm max}},\ \sqrt[4]{\frac{12l_{ij}^2F_i}{c \pi^2 E}} \right).
\label{eq:det_tb}
\end{eqnarray}
The procedure sets $\Delta t = 0.1$ mm, which is the thickness of a commonly available plate.
In Step 3, $k=0$.
If $t_{ij \alpha} > t_{ij \beta}$, then the procedure is described as follows: \\
\noindent (a) If $j \sigma_{\rm lim}$, then the procedure is described as follows: \\
\noindent (a) If $k