Modeling and Improvement for String-Based Drive Mechanism in Insect-like Flapping Wing Micro Air Vehicle

Abstract

Most of the driving mechanisms used in micro flapping wing vehicles are gear and linkage transmission mechanisms, which convert the unidirectional rotation of the motor into the reciprocating flapping of the driving mechanism. However, gear and linkage transmission mechanisms occupy a larger space and weight with certain energy losses. This article introduces a drive mechanism that is different from gear and linkage mechanisms, namely the string-based mechanism. In this study, the working principle and various parameters in string-based mechanisms are analyzed, and the flapping trajectory and amplitude of the mechanism is simulated. Following that, this article proposes an improved method of the cam mechanism, designs the trajectory of a cam mechanism, and a physical design is carried out with a driving mechanism width of 22 mm. Subsequently, the experiments are conducted to compare the flapping trajectory of the actual mechanism with the simulation trajectory, thereby verifying the simulation accuracy. The results prove that, in a string-based mechanism, the ratio of the radius of the pulley to the installation radius of the double-layer bearings has a direct impact on the flapping amplitude, and optimizing the design of cam mechanisms can effectively reduce the tensile and relaxation phenomena in string-based mechanisms. The above conclusion was verified in physical experiments, where the modified cam mechanism effectively reduced the stuck phenomenon in the string-based mechanism.

1. Introduction

In recent years, micro aerial vehicles (MAVs) have gained significant attention in the public domain. Their applications range from common activities like aerial photography, air shows, and formation performances [1,2] to military operations such as surveillance and suicide attacks, thus highlighting their significance in this emerging industry. However, the majority of MAV activities are centered around quad copter drones or micro unmanned helicopters, with flapping-wing MAVs (FMAVs) being rarely utilized. This stands in stark contrast to the majority of flying creatures in nature that employ flapping wing flight [3]. This raises the question: why has the flapping wing flight mode, which is widely used by flying animals, been largely overlooked in the growing MAV industry?

The answer to this question is also related to the challenges that the flapping aircraft needs to face to achieve stable and controlled flight, and a very important factor in this is the design of the drive mechanism. In the design of a drive mechanism, the largest difference between a flapping aircraft and the rotorcraft is the power supply for a rotating wing, which is an example of reciprocating flapping, while the power supplied for the rotorcraft is an example of unidirectional rotating blades. In the most common drive mode, i.e., motor drive, both direct-drive rotorcraft or non-direct-drive rotorcraft all convert the one-way rotation of the motor into the one-way rotation of the blades, which is simpler and more efficient. But the reciprocating motion features in the FMAV require a conversion in the motor unidirectional rotation to wing flapping, which is complex and inefficient [4,5,6]. To acquire the ability of flapping flight, in nature, birds use contractile pectoral muscles [7,8], and insects use the resonant thorax and hydraulic wing veins [9,10,11] for better flapping flight (which is not practical for design in FMAVs). Therefore, the drive mechanism has higher requirements in a mechanism design that is based on the purely mechanical method of achieving flapping to achieve the flying of FMAVs.

The mechanisms employed by existing flapping MAVs to achieve wing flapping can be broadly categorized as follows: gear-bar linkage-based mechanisms [12,13,14,15,16,17,18,19,20,21,22], spring-bar linkage-based mechanisms [23]; string-based mechanisms [24,25,26,27], ‘Smart Composite Microstructures’ (SCM) and voltage-driven piezoelectric bimorphs mechanisms [28,29,30,31], and motor direct drive mechanisms [32,33]. Among these, the voltage-driven piezoelectric bimorph mechanism boasts a high flapping speed but suffers from a low driving capacity. At present, it is solely utilized in millimeter-scale insect-like flapping wing aircraft. On the other hand, motor direct-drive mechanisms necessitate controlling the motor to reverse, thus affording flexible control. However, their low efficiency hinders practical applications, which is typically used by Purdue Hummingbirds [33]. The gear-bar linkage-based mechanism, spring-bar linkage-based mechanism, and string-based mechanism are commonly employed in decimeter-level to meter-level flapping wing aircraft, with the string transmission mechanism exhibiting unique advantages over the other two. The string, with an appropriate Young’s modulus, is not as rigid as the gear-bar linkage-based mechanism, and it exhibits slight ductility in the face of vibration and resonance, thus safeguarding the mechanism and ensuring a smoother flapping trajectory. Furthermore, compared to the spring-bar-based mechanism, the string-based mechanism boasts greater flexibility without affecting the preset flapping trajectory, thus rendering it effective and reliable for flapping flight control. Simultaneously, the advantages of the string mechanism have been demonstrated in current prototypes, as exemplified by the nano Hummingbird [24], which achieves relatively stable hovering with a string mechanism (as evidenced by existing experiments). These prototypes illustrate the feasibility of practical applications for string-based flapping aircraft.

