 Research Article
 Open Access
 Published:
Proposal of forceps force limiter design using leaf spring buckling
ROBOMECH Journal volume 9, Article number: 3 (2022)
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 \(1.5 \times 10^{1}\) 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.
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 [3,4,5]. 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 [6,7,8] 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.
Force limiter mechanism
In the prototype model of the force limiter, the ease of a leaf spring attachment is significant. However, if the structure with a leaf spring as shown in Fig. 1 is adopted, it requires linear slider mechanisms, which results in larger size for the forceps.
Let us consider the mechanism of force limiter as shown in Fig. 2. 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. 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. 2 (b)). 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.
Leaf spring buckling
Figure 3 shows the coordinate system of the leaf spring. The bending formula is given by the following expression:
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 xaxis 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 crosssectional area of the leaf spring.
Because the cross section of the leaf spring is a rectangle, I is given by
where b and t denote the width and thickness of the leaf spring, respectively.
\(M'(s)\) denotes the shearing force at s. By differentiating (1), the following equation is obtained:
where \(F_b\) denotes the buckling load.
Equation (3) is analogous to the equation of motion of a nonlinear pendulum, whereby the buckling load corresponds to the period of the pendulum. Let \(\theta _\mathrm{max}\) be \(\mathrm{Max} \theta (s)\). It is known that the period of a nonlinear pendulum is given by the complete elliptic integral of the first kind and depends on \(\theta _\mathrm{max}\) [13]. This study assumes \(\theta _\mathrm{max} \le 45\ ^{\circ }\), 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:
where l denotes the length of the leaf spring and c denotes the coefficient of fixity. From (2) and (5), 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:
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
\(L_{3i}\) and \(L_{4i}\) are determined using
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
where the initial conditions are as follows:
The boundary conditions are as follows:
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 \(t \ge t_\mathrm{min}\).

(b)
The maximum stress
Let \(\sigma (s)\) be the maximum stress at s, then \(\sigma (s)\) is given by
where Z denotes the modulus of the section, which is given by
From (1), (2), (15), and (16), \(\sigma (s)\) is calculated by the following equation that does not include M(s):
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
where
(c) The limitation of b
From (2) and (5), if t is determined, then b can be calculated by
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
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
Let \(t_{ijk}\) be t with respect to \(l_{ij}\). Hence, \(t_{ijk}\) must satisfy
where \(k=0,1 \ldots n_{ij} \ (n_{ij} < (t_{ij \beta }  t_{ij \alpha }) / \Delta t)\),
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 halfsize 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
Simulations and experiments
Specification of the force limiter
To verify the effectiveness of the proposed method, the forceps specifications were set as follows.

Referring to the maximum range of the force sensor [14] for gripping the intestine, the gripping force of the force sensor was set to \(F_\mathrm{in}=6\) N.

It was assumed that the gripping force of the distal side was equal to that of the proximal side due to the installation of a curved slider mechanism as will be described in the calibration section.
Based on the above specifications, the parameters were set and materials chosen as follows:

The values of \(l_\mathrm{max}\) and \(b_\mathrm{max}\) were set to 180 and 30 mm, respectively, as determined based on the length of the middle finger and half the length of the thumb, according to the database of Japanese hand sizes [15].

The linkage parameters were set as (\(L_1\), \(L_2\)) = (2, 60) mm.

The range of \(L_{3}\) was set as (\(L_\mathrm{3min}\), \(L_\mathrm{3max}\)) = (24, 36) mm. These values were determined based on the length of \(L_{2}\). \(\Delta x_\mathrm{in}\) was set as 8.0 mm.

Strong surgical stainless steel (SUS631, with Young’s modulus of 200 GPa, Poisson’s ratio of 0.28 and \(\sigma _\mathrm{lim}=1030\) MPa (proof stress)) was selected as the material for the leaf spring.

The minimum thickness of the commonly available plate was set as \(t_\mathrm{min}=0.1\) mm.
SolidWorks 2016 was used to conduct the FEM for the stress analysis.
Spring constant of the buckled leaf spring.
The spring constant of the buckled leaf spring was confirmed through simulations and experiments. Table 1 shows the candidates of \(L_{3}\) and \(L_{4}\). The value of l was the smallest for the 4th candidate, i.e., 50 mm. The linkage parameters were set as (\(L_{3}\), \(L_{4}\)) = (33, 27) mm. Table 2 shows the specifications of the leaf spring. Figure 6 and Table 3 show the deformed shape of the leaf spring with \(\Delta x_i = 6.5\) mm; this simulation converged to (x(l), y(l)) = (43.5, 0) mm and (\(x'(l)\), \(y'(l)\)) = (1, 0). Using the FEM, \(\sigma _\mathrm{max}=997\) MPa, which is lower than \(\sigma _\mathrm{lim}\) (see Fig. 7) . From Fig. 7, little difference of the stress was observed in the width direction.
Furthermore, the spring constant of the leaf spring was confirmed by experiments. Figure 8 shows the experimental setup for the force measurement. A force gauge (AIKOH model RZ20) was attached on a linear stage, which deformed the leaf spring. To separate the force gauge from the momentum load, the leaf spring was fixed to the arm device as shown in Fig. 8.
Figure 9 shows the result of the leaf spring buckling experiment. From the simulation results, the spring constant of the leaf spring was estimated to be \(7.9 \times 10^{2}\) N/mm, which is about twosevenths of that of a weak coil spring (\(2.8 \times 10^{1}\) N/mm for the MISUMI AWA 830 spring of outer diameter 8 mm, length 50 mm at 6.7 N, and maximum load of 8.53 N). From the experimental results, the spring constant was estimated as \(1.5 \times 10^{1}\) N/mm in the range \(0.5 \le \Delta x_i \le 6.5\) mm and \(1.0 \times 10^{1}\) N/mm in the range \(2.0 \le \Delta x_i \le 6.5\) mm. Although the experimental result was different from the simulation result, the spring constant of the buckled leaf spring was less than that of the weak coil spring.

