A New Planar Overconstrained Mechanism Generated by Merging Two Symmetric Watt’s Six-Bar Linkages that Perform Gear-Like Motion

In recent years, the concept of a foldable smartphone, commonly referred to as a foldable phone, has gained substantial traction in the cellphone industry. In November 2018, the Chinese startup Royole made headlines by releasing the first commercially available foldable smartphone featuring an OLED display—the Royole Flexpai. Subsequently, in Feb. 2019, Samsung officially introduced the Galaxy Fold during its media event at the Mobile World Congress. Following these groundbreaking releases, numerous companies, including Motorola, Huawei, Xiaomi, OPPO, Vivo, Honor, and Google, have entered the foldable phone market, offering a variety of products now accessible in stores.

However, a critical component of foldable phone technology is its mechanical hinge, primarily comprising intricate mechanisms. Consequently, a surge in the number of foldable phone hinge-related patents has been observed since 2015, with notable companies such as Apple, Microsoft, Amphenol, LG, and those previously mentioned in the list of foldable phone manufacturers staking their claims. This underscores the paramount importance of the folding hinge in the evolving landscape of mobile technology.

The required folding motion can be easily achieved using a gear pair with a ratio equal to $−1$⁠. Due to space constraints, some companies select for four small gears to achieve synchronized folding motion, as seen in examples from Samsung [1] and OPPO [2], as shown in Figs. 1 and 2. From a mechanistic standpoint, gears may be the simplest components capable of executing the folding motion. However, as companies strive to minimize the size of the phone, issues related to strength emerge with small-sized gears. Consequently, Fine Technix, one of the hinge providers, proposed an alternative way by employing cam and slider mechanisms as replacements for gears in the product of Samsung Flip 4 and Fold 4 models, as shown in Fig. 3. This mechanism can be conceptualized as two cylindrical cams sharing a common sliding follower. Therefore, designing a hinge mechanism that is both robust and enduring remains an ongoing design challenge to be addressed.

While analyzing hinge mechanisms from numerous patents and providing feedback to the design team of the collaborative hinge-related company, a particular mechanism from patent CN 114584638A raised significant concerns (Fig. 4). Based on the description of each part and kinematic pair in patent CN 114584638A [4], a portion of the folding hinge is illustrated in Fig. 5 as a schematic kinematic diagram [5]. The numbers shown in Fig. 4 are the part numbers labeled in the patent, and we identify each rigid body and relabel them with link numbers (⁠$Li$⁠) in parentheses in Fig. 5.

Because the writing style of patents differs from that of journal papers, they do not explicitly describe how each component works as a mechanism and dimensions are also unknown. It also require several figures to explain how the hinge works, which is not the focus of this article. Hence, we carefully read each description and interpreted the mechanism into the kinematic diagram shown in Fig. 5. For further details, one may refer to the patent [4].

A zero-degree-of-freedom mechanism is not movable and is definitely not functional as a folding hinge. It is strange that the company claims such mechanism. Despite the possibility that the mechanism described in the patent may be nonfunctional or inaccurately documented, the company claimed that the design offers stable benefits. This anomaly raised our curiosity: Could it potentially be an overconstrained mechanism? If so, this could be an interesting kinematic problem that no one has discussed earlier.

Upon further investigation, it was discovered that Simionescu and Smith [6] proposed an overconstrained mechanism consisting of 9 links and 12 class I joints by combining two Watt II linkages. However, their approach involved utilizing function cognates to merge two 7R Watt II linkages or two 3RT3R Watt II linkages, which requires scaling and rotational method. These mechanism configurations differ significantly from the symmetric configuration shown in Fig. 5. Nevertheless, their methodology illuminates the potential feasibility of the overconstrained mechanism depicted in the patent.

In mechanism science, determining a mechanism’s degrees-of-freedom is crucial. The Chebychev–Grübler–Kutzbach criterion [7,8] is widely used for this purpose, assessing mobility without considering geometry. However, it falters with overconstrained mechanisms, where unique geometry challenges the criterion’s applicability, leading to mobility equation inconsistencies.

Famous spatial overconstrained mechanisms are identified by scientists around the 20th century, encompass Sarrus [9], Bennett [10,11], Bricard [12], Myard [13,14], Goldberg [15], Franke [16], Harrisberger [17], Waldron [18], Schatz [19], Hon-Cheung and Baker [20,21], Wohlhart [22], Lee and Yan [23], Mavroidis and Roth [24,25], as well as Song and Chen [26,27].