This article aims to analyze the influence of different design parameters on the flapping trajectory and amplitude in string-based mechanisms, as well as seeks to simulate the defects present in circular rollers. Through utilizing simulated data on circular rollers, a cam mechanism is devised and the simulation method used is explained in detail. Experiments were conducted to measure the flapping trajectory of the mechanism designed, which was based on the parameters, and the comparison with simulation results proved the rationality of the structural mechanism design.

2. Theoretical Mechanism Design and Model

The string-based flapping mechanism was originally used to generate wing flapping for the nano Hummingbird [24], and this was then imitated by the team from SJTU [27]. As the most classical and mature example of flapping wing vehicles that use the string-based mechanism, the nano Hummingbird and its drive mechanism were comprehensively demonstrated in the paper of [24]. This study will replicate and validate the mature platform based on the string-based mechanism of the nano Hummingbird, and the SolidWorks principle diagram is shown in Figure 1, where the main parameters are shown in detail. As shown in Figure 1, drive strings are two independent strings, which wrap around the double-layer bearing on the center gear. Two ends of one drive string are fixed on the left roller while the two ends of the other string are fixed on the right roller. As shown in Figure 2, the rotation direction of the central gear is counterclockwise. In Figure 2a,b, the position of the double-layer bearing is approximately above the connecting line between center of left and right roller. At that time, the right string is pulled by the double-layer bearing and the left string is pulled by the phase keeper strings, and the flapping direction of the left and right wings is downward. In Figure 2c,d, the position of the double-layer bearing is approximately below the connecting line between the center of the left and right roller. At that time, the left string is pulled by the double-layer bearing and the right string is pulled by the phase keeper strings, and the flapping direction of the left and right wings is upward. During the rotation of the center gear, only one side of the string is pulled to maintain a taut state, which causes the rotation of the roller. Meanwhile, another string is not directly pulled by the center gear, which will become loose in the absence of other drives. Therefore, two additional strings are required to connect the left and right rollers to keep the two rollers turning at the same angle, which will result in pulling another string taut (and these are referred to as phase keeper strings). In a theoretical design, the left and right wings need to have the same flapping angle to prevent asymmetry during flapping. Therefore, the rollers on both sides need to have the same radius to ensure they can have the same rotation angle during the rotation of the center gear.

As shown in Figure 3, a Cartesian coordinate system was established, and the origin was the center of the center gear. The vector from origin to the center of the right roller, which was named $O_r$, was the positive direction of the x-axis. The center of the left roller was on the negative direction of the x-axis. The center of the double-layer bearing rotated on the center gear with a radius of $r_0$, with coordinates of $(x_b,y_b)$, and a radius of $r_3$. We then took the taut left string as an example, which had a total length of L and was wrapped around the double-layer bearing with an arc of $BC$, as well as a little arc on the left roller (which was expressed as $AD$). There, the point $ABCD$ was a tangent point of the lines $AB$ and $CD$, as well as of the left roller and double-layer bearing. The total length of the left string can be express in Equation (1).

$$L=AB+CD+BC+AD+AE$$

(1)

Here, point E is the intersection of the two ends of the left string on the left roller, which controls the rotation of the left roller when the left string is being tautly pulled. And the coordinates of point A, B, C, and D can be solved by the tangent equation of the two circles. To illustrate this, point A can be taken as an example, which can be express as Equation (2).

$$\left(x_l-r_1\sqrt{1+\frac{1}{({k_{AB})}^2}},y_l+\frac{r_1}{k_{AB}}\sqrt{1+\frac{1}{({k_{AB})}^2}}\right)$$

(2)

And $k_{AB}$ means that the gradient of line $AB$, where b means the intercept of line $AB$, can be solved by Equation (3).

$$\frac{\left|k_{AB}·x_1-y_1+b\right|}{\sqrt{{\left(k_{AB}\right)}^2+1}}=r_1,\frac{\left|k_{AB}·x_3-y_3+b\right|}{\sqrt{{\left(k_{AB}\right)}^2+1}}=r_3$$