Note 1:
This procedure adjusted the value of \(L_1\) to minimize the absolute value of the spring constant of the leaf spring.

Note 2:
In the experiments, a stage of a milling machine (TOYO ASSOCIATES: Little Milling 9) was used instead of a linear stage.
Calibration of the gripping force limitation
Figure 10 shows a prototype of the forceps with the force limiter. By using the curved slider mechanism of the proximal side, the displacement of the distal side gripper is equal to that of the proximal handle. To manufacture the prototype easily, the following material was selected:

The proximal handle was built of aluminum alloy (A5052, A6063).

The distal side gripper was built by 3D printing (Keyence: AGILISTA3200, Material: ARM2).

Slide bearings (OILES # 80 flange bush) were installed in the rotating parts.
Figure 11 shows the experimental setup for gripping force measurement. The handle of the proximal side is manually driven, and a force gauge measures the maximum force on the distal side of the gripper. This test was performed 10 times, and the average gripping force was found to be 4.0 N, which was about twothirds of the theoretical value.
The variation can be attributed to friction and lack of rigidity in the transmitting mechanism. Friction at sliders and joints decreases the gripping force. Furthermore, because of the lack of rigidity, the phase of the slider is shifted, which also decreases the gripping force.
To increase the threshold value of the gripping force, let us consider adjustment of b, after assembling the mechanism. Referring to (20), b should be set as 42 mm, which is one and a half times larger than 28 mm. Instead of adjusting b, the additional leaf spring ((l, b, t)=(50, 14, 0.1) mm) was installed in the gripper, as shown in Fig. 12. The gripping force increased to 6.1 N, which was approximately the same as the theoretical value.
Ex vivo experiments
Ex vivo experiments were performed to confirm that the developed forceps could limit the gripping force. A piece of chicken breast (mass: 24 g) was used as a dummy organ. In the preliminary experiments of chicken breast, the gripping scar was observed where the force was greater than 11.4 N. This value was about twice as large as the maximum range of the force sensor [14]. The proximal side handle was manually driven until the force limiter began to function. For comparison, a force gauge (AIKOH model RZ20) pushed the dummy organ by using the distal gripper part.
Figure 13 shows the results of the ex vivo experiments of gripping the chicken breast when the force limiter was functioning. No slips were observed during the experiments. Figure 14 shows the chicken breast after the gripping experiments. Figure 14 also shows the scars caused by the force gauge, in the preliminary experiments. No scar to the dummy organ was observed in the gripping experiments, whereas a scar was observed where the force was greater than 11.4 N.
Microscopic observations were performed to evaluate the damage to the dummy organs. The dummy organs were fixed in 4% paraformaldehyde for 10 min, embedded in 3% agar, and then frozen and sectioned in a cryostat (18 \(\upmu\)m thick, at 25\(^{\circ }\)C). Prior to microscopic observations, the dummy organs were stained with hematoxylin–eosin.
Figure 15 shows microscopic images of the dummy organ. When no force was applied, the cells were \(>0.3\) mm in length (see Fig. 15A), whereas the gripper caused damage to the tissue when the applied force was \(\ge 7.4\) N, which manifested as small voids \(<100\) \(\upmu\) m in length (see Fig. 15B–D). The tissue was minimally damaged when the force limiter was implemented (see Fig. 15E). These results indicate that the proposed method can limit the gripping force.
Discussion
Applied force at the handle
In the ex vivo experiments, \(F_\mathrm{in}\) was measured during gripping. Figure 16 illustrates the experimental setup for measuring the \(F_\mathrm{in}\). The dummy organ was placed at the distal side of the gripper, and a force gauge was used to push the movable handle. For comparison, the applied force was measured when the gripper was locked. The \(F_\mathrm{in}\) on the dummy organ was 8.68 N, whereas that of the locked gripper was 8.89 N. The small difference in the values indicates that the gripping force applied to the dummy organ was \(\sim 6.1\) N, as designed.
Design of \(F_\mathrm{in}\)
In the ex vivo experiments, little damage to the dummy organs was observed when the force limiter was applied, while some damage was observed in the absence of the limiter when the force was \(\ge 7.4\) N. These results indicate that when \(F_\mathrm{in}\) was set as 6.0 N, the proposed method prevented damage to the dummy organs. Although these results are limited to the dummy organ, it is expected that the proposed method would prevent damage to other organs if \(F_\mathrm{in}\) is properly adjusted.
Spring constant of the proposed method