Some scientists provided analytical approaches or modular approach to discover more overconstrained mechanism, such as Pamidi et al. [28], Baker [29–31], Huang and Sun [32], Dai et al. [33], Li and Schicho [34], Ye et al. [35], Pfurner et al. [36], Zhang and Dai [37], Lu et al. [38–40], Ramadoss et al. [41], and Hsu et al. [42–44]. These studies emphasize the importance of implementing a systematic methodology within the design process to mitigate the risk of designers overlooking the presence of mechanisms that deviate from adherence to the Grübler–Kutzbach criterion. However, a large amount of the research focused on finding spatial overconstrained mechanisms that ignored special geometry for planar overconstrained mechanisms that may be industrially useful.

Dijksman [45] proposed drawing pole configurations to find planar overconstrained linkages. He demonstrated the method through Hart’s straight-line mechanism, Kempe’s focal linkage, and the linkage with its Desargues’ pole configuration that permanently upheld. Simionescu and Smith [6] proposed a method to generate overconstrained mechanisms by merging two Watt II function cognates (two different Watt II linkages have identical input–out relationships). The Watt II linkage could be either 7R or 3RT3R (a prismatic pair connect with the ground link). Simionescu and Smith [46] later proposed the overconstrained mechanism generated by either two Stephenson II or two Stephenson III cognates. They took advantage of the cognates that generate identical coupler curves to reduce the kinematic pair constraints on the coupler link. The concepts of using function or coupler curve cognates to find overconstrained mechanism inspired the research of this article, and it may extend to find more overconstrained mechanism with similar concepts.

In Fig. 5, with $L1$ as ground link, one may consider the nine-bar linkage separately into two Watt’s six-bar linkage: one with links (123456) and one with links (167892). The two Watt’s six-bar linkage share link 2 and link 6 and their link dimensions are symmetrical with the $y$-axis. However, the two linkage would typically exhibit distinct input–output relationships, resulting in a combined mechanism with zero mobility. That is why the mobility of the mechanism in Eq. (1) is zero. Nevertheless, within a certain range, if a special case could make the input–output relationships of link 2 and link 6 identical for both Watt’s six-bar linkages, they will undergo synchronized motion, allowing the merged mechanism to move with a single degree-of-freedom. To make it simple, for the first Watt’s six-bar linkage (123456), link 2 is the input link and link 6 is the output link. Symmetrically, for the second Watt’s linkage (167892), link 6 and link 2 are the input and output link, respectively. To have single degree-of-freedom, the function between link 2 and link 6 must be reversely identical. Therefore, the only possibility is when link 2 and link 6 have gear-like folding motion with $velocity ratio=−1$⁠.

Considering the links (123456) in Fig. 5 as a Watt’s six-bar linkage and depict it in Fig. 6. One may see that $L1$ and $L3$ are ternary links; $L2$⁠, $L5$⁠, and $L6$ are binary links; and $L3$ is a slider. In the context of the Watt’s six-bar linkage depicted in Fig. 6, if both link 2 and link 6 demonstrate this gear-like folding motion with velocity $ratio=−1$⁠, their input–output relationship becomes (⁠$ωout/ωin$$=−1$⁠), rendering it reversible (⁠$ωin/ωout=−1$⁠). Subsequently, by ensuring that the second Watt’s six-bar linkage (167892) possesses symmetrical dimensions with respect to the midline of L1, the merged mechanism in Fig. 5 would become an overconstrained mechanism. It is worth noting that the condition discussed in this paragraph is sufficient but not necessary because the symmetrical condition was added.

Given this condition, if one can synthesize a Watt’s six-bar linkage where (⁠$ωout/ωin=−1$⁠) holds within a specified range, the nine-bar overconstrained mechanism can be synthesized. In the context of the folding hinge application, this relationship should ideally be satisfied over a rotational span of at least 90 deg.

Despite an extensive search through related literature, no articles were found discussing such six-bar or related linkages. Besides, in the domain of exact function generation synthesis, it has been established that up to five precision positions can be synthesized for a four-bar linkage [47] and up to 11 precision positions for a six-bar linkage [48,49]. Because of the limit, this article tries to use optimization method to see if there exist such special Watt’s six-bar linkages.