(3)

Equation (3) has four sets of a solution, and they should be selected correctly. Furthermore, the coordinates of point $ABCD$ can be solved, and the length of $AB$, $CD$, $BC$, and $AD$ can be calculated. Moreover, the length of arc $AE$ can be expressed by Equation (4).

$$AE=\frac{1}{2}\left(L-AB-CD-BC-AD\right)$$

(4)

The coordinates of point E in the polar coordinates were established with the center of the left roller as the origin, which can be expressed as Equation (5).

$$\left(r_1,arctan\frac{y_A-y_l}{x_A-x_l}+\frac{AE}{r_1}\right)$$

(5)

With the rotation of the double-layer bearings on the center gear, the polar coordinates of point E kept changing, and the flapping angle and speed of the flapping wing fixed on the roller were also found to be constantly changing. The calculated trajectory of point E was in an ideal state when the string was in a tight state.

During the rotation of the double-layer bearings on the center gear, the center of the bearing and direction of rotation constantly changed. This caused the string to become either too loose or too tight, thus causing stuck or slack phenomena within the mechanism.

3. Data Choice Based on the Mechanism Design

Considering that the overall mechanism of flapping wing vehicles is relatively small, there are size requirements for the flapping mechanism. Assuming that the maximum transverse span allowed is $D_{span}$, the maximum longitudinal length is $D_{width}$, and that there is no collision between the roller group and the bearing group, the four parameters mentioned above need to meet the design scale constraints of the flapping mechanism as system of Equations (6).

$$\left\{\begin{array}{c}\begin{array}{c}2\left(r_0+r_3\right)\le D_{width}\\ r_1=r_2\le D_{width}\end{array}\\ \begin{array}{c}2\left(d+r_1\right)\le D_{span}\\ r_0+r_3+r_1<d\end{array}\end{array}\right.$$

(6)

As shown in Figure 4—when considering the inertial effects present during flapping after installing wings, the non-complete structural stiffness, and the stretchability of actual strings—it is necessary to incorporate redundancy into the flapping angle design to avoid the vibrations caused by wingtip collisions. Assuming the maximum allowable flapping angle allowed by the mechanism is ${\theta }_{allow\_max}$ and the redundancy ratio is $k_r$, the maximum flapping angle ${\theta }_{f\_max}$ that needs to be considered during the design process should satisfy the Inequality (7).

$${\theta }_{f\_max}\le {k_r\theta }_{allow\_max},$$

(7)

where $k_r$ is an empirical parameter that is obtained by combining structural parameters such as the flapping frequency $f_F$, wing material, and physical experiments, with a value range of 0.7 to 0.85. Thus, ${\theta }_{f\_max}$ can be obtained through the simulation of the design parameters $r_0$, $r_1$, $r_3$, and d, which are detailed in Section 2. ${\theta }_{allow\_max}$ is calculated through structural design parameters, and it is given by Equation (8).

$${\theta }_{allow\_max}=2\left(\pi -2arccos\frac{d}{l_w+r_1}\right),$$

(8)

where $l_w$ is the wing length. Together with the other parameters related to structural design, it forms the flapping span $D_{flapping\_span}$, which needs to meet the constraint condition of the whole machine size as Inequality (9).

$$2\left(l_w+d+r_1\right)\le D_{flapping\_span}.$$

(9)

In summary, the complete design constraint conditions for the string drive flapping mechanism can be expressed as system of Inequality (10).

$$\left\{\begin{array}{c}\begin{array}{c}2\left(r_0+r_3\right)\le D_{width}\\ 2r_1\le D_{width}\\ 2\left(d+r_1\right)\le D_{span}\end{array}\\ \begin{array}{c}{r}_0+r_3+r_1<0.75d\\ 112.5\frac{r_0}{r_1}+0.3\le 2k_r\left(\pi -2arcos\frac{d}{l_w+r_1}\right)\\ 2\left(l_w+d+r_1\right)\le D_{flapping\_span}\end{array}\end{array}\right.$$

(10)

where ${\theta }_{f\_max}=112.5\frac{r_0}{r_1}+0.3$ is obtained in the subsequent sections but cited here in advance for the parameter design of the flapping mechanism.

Regarding the equation above, the values of the various constraints are shown in

Table 1. Characteristics of design parameters.

Design Parameters	Data
$D_{width}$	7.5 mm
$D_{span}$	25 mm
$D_{flapping\_span}$	160 mm
$k_r$	0.7

.

