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 [
1] or manipulators [
2] have been developed.
One problem with these surgical instruments is the transmission of force information. Some surgical robots [
1] lose the force information to the surgeons. 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 [
3‐
5]. However, equipping forceps only with force sensors does not eliminate accidents owing to the use of excess force. Furthermore, surgeons with little experience in MIS have difficulty controlling the force acting on the instruments, even if surgical manipulators with force feedback are used.
To prevent such accidents, force limiters for forceps [
6‐
8] have been developed. If the gripping force exceeds the threshold value, the coil spring begins to deform, as shown in Fig.
1. 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.
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 [
9] 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. Previous studies have examined mechanisms that use buckling; for example, some personal computer keyboards employ buckled coil springs [
10], leaf springs are used in constant load springs [
11], and a force limiter for a toothbrush employs a leaf spring [
12]. 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 independent adjustment of its thickness and width, based on the stress and force values. The aim of this research is to confirm the principle of the gripping force limitation using leaf spring buckling.
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, as shown in Fig.
5. 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:
Step 1: Determine the range of \(L_3\) and \(L_4\).
Let
\(F_\mathrm{in}\) be the input force to the linkage and
\(\Delta x_\mathrm{in}\) be the base handle displacement in the
x direction. This step provides
\(F_\mathrm{in}\) and
\(\Delta x_\mathrm{in}\). The designed buckling load
\(F_i\) and displacement of the leaf spring
\(\Delta x_i\) are calculated using
$$\begin{aligned} F_i= & {} \frac{L_{3i}}{L_{4i}}F_\mathrm{in} \end{aligned}$$
(6)
$$\begin{aligned} \Delta x_i= & {} \frac{L_{4i}}{L_{3i}} \Delta x_\mathrm{in}. \end{aligned}$$
(7)
\(L_{3i}\) and
\(L_{4i}\) are determined using
$$\begin{aligned} L_{3i}= & {} L_\mathrm{3min} + \frac{L_\mathrm{3max} - L_\mathrm{3min}}{m_1} i \end{aligned}$$
(8)
$$\begin{aligned} L_{4i}= & {} L_2 - L_{3i}, \end{aligned}$$
(9)
where
\(i=0,\ 1 \ldots m_1\). In this range, the value of
\(L_3\) that minimized
l is determined.
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 \(l_i > \Delta x_i\). \(l_\mathrm{max}\) denotes the maximum length, which is limited by the handle size. Thus, \(l_i \le l_\mathrm{max}\). Hence, l must be sampled in the range \(\Delta x_i < l \le l_\mathrm{max}\). Let \(l_{ij}\) be the jth sample of \(l_i\). Then, \(l_{ij} = \Delta x_i + \frac{l_\mathrm{max} - \Delta x_i}{m_2}j\), where \(j=0,1 \ldots m_2\). In Step 2, \(j=0\).
Step 3: Simulate the deformed shape.
The linear constitutive relationship is given by
$$\begin{aligned} \left[ \begin{array}{c} x'(s) \\ y'(s) \\ \end{array} \right] = \left[ \begin{array}{c} \cos \theta (s) \\ \sin \theta (s) \\ \end{array} \right] , \end{aligned}$$
(10)
where the initial conditions are as follows:
$$\begin{aligned} (x(0),\ y(0))= & {} (0,\ 0) \end{aligned}$$
(11)
$$\begin{aligned} \theta (0)\,=\, & {} \mathrm{atan2}(y'(0), x'(0)) \ \end{aligned}$$
(12)
$$\begin{aligned} (x'(0),\ y'(0))= & {} (1,\ 0). \end{aligned}$$
(13)
The boundary conditions are as follows:
$$\begin{aligned} (x(l_{ij}),\ y(l_{ij}))= & {} (l_{ij}-\Delta x_i,\ 0) \end{aligned}$$
(14)
Equations (
3) and (
10) 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:
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 \(A_{ij}\).
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.
Step 4: Determine the range of t at \(l_{ij}\).
The constraints on
t are described as follows:
(a)
The lower limit of t Let \(t_\mathrm{min}\) be the lower limit of t and \(t \ge t_\mathrm{min}\).