Because the analytical approach of synthesizing linkages have precision positions limit. With the technology development, using optimization method to synthesize different mechanisms become significant research topics in the field. Kelaiaia et al. [50,51] reviewed 128 articles about optimal design for parallel manipulators with different objectives. They also proposed a design method for parallel manipulator optimization. Kang et al. [52] compared the accuracy and efficiency of different metaheuristic optimization algorithms for path generation problems. Based on their review, there are two main categories of optimization algorithm. One is the traditional gradient-based algorithm, for example, the gradient method [53–55], Guass–Newton method [56], tabu-gradient method [57], and ant-gradient method [58]. Another category is the population-based metaheuristic global optimization algorithm, which can be subdivided into two subcategories. First is the swarm intelligence-based algorithms, such as particle swarm optimization [59], cuckoo search [60], krill herd [61], and their improved variants. Second is the evolutionary-based algorithms inspired by biological evolution, such as the differential evolution (DE) algorithm [62], genetic algorithm [63], teaching learning-based optimization algorithm [64], and their improved variants.

It is well known that different algorithms have varying strengths and are suitable for different applications. Previous studies indicate that DE and its variants, such as EPSDE [65] and L-EPSDE [52], demonstrate the best performance for linkage synthesis problems. In addition, python offers a readily available package for differential evolution, which reduces technical barriers. Consequently, this article aims to use the differential evolution algorithm to synthesize a six-bar linkage that exhibits a gear-like input–output motion.

This section gives the setting of how to calculate the kinematic properties of Watt’s six-bar linkage with a prismatic joint and uses the global optimization algorithm to find the dimensions of the linkage that satisfy the requirements.

Notice that $d2$ in Eq. (2) is the variable for the slider to slide. $ri$⁠, for $i=,0…7$⁠, is the constant that represents each link length. $θ2$ through $θ6$ are the variables that represent angles of each link that rotates. $θ1$ and $θ7$ are constant because they are attached to the fixed link.

Let $θ2$ be the input angle, one can solve $θ4$ by the first loop equation. By using $θ4$⁠, one can solve $θ6$ as the output angle. The detailed mathematical derivation of how to solve the angles are shown in Appendix A. The python program basically put the derivation into calculation code. However, if one is familiar with the four-bar linkage, an input angle has two output solutions, and hence, the Watt’s six-bar linkage will have total four possible solutions. During the testing, one of the sign (solution) is chosen so the linkage will fit the requirements, and the program only runs this sign for the optimization algorithm. A sign change means that the linkage either encounters singularity or it needs reassemble to achieve the configuration. Hence, choosing only a solution is rational for this synthesis problem.

Now that the angles have been determined, Eqs. (4) and (5) form systems of two linear equations each. To simplify the calculations, let $ω2=1$ represent the constant input angular velocity. Utilizing Cramer’s rule, $d2˙$ and $ω4$ can be readily solved. By substituting $ω4$ back into Eq. (5), the values for $ω5$ and $ω6$ can also be determined.

Based on the description in Sec. 1.2, the first goal of this article is to design a Watt’s six-bar linkage with a consistent (⁠$ωout/ωin=−1$⁠) ratio across a rotational range of at least 90 deg.

In addressing function generation challenges, the conventional approach involves employing Chebyshev spacing [47] to select precision points, thereby minimizing structural errors. However, when using an optimization approach, one can only make the output values as close to the desired values as possible at each control position. Because the requirement in our case is rare in the literature, we initially used a finite set of control points to explore the feasibility.

Table 1 presents the finite control positions of the input angle (in deg) employed in synthesizing the Watt’s six-bar linkage over a 100-deg range. The choice of 100 deg, instead of 90 deg, ensures a broader rotational span that meets the specified requirements. The subsequent details outline the key parameter set for the optimization problem.

The objective function for linkage synthesis in function generation is generally formulated to measure the difference between the desired output and the output produced by the linkage over a range of input values.

Because the requirement is to have (⁠$ω6/ω2=−1$⁠) in the $100$ deg range of the input angle $θ2$⁠, it is desired to have $ω6+ω2=0$ at every control position. Let $ω2$ be set to 1 as the constant input angular velocity. Here is the detailed description:

• Desired function: $ω6=−ω2$

• Output function: $ω6i(x,θ2i)$ is the angular velocity of link 6 from the actual synthesized linkage, which can be calculated by Sec. 2.2.