The complete constraint equation is comprehensive, but there are many considerations that need to be taken into account (which is not in line with the simplification strategy in structural design). Therefore, it is necessary to simplify the equation by incorporating empirical parameters in the design. In this case, the maximum flapping angle was fixed at ${\theta }_{f\_max}=160{}^{\circ }$, which means $r_0=1.42r_1$, and the wing length was determined to be $l_w=65$ mm. By simplifying the complete equation, we obtain the system of Inequality (11).

$$\left\{\begin{array}{c}\begin{array}{c}2\left(1.42r_1+r_3\right)\le D_{width}\\ 2r_1\le D_{width}\end{array}\\ \begin{array}{c}2\left(d+r_1\right)=D_{span}\\ 2.42r_1+r_3<0.75d\end{array}\end{array}\right.$$

(11)

For the simplified constraint equation, according to the processing scale requirements, a minimum value of 0.8 mm was added for the radius of the roller group and the radius of the bearing group as system of Inequality (12).

$$\left\{\begin{array}{c}{r}_1\ge 0.8\mathrm{mm}\\ r_3\ge 0.8\mathrm{mm}\end{array}\right.$$

(12)

Thus, the valid combination range of $r_1$ and $r_3$ was obtained, as shown in the Figure 5. Based on the valid combination range of $r_1$ and $r_3$ in Figure 5, appropriate parameters were selected. Taking into account the supporting fittings in the mechanism, such as the position of the connecting columns and the thickness of the support surface, etc., the parameters of the string drive flapping mechanism that met the design requirements could be completed. The final values are shown in

Table 2. Characteristics of the design parameters.

Design Parameters (mm)	Data
Radius of rollers $r_1$	1.2
Radius of bearing $r_3$	1.9
Radius of center gear $r_0$	1.7
Distance of center of circles d	10.15
Length of strings $l_w$	65

.

Through analyzing the various constraints in the drive mechanism, as well as by selecting the corresponding design parameters within the obtainable range of values, the corresponding final design parameters could be obtained and a subsequent analysis could be conducted based on the obtained parameter values.

4. Data Analysis of Mechanism Design

Based on the derivations presented above, simulations and physical studies could then be conducted using the parameters listed in Table 2. The flapping angle of the rollers was analyzed under ideal conditions, wherein the string was fully taut through the rotation of the center gear. The total length of the strings was not affected by the flapping angles as their lengths could be adjusted by bypassing the left and right rollers. A string with a length of 34.7 mm was chosen for this study. We assumed that two strings are always in an ideal state (i.e., tight state) without considering the strings used to keep them moving in the same phase. The angles of the right and left rollers are shown in Figure 6.

Based on the initial parameters in Table 2, we made changes to each parameter to observe its influence on the flapping amplitude and trajectory. Our results, as shown in Figure 7, Figure 8, Figure 9 and Figure 10, indicate that the radius of the rollers and center gear had a significant impact on the flapping amplitude and trajectory, while changes to the radius of the double-layer bearings and length between the center gear and rollers had little effect.

The influence of the parameters on the flapping angle can also be understood from their physical meaning. Firstly, the radius of the double-layer bearing assembly and the installation position of the roller assembly with respect to the center gear had a relatively small impact. Since the strings wrapped around the double-layer bearings, they could be approximated as two tangents. The change in the radius of the double-layer bearing had a minimal impact on the length of these tangents; therefore, it also had minimal impact on the flapping angle. Similarly, the installation position of the roller with respect to the center gear mainly affected the total length of the string, and a change in this value had minimal impact on the length of the tangents. Therefore, both the radius of the double-layer bearing and the installation position of the roller had minimal impact on the flapping angle. On the other hand, when considering both the radius of the roller and the installation radius of the double-layered bearing, we can imagine that the string travels an equal distance around both circles, with the resulting angle change being related to the radius of both circles.

Based on the simulation, changing the width of the flapping mechanism during the design process will not cause significant changes to the final flapping angle. Additionally, we do not need to worry about the small size of the double layer bearings, which is a challenging size to achieve in a micro-scale design in structural development. The flapping trajectory and amplitude could be adjusted by changing the radius of the rollers and center gear, which means the ratio of the rollers to center gear had the greatest influence on the design of mechanisms.