The spring constant of the buckled leaf spring was estimated as \(1.5 \times 10^{1}\) N/mm in the range \(0.5 \le \Delta x \le 6.5\) mm. These results indicate that the spring constant of the proposed force limiter is less than that of a normal coil spring. Although this study examined a manual manipulator’s grip, the proposed method can be applied to the joints of a surgical manipulator.
Discrepancy of the buckling force and the spring constant
In the \(0.5 \le \Delta x \le 6.5\) mm range, the experimentally obtained spring constant was larger than that obtained via simulation. On the other hand, the buckling load was lower in the experiment than in the simulation, as shown in Fig. 9. Although these discrepancies occurred, the spring constant of the proposed method is lower than that of a weak coil spring, and the disparity in the buckling load is adjustable in Step 8.
These discrepancies can be attributed to the fixation method applied at the edges of the leaf spring. Owing to the lack of rigidity and backlash of the bearings, the angle at the edge is not always zero, as shown in Fig. 17. Both edges cannot support momentum loads when \(\Delta x\) is lower than a specific value, whereas they can support momentum loads when \(\Delta x\) is larger than a specific value, resulting in a discrepancy between the simulated and experimentally obtained spring constants. The angle at the edge increases the imaginary length of the leaf springs, decreasing the buckling load and the spring constant.
Gripping position at the handle
In the proposed mechanism, the threshold value of the force limiter depends on the gripping position of the base and the movable handle. Thus, the threshold value error arises from the gripping position error of the handles. To reduce the threshold value error, it is expected that the handle shape reduces the gripping position error. Owing to the ease of attaching the experimental tools, the developed handle shape is straight, as shown in Fig. 10. Therefore, to reduce the gripping position error, the handle shape should be improved, as shown in Fig. 2.
Limitation of \(\sigma _\mathrm{max}\)
To satisfy \(\sigma _\mathrm{max} < \sigma _\mathrm{lim}\), the base handle should employ a displacement limiter. Therefore, a locking mechanism can be used as the displacement limiter. Figure 18 shows the proposed locking mechanism, which is based on a traditional Japanese pothook, Jizaikagi [16]. If the displacement of the base handle exceeds the threshold value, the rod comes into contact with the base handle. The friction caused by this contact provides the locking mechanism, and the locking force increases with the gripping force at the handles. The authors developed and confirmed its mechanisms in experiments [16].
Conclusions
To decrease the change in the force with respect to the displacement, the absolute value of the spring constant of a force limiter should be reduced. This paper proposed the use of a leaf spring buckling mechanism as a force limiter for forceps. The advantage of this study is the ease of adjusting the designed parameters. Specifically, the use of a leaf spring allows the thickness and width to be adjusted to the stress and force independently. Based on the simulations and experiments performed, the following findings were confirmed:

From the simulation, the spring constant of the leaf spring was estimated as \(7.9 \times 10^{2}\) N/mm, which was twosevenths that of the weak coil spring (MISUMI AWA 830).

From the experiments, the spring constant of the buckled leaf spring was estimated as \(1.5 \times 10^{1}\) N/mm.

In the calibration stage, an additional leaf spring was installed in the force limiter. This caused the threshold value of the force limiter to increase from 4.0 to 6.1 N.

In the ex vivo experiments, minor damage to the dummy organ was observed with a microscope.
These results indicate that the spring constant of the proposed force limiter is lower than that of a normal coil spring and the use of leaf spring can reduce the damage caused to the dummy organ. Additionally, locking mechanisms that can be used to limit the stress in the leaf spring, were considered. Future studies will focus on improving the force limiter mechanism and performing in vivo experiments.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.
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We would like to thank Editage (www.editage.com) for English language editing.
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All work except ex vivo experiments was conducted by SN. In the ex vivo experiments, YT performed HE staining and microscopic observation. Analysis of HE staining and microscopic observation was conducted SK and TI. The other work about the ex vivo experiments was performed by SN. All authors read and approved the final manuscript.
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Noda, S., Tokuoka, Y., Kuriu, S. et al. Proposal of forceps force limiter design using leaf spring buckling. Robomech J 9, 3 (2022). https://doi.org/10.1186/s40648022002188
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DOI: https://doi.org/10.1186/s40648022002188
Keywords
 Leaf spring
 Buckling
 Forceps
 Force limiter