• Error function: At $i$th position, given $θ2i$⁠, the error between the desired angular velocity and the actual velocity can be presented as follows:

  $E(x,θ2i)=ω6i(x,θ2i)+ω2$

  (7)
• Objective function: The objective function aggregates the error for each control positions:

  $f(x)=∑i=1N(ω6i(x,θ2i)+ω2)2$

  (8)

• Optimization goal: The goal is to find the design parameter x that minimize the objective function.

In optimization synthesis problems for linkages, various constraints can be introduced, such as the Grashof criterion, circuit or branch defect constraints, or transmission angle constraints. However, the folding hinge problem does not necessitate crank requirements, making the Grashof criterion unnecessary. Furthermore, our calculation method is exempt from circuit defects, though choosing the sign in Eqs. (A2) and (A7) manually is necessary to achieve a desired configuration. While transmission angle constraints may be essential for practical hinge design, this article aims to establish the concept of overconstrained mechanisms and does not incorporate transmission angles into consideration.

During testing, it was observed that the Watt’s six-bar linkage encounters singularity and exhibits two degrees-of-freedom when $d2=0$⁠. Consequently, a constraint is imposed to ensure $d2>10$ for $N$ finite positions. If this constraint is not met, the algorithm will not record the result.

Another crucial constraint involves determining the range for each parameter, referred to as “bound” in SciPy.Optimization. Without setting these bounds, the optimization process becomes unmanageable due to an overwhelming number of possibilities and the absence of constraints.

The first concern when setting the bounds is that each link dimension should not be too small. If they are too close to zero, it becomes a singular case, and the linkage cannot be formulated. In addition, a small dimension may result in interference problems that are impractical for assembly. Hence, based on the 3D-printed prototype consideration, each link length was set to be greater than 10. Second, if the link is too long, it will take up too much space and may become unacceptable in the folding hinge application. Therefore, the maximum dimensions for $r4$⁠, $r5$⁠, and $r6$ were set to be 250. The maximum bounds for $r1$ and $r42$ were set to be smaller because we wanted these two dimensions to take up less space, making them more suitable for hinge applications. It is worth noting that these bound settings were based on testing experience with this problem and aimed at finding a rational solution that satisfies the goal. These settings are not mandatory or strict for synthesizing the Watt’s six-bar linkage proposed in this article. If the bound settings prevent achieving better results, they can be adjusted. The bounds for each design parameter are detailed in Table 2, established through several testing iterations.

The differential evolution algorithm is selected for its commendable performance in linkage synthesis, as indicated in the literature [52,66], and its user-friendly implementation in python. The application of the algorithm is straightforward, and one can readily employ it by consulting the documentation provided in the scipy.optimize.differential_evolution package [67] on its website. Various optimization algorithms may be capable of solving the proposed problem. However, the primary concern of this article is the accuracy of the results, which are intended for use in overconstrained mechanisms, rather than the convergence speed or detailed algorithm performance. Based on the numerical results, the differential evolution algorithm is believed to be effective in solving the proposed problem.

Based on the established parameters, the python program was implemented in Google Colab. The maximum number of iterations was set to 1000, and the tolerance was set to 0.001, while other optional settings were kept at their default values. If the result did not converge within 1000 iterations, the data were recorded and used as the new initial guess for another run. This process was repeated until the algorithm converged, at which point the data were recorded. Figure 7 shows the convergence process of the optimization of the $N=5$ example. The plot shows a steep decline in the objective function value during the first 50 iterations, indicating that the algorithm quickly identifies a promising region of the search space. However, with our tolerance setting, it did not meet the successful convergence requirement. Hence, the right plot of Fig. 7 shows the new objective function values using the previous final data as the initial guess. It was repeated until succeed. In different cases, the steep decline trend is similar, and the result seems stable and promising.

As indicated in Table 4, the objective function (Eq. (8)) can be minimized significantly with the parameters shown in Table 3. The small error is deemed acceptable for the problem at hand.

Using the parameters derived from the optimization algorithm, the input–output relationship can be visualized in Fig. 8. The results exhibit a highly linear trend closely resembling the $−1$ gear ratio motion. For achieving synchronized folding motion, one can introduce an offset to the vector on link 6. Referring to Fig. 9, link 2 and link 6 exhibit gear motion, wherein the vector a on link 2, rotating by $θ$ to a’, corresponds to the vector b on link 6 rotating negatively by $θ$ to b’. Consequently, the offset angle $ϕ2$ in the data of Fig. 8 is determined to be 67.8588 deg. Guided by this concept, a comparison will be made between the theoretical angle for vector b’ and the angle generated by the synthesized linkage.