To visually demonstrate the impact of the ratio of the installation position of the bearing group to the radius of the roller group on the flapping trajectory, we defined the ratio as $k_r$. Figure 11 shows the simulation of the influence of the varying $k_r$ on the flapping trajectory, which can be seen that the flapping amplitude increased linearly with the increase in $k_r$, which overall followed a straight line trend. To approximate it, we obtained a linear function relationship between the flapping amplitude ${\theta }_{f\_max}$ of the string drive mechanism and the ratio $k_r$ of the installation position of the bearing group to the radius of the roller group, which was given by the Equation (13).

$${\theta }_{f\_max}=K·k_r+B=112.5k_r+0.218,$$

(13)

where K and B represent the slope and the intercept of the fitting line, respectively. The values for this equation were obtained when the radius of the roller group was 1.285 mm, the radius of the bearing group was 1.785 mm, and the installation position of the roller group was 10.15 mm. To verify this, we attempted to change these values and compared the impact of the different values on K and B, thereby forming a parameter comparison, as can be seen in

Table 3. Characteristics of the design parameters.

${\mathit{r}}_1$ (mm)	${\mathit{r}}_3$ (mm)	d (mm)	K	B
1.285	1.785	10.15	112.5	0.218
1	1.785	10.15	112.6	0.1258
1.3	1.785	10.15	112.5	0.2241
1.6	1.785	10.15	112.3	0.3595
1.285	1.4	10.15	113.3	0.1323
1.285	1.8	10.15	112.5	0.2218
1.285	2.2	10.15	111.5	0.3375
1.285	1.785	8	111	0.6161
1.285	1.785	10	112.5	0.233
1.285	1.785	12	113.2	0.1065

. It can be seen that changing the radius of the bearing group and the installation position of the roller group has a very weak effect on the slope of the fitting line. The difference between the value of the intercept of the fitting line and the slope was two orders of magnitude, which had a minimal impact. Therefore, the linear function relationship between the flapping amplitude ${\theta }_{f\_max}$ of the string drive mechanism and the ratio $k_r$ of the installation position of the bearing group to the radius of the roller group can be expressed as Equation (14).

From the simulation analysis of the angle change, when the radius of the left and right rollers where under ideal tight conditions, it can be seen that the peak value was taken when the phase of the center gear was about 171° and 351°, which was not the ideal values of 180°and 360°. This difference was due to the distance of the center gears between the rollers, which tends to be zero when the distance of the center of circles increases toward infinity.

$${\theta }_{f\_max}=112.5\frac{r_0}{r_1}+0.3$$

(14)

The simulation above is based on the assumption that both strings are tight. However, the actual situation was that, due to the soft nature of the strings, the phase keeper strings had to be used to keep the strings tight. Based on the connection method and parameters shown in Table 2, when the phase of the center gear was from −9° to 171°, the left roller was pulled directly by the center gear and right roller was pulled by the phase keeper strings, which meant that the change in value in the left rollers was the same as in the right rollers. Nevertheless, it can be seen that a difference exists in the simulated phase of the left and right rollers when the strings are supposed to be tight, which resulted in a stuck condition and delay during the rotation of the center gear.

This is meaningful in the design of an actual flapping mechanism. When there was a stuck condition, the strings on both sides reached the tightest state. Although it could be successfully turned over, it still had an impact on the tightness of the mechanism and the connection of the strings and rollers. This situation will thus reduce the life of the mechanism and result in unsmooth flapping during the working of the mechanism. Based on the above experiments and analysis, structural improvements need to be proposed to reduce or even eliminate the impact of this stuck condition, and the cam mechanism is an improvement based on the above ideas.

5. Cam Mechanism

In previous analyses, it was observed that when both rollers on either side are circular, there exists a discrepancy between the angles generated by the ideal ’unwinding’ of the string on one roller and the ideal ’winding’ of the string on the other. The presence of a synchronous string necessitates equal rotation angles for both side rollers, which results in a conflict that is manifested as the ’over-tightening’ phenomenon of the string and the stagnation phenomenon in physical experiments.

To address this issue, the NANO Hummingbird team introduced a patented solution in 2018 [34] involving the use of non-circular rollers, specifically where a cam-like mechanism was referred to as a cam (shown in Figure 12). This design was also adopted by the team of SJTU [27] to effectively mitigate the stagnation phenomenon in string transmission systems with circular rollers. However, their patent and study did not provide insights into the design rationale and parameter analysis of the non-circular cam profile, while Cardona’s work on cam [35,36] and Zhu’s research [37] gave us inspiration for deriving cam trajectories. In this section, we start with the conceptual design of the circular roller mechanism, then we elucidate the methodology for designing the non-circular cam mechanism and mathematically model it. In conjunction with the relevant parameters from the circular roller flapping mechanism, we analyze the expression of the non-circular cam mechanism, which is abbreviated as the cam mechanism in the subsequent content.