Let
\(\sigma (s)\) be the maximum stress at
s, then
\(|\sigma (s)|\) is given by
$$\begin{aligned} |\sigma (s)| = \frac{|M(s)|}{Z}, \end{aligned}$$
(15)
where
Z denotes the modulus of the section, which is given by
$$\begin{aligned} Z=\frac{bt^2}{6}. \end{aligned}$$
(16)
From (
1), (
2), (
15), and (
16),
\(\sigma (s)\) is calculated by the following equation that does not include
M(
s):
$$\begin{aligned} |\sigma (s)| = \frac{|\kappa (s)|Et}{2}. \end{aligned}$$
(17)
According to (
17),
\(\sigma (s)\) does not depend on
b but depends on
\(\kappa (s)\),
E, and
t. Let
\(\sigma _\mathrm{lim}\) be the proof stress. Because
\(|\sigma (s)| \le \sigma _\mathrm{lim}\),
t must satisfy
$$\begin{aligned} t \le \frac{2\sigma _\mathrm{lim}}{ E \kappa _\mathrm{max}}, \end{aligned}$$
(18)
where
$$\begin{aligned} \kappa _\mathrm{max} = \mathrm {Max} | \kappa (s) |. \end{aligned}$$
(19)
(c) The limitation of
b
From (
2) and (
5), if
t is determined, then
b can be calculated by
$$\begin{aligned} b=\frac{12l_{ij}^2 F_i}{c \pi ^2 E t^3}. \end{aligned}$$
(20)
Let
\(b_\mathrm{max}\) be the maximum
b, which is determined by the gripper size. Then
b must satisfy
\(b \le b_\mathrm{max}\), thus
$$\begin{aligned} t \ge \root 3 \of {\frac{12l_{ij}^2F_i}{c \pi ^2 b_\mathrm{max} E}} . \end{aligned}$$
(21)
If
\(t > b\), then (
20) cannot be used to determine
b because the buckling direction in this case is orthogonal to the direction shown in Fig.
2. Hence, to satisfy
\(t \le b\),
t must satisfy
$$\begin{aligned} t \le \sqrt[4]{{\frac{12l_{ij}^2F_i}{c \pi ^2 E}}}. \end{aligned}$$
(22)
Let
\(t_{ijk}\) be
t with respect to
\(l_{ij}\). Hence,
\(t_{ijk}\) must satisfy
$$\begin{aligned} t_{ij\alpha }\le t_{ijk} \le t_{ij\beta }, \end{aligned}$$
(23)
where
\(k=0,1 \ldots n_{ij} \ (n_{ij} < (t_{ij \beta } - t_{ij \alpha }) / \Delta t)\),
$$\begin{aligned} t_{ijk}= & {} t_{ij \beta } - k \Delta t \end{aligned}$$
(24)
$$\begin{aligned} t_{ij \alpha }= & {} \mathrm {max} \left( t_\mathrm{min}, \sqrt[3]{{\frac{12l_{ij}^2F_i}{c \pi ^2 b_\mathrm{max} E}}} \right) \end{aligned}$$
(25)
$$\begin{aligned} t_{ij \beta }= & {} \mathrm {min} \left( \frac{2\sigma _\mathrm{lim}}{E \kappa _\mathrm{max}},\ \sqrt[4]{{\frac{12l_{ij}^2F_i}{c \pi ^2 E}}} \right) . \end{aligned}$$
(26)
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:
(a)
If \(j<m_2\), the procedure increases j by one and returns to Step 3.
(b)
If \(j=m_2\), the procedure excludes ith candidate.
Step 5: Determine
\(b_{ijk}\).
Let
\(b_{ijk}\) be
b at
\(t_{ijk}\), then
\(b_{ijk}\) is calculated from (
20).
Step 6: Final stress confirmation.
The maximum
\(|\sigma (s)|\) obtained by (
17) is a little less than that obtained by the finite element method (FEM). Therefore, the final stress confirmation is conducted by the FEM. Let
\(\sigma _\mathrm{max}\) be the maximum stress obtained by the FEM. Because the leaf spring buckles symmetrically, the half-size analysis can decrease the calculation time. If
\(\sigma _\mathrm{max} \le \sigma _\mathrm{lim}\), then the leaf spring design is completed. If
\(\sigma _\mathrm{max} > \sigma _\mathrm{lim}\), then the procedure is described as follows:
(a)
If \(k<n_{ij}\), the procedure increases k by one and returns to Step 5.
(b)
If \(k=n_{ij}\) and \(j<m_2\), the procedure sets \(k=0\), which increases j by one and returns to Step 3.
(c)
If \(k=n_{ij}\) and \(j=m_2\), the procedure excludes ith candidate.
Step 7: Candidate selection.
In this step, the minimum value of \(l_{ij}\) is selected. If there are no candidates of \(l_{ij}\) remaining, then the procedure returns to Step 1 and the values of \(F_\mathrm{in}\) and \(\Delta x_\mathrm{in}\) are modified.
Step 8: Final adjustment of b.
Let
\(F_\mathrm{gripreal}\) and
\(F_\mathrm{gripreal}\) be the designed and real gripping forces at the distal side, respectively. If
\(|F_\mathrm{gripdes} - F_\mathrm{gripreal}|> \varepsilon\), then the process must be repeated from Step 1 using modified
\(F_\mathrm{in}\) and
\(\Delta x_\mathrm{in}\) values. However, this would require remanufacturing the base handle, which is impractical. To avoid this, it is important to adjust the value of
b. Let
\(\Delta b\) and
\(\Delta F_\mathrm{in}\) be the differences between
b and
\(F_\mathrm{in}\) values respectively. This study assumes
\((F_\mathrm{gripdes} - F_\mathrm{gripreal}) \propto \Delta F_\mathrm{in}\), and
\(\Delta b\) is given by
$$\begin{aligned} \Delta b = \frac{12l_{ij}^2 }{c \pi ^2 E t^3} (F_\mathrm{gripdes} - F_\mathrm{gripreal}). \end{aligned}$$
(27)
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