Figure 10 illustrates the error between the theoretical angle of the gear motion and the actual angle of link 6 post-offset. Given that the errors are within the scale of $10−3$⁠, which is remarkably small, the maximum error is also documented in Table 4. It is worth noting that the error appears to have a common pattern in Fig. 10, and it is not affected by the control point locations. The author expected the error to be zero near the control points and to fluctuate between them; however, this is not the case. The error peaks for the four linkages occur around 20 and 85 deg, and the trough occurs around 55 deg. This may result from the similar configurations of the four linkages. From Table 4, one may notice that $θ1$⁠, $ϕ1$⁠, $r42$⁠, and $r4$ for the four linkages have close values. This may be a limitation of the optimization algorithm. Despite trying different initial guesses, the results turn out to be similar. It is unknown if there are any other good solutions far from the obtained results.

On the other hand, if different control points are used, the objective function values cannot be compared. One may notice that the $N=5$ case has a smaller $f(x)$ but a larger maximum error compared to other cases. Therefore, to quantify the error throughout the range, the root-mean-square deviation (RMSD) is calculated over a 100-deg range using Eq. (9).

Because the four tested results exhibited a common pattern in the error distribution, adding more control points should provide a better description of the problem. We set 101 control points (⁠$N=101$⁠) evenly distributed in the range and executed the optimization algorithm.

However, after several tests, it was noticed that the converged result had $r42$ and $r6$ equal to 10, the minimum bound value. If the value is too small, the linkage will encounter singularity and interference problems. For example, if $r6$ equals zero, the four-bar loop degenerates into a three-bar loop, and the output link cannot be driven at all. We also tried setting the lower bound to a smaller value, such as 1, but the result did not improve significantly. Hence, based on interference and feasibility considerations, we reduced the minimum bound to 6 and reran the program.

The dimensions of the synthesized linkage are shown in Table 5. The error results are shown in the last row of Table 4 and plotted in Fig. 11. It can be seen that the maximum error and RMSD are smaller, but not significantly so. Hence, the theoretical Watt’s six-bar linkage with a perfect velocity ratio of $−1$ remains unknown.

The overconstrained model for the minimized error design (⁠$N=101$⁠) was built in solidworks, and the motion simulation video is demonstrated online.¹

After the desired six-bar linkage is synthesized, one may tries to assemble the overconstrained mechanism depicted in Fig. 5. It is common for actual linkages or mechanisms to incorporate clearances for each kinematic pair. If the clearance errors surpass the errors illustrated in Fig. 10, the linkage can exhibit movement and possesses a single degree-of-freedom. Consequently, strictly speaking, the linkage is not a theoretically overconstrained mechanism but a practically overconstrained mechanism.

In designing the mechanism, the second Watt’s six-bar linkage (167892) shares the same dimensions as the first six-bar linkage (123456) but is symmetrical with respect to the $y$-axis.

To confirm the mobility of the synthesized linkage, the parameters for $N=5$ case were employed to create an assembly model in solidworks, as depicted in Fig. 12. Despite the extremely small error for the mechanism, the model becomes immovable after applying all kinematic pair constraints. However, releasing the tangent contact constraint for one sliding pin grants the linkage a single degree-of-freedom. Due to the small error, it can be observed that the sliding pin without the constraint remains closely within the groove in the $100$ deg range.

It is recognized that practical mechanisms always involve small clearances between kinematic pairs [68,69]. The accumulated error from each joint clearance should surpass the minimal error shown in Fig. 10. Consequently, the mechanism should exhibit mobility in the real world. To verify the functionality of the overconstrained mechanism, a 3D model for N=5 case is printed and assembled using nuts and bolts. The image of the 3D-printed model is presented in Fig. 13 at four positions, and it moves smoothly with one degree-of-freedom. One may see the motion video online.²

However, the synthesized linkage encounters singularity when $θ2$ is approximately 115.81 deg, causing $r42$ and $r5$ to align in a straight line. To prevent this issue, one should avoid folding the mechanism beyond 115.81 deg. Nevertheless, if the linkage’s operation is limited to rotate between 0 and 100 deg, it qualifies as an acceptable overconstrained mechanism.