In Figure 13, a detailed motion schematic of the cam mechanism is shown. As illustrated in Figure 13, the diagram depicts the complete cycle of motion for an optimized non-circular cam based on circular rollers. Using counterclockwise rotation as an example, it primarily comprises the following four operational states: in Figure 13b–d, the right roller is driven by the central gear along a non-circular path, while the left roller is pulled by a synchronous string along a circular path;in Figure 13a,b,h, the right roller is driven by the central gear along a circular path, while the left roller is pulled by a synchronous string along a non-circular path;in Figure 13f–h, the left roller is driven by the central gear along a non-circular path, while the right roller is pulled by a synchronous string along a circular path; and in Figure 13d–f, the left roller is driven by the central gear along a circular path, while the right roller is pulled by a synchronous string along a non-circular path. This schematic provides a comprehensive overview of the cam mechanism’s dynamic behavior during its complete cycle of motion.

In the previous section, it was demonstrated, through simulation, that the radius of the double-layer bearings’ assembly has a minimal impact on a string-based mechanism. In this section’s analysis, we will treat it as a point, disregarding the influence of the bearing assembly’s radius, as shown in Figure 12. The total length of the red string and the blue string remained constant. Taking the red string in Figure 12 as an example, its length can be expressed as Equation (15).

$$L_{left}=\widehat{O_{s}B}+\stackrel{-}{BC}$$

(15)

where $\widehat{O_{s}B}$ represents the winding length of the string on the circular and non-circular tracks, and $\stackrel{-}{BC}$ represents the length of the string under ideal tension conditions. Under this assumption, in combination with the relevant parameters mentioned earlier, we simulated the change in the deflection angle of the string on the circular roller under ideal tension conditions and obtained a simulation diagram, which is shown in Figure 14. It can be observed that there is a significant difference in the angles between the left and right rollers. This difference also exists in simulations that consider the double-layer bearing radius.

Through simulation, it can be determined that both rollers simultaneously reach extremities at approximately ${\alpha }_1=-7.2{}^{\circ }$ and ${\alpha }_2=172.8{}^{\circ }$, which correspond to the angles of the central gear when the left and right roller switch between active and passive stretching. In the context of this switching process, there exists a transition angle ${\alpha }_0$ between the non-circular and circular profiles, thus ensuring that one of the left or right roller is stretched along a circular or non-circular path while the other is stretched along a non-circular or circular path by the synchronous string. This angle, ${\alpha }_0$, should result in equal rotation angles for the left and right rollers; hence, it is the midpoint between ${\alpha }_1$ and ${\alpha }_2$, i.e.,

$${\alpha }_0=\frac{1}{2}\left({\alpha }_1+{\alpha }_2\right)=82.8{}^{\circ }$$

(16)

When the central gear is at this angle, it remains at a tangent to the circular paths on both the left and right rollers. Upon a counterclockwise rotation, the left roller retracts the string along the non-circular path while the right roller releases it along the circular path. Conversely, during clockwise rotation, the left roller releases the string along the circular path while the right roller retracts it along the non-circular path. Furthermore, considering the structural symmetry, it becomes evident that the simulation results for ${\alpha }_1$ to ${\alpha }_2$ and ${\alpha }_2$ to ${\alpha }_1+360{}^{\circ }$ yield identical non-circular cam profiles. Therefore, for ease of analysis, this paper will focus on the analysis of the non-circular cam profile as the central gear rotates from ${\alpha }_0$ to ${\alpha }_2$.

Figure 15 shows a representation diagram of the cam mechanism, in which the direction of the connection between the center and the intersection point of the circular trajectory and the cam trajectory is taken as the positive direction of the polar axis, and where the direction of the cam trajectory is taken as the positive direction of the angle. A polar coordinate system is then established, and the cam trajectory can be expressed as Equation (17).

$$\rho =\rho \left(\theta \right),\rho \left(0\right)=r_1$$

(17)

where $\theta$ is the angle between the positive direction of the polar axis and the trajectory. As shown in Figure 16, the initial position was set as the central gear rotation angle ${\alpha }_0$. At that moment, the intersection point $B_0$ of the left roller’s circular trajectory and the cam trajectory was at a tangent to the line $B_0{C}_0$. Similarly, the intersection point of the right roller was also at a tangent. At that time, the initial angles ${\theta }_0$ and ${\delta }_0$ of the two cams could be solved through the tangential equation of the circle, and the initial length $m_0$ of the string could also be calculated. The values of the above parameters are shown in

Table 4. Characteristics of the design parameters.