To ensure that the generated linkage is not a singular case, the overconstrained mechanism for $N=11$ is also 3D printed. The four positions of this mechanism are illustrated in Fig. 14, and a video demonstrating its single degree-of-freedom motion can be viewed online,³ providing evidence of its mobility.

Foldable phone hinges require an elegant folding motion for the floating plate to support the flexible screen effectively. Without specialized motion, the flexible screen may break or develop noticeable creases. To achieve such motion, synchronized rotation pairs are necessary. Gearing is the most common mechanical component to accomplish this, but due to the need for small sizes, engineers are seeking better alternative solutions.

Inspired by patent CN114584638A, this article proposes the first Watt’s six-bar linkage to achieve an approximate gear-like motion for certain range. Even without combining it with a symmetrical counterpart, the Watt’s six-bar linkage itself could serve as an alternative hinge design. For example, Samsung uses a cam for synchronized motion in their products, which also results in an small angle difference when folding.

When combined with a symmetrical Watt’s six-bar linkage, the resulting nine-bar linkage forms a novel overconstrained mechanism that has not previously appeared in academic literature. This mechanism has potential applications not only in foldable phone hinges but also in any situation requiring synchronized folding motions.

To make the proposed overconstrained mechanism movable, it requires specific parameter ratios, as shown in our results. Furthermore, we can only achieve an approximate gear-like folding motion, not a perfect one. Without any joint clearance, the mechanism remains unmovable. Therefore, if the application, such as a folding hinge, has additional dimensional restrictions, it will be difficult to apply the proposed mechanism. This article explores the mobility potential of the mechanism from patent CN114584638A. However, the precise dimensions of such mechanism that can put into the small space remain unknown.

• Based on the results, we can synthesize Watt’s six-bar linkage with gear-like input-output motion. However, its feasibility in the folding hinge application remains unknown. By incorporating the requirements from the partnered hinge company, restrictions can be added, and the newly designed mechanism can be analyzed. It is important to note that the design may become the property of the company, which is confidential.

• This article asserts that the proposed mechanism can move due to clearance. Although the error is extremely small and the prototype was created to validate this claim, further analysis of acceptable error margins will be necessary if the linkage is used in more precise applications. Future error analysis research should apply the methods from Refs. [68,69] to the proposed mechanism.

• Because this article resembles the original design figure from the patent, it explores one type (solution) of the mechanism to achieve the input–output motion with a velocity ratio of $−1$⁠. Other types of mechanisms could be investigated, including the possibility of replacing some revolute joints with prismatic joints.

This study has successfully synthesized a novel planar overconstrained mechanism by merging two symmetric Watt’s six-bar linkages. Utilizing a differential evolution optimization algorithm, we achieved a linkage with gear-like input–output motion. Comprehensive error analysis demonstrated the precision of the synthesized mechanism, revealing minimal discrepancies between theoretical predictions and actual angles. The 3D-printed prototype validated its real-world functionality, showcasing smooth motion with a singular degree-of-freedom and confirming the mechanism’s potential for practical applications.

Our contributions are significant and threefold:

1. Innovative synthesis: For the first time, we synthesized a Watt’s six-bar linkage capable of approximate gear-like motion for $100∘$⁠.

2. Novel mechanism: We introduced a new planar overconstrained mechanism along with a detailed methodology for its synthesis.

3. Practical application: We proposed a viable design for foldable phone hinges, meeting stringent requirements for compactness and durability.

This research not only advances the theoretical understanding of overconstrained mechanisms but also demonstrates their practical feasibility, offering innovative solutions for the design of compact and versatile devices.

Looking ahead, further research is necessary to thoroughly evaluate the mechanism’s feasibility in real-world folding hinge applications. Detailed clearance error analysis will be essential for applications that require higher precision, ensuring the mechanism possesses single degree-of-freedom.

Special thanks to Syncmold Enterprise Corp for the project that inspired the idea for this article.

• The National Science and Technology Council of Taiwan (Grant No. NSTC 112-2222-E-027-004-MY2).

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

$d2$ =

the distance of the slider

$ri$ =

the dimensions labeled in Fig. 6 

$θi$ =

angle of the $i$th link with respect to the $x$-axis

$ωi$ =

angular velocity of the $i$th link