Initial Parameters (mm)	Data
Initial angle of left roller ${\alpha }_0$	1.2
Initial angle of right roller ${\delta }_0$	1.9
Initial length of tangent $m_0$	1.7

.

Due to the clockwise rotation that followed, the right roller was directly pulled by the central gear, so the flapping angle of the entire mechanism as a function of the central gear rotation angle $\alpha$ was the same as that generated by the circular trajectory of the right roller. Therefore, the positional mapping relationship between the rotation angle of the left roller and the central gear angle $\theta =f\left(\alpha \right)$ could be obtained. The Curve Fitting toolbox in MATLAB was used to fit the simulated data. The mapping equation could be fitted in the form of first-order Fourier series, which can be expressed as Equation (18).

$$\begin{array}{cc}\hfill \theta & =f\left(\alpha \right)=a_0+a_{1}cos\left(\omega \alpha \right)+b_{1}sin\left(\omega \alpha \right)\hfill \\ \hfill & =69.32-81.68cos\left(0.01475\alpha \right)+54.58sin\left(0.01475\alpha \right)\hfill \end{array}$$

(18)

Based on the initial angle, when the central gear rotates to an angle $\alpha$, where ${\alpha }_0<\alpha <{\alpha }_2$, the left roller rotates to an angle $\theta$. At that time, point B was found to be the intersection point of the tangent line $BC$ and the cam mechanism, and its angle was $\phi$. Its value is related to both the central gear rotation angle $\alpha$ and the graph trajectory $\Gamma$, and it can also be expressed as $\Gamma (\alpha ,\rho )$. The parameters in this cam mechanism should satisfy the system of Equation (19).

$$\left\{\begin{array}{c}{m}_0=m+\widehat{AB}\\ \theta \left(k_{\widehat{B}}\right)=\theta \left(k_{BC}\right)\end{array}\right.$$

(19)

where the two equations are presented to describe the fundamental relationships governing the motion of the string on the cam mechanism. The first equation signifies that the total length of the string remains constant. The second equation expresses the tangential condition at point B, where the angle of the arc $\widehat{AB}$ is equal to the angle of taut string $\stackrel{-}{BC}$, thus representing the tangential relationship between the arc and the straight line. The coordinates of the key points were defined as follows: Point A: $\left(-d+r_{1}cos\left(\theta \right),r_{1}sin\left(\theta \right)\right)$; Point B: $\left(-d+r_{1}cos\left(\theta \right),r_{1}sin\left(\theta \right)\right)$; Point C: $\left(r_{0}cos\left(\alpha \right),r_{0}sin\left(\alpha \right)\right)$; and Point D: $\left(-d,0\right)$. It can be seen, through analysis, that there are two parameters to be solved and two equations, which theoretically allows for analytical solutions. However, due to the presence of many non-homogeneous square root terms in the equations that need to be resolved, it is not possible to provide formulaic solutions. Therefore, it is necessary to use a micro-element method for a step-by-step iteration to obtain numerical solutions for the cam trajectory.

Before numerically solving the cam profile by using the differential element method, let us represent the previous analysis in a more intuitive mathematical manner. The simplified diagram, as described through mathematical relationships, is shown in Figure 17.

Consider Point A, which moves along a known trajectory with Point D as its center and $r_1$ as its radius. Additionally, Point C moves along a known trajectory with the origin of the coordinate system as its center and $r_0$ as its radius. Now, there exists a curve $\Gamma$ attached to Point A, which moves in conjunction with the rotation of Point A. During the motion of Points A and C, this curve satisfies both the tangential relationship in the formula and the constraint of the constant total length. Point B is the intersection point of the $\Gamma$ curve and the line BC, thereby serving as both a point on the $\Gamma$ curve and a point where the tangent of curve $\Gamma$ intersects with BC. The objective was to determine the expression for the $\Gamma$ curve. Given that Point B lies on the tangent of the curve, and that the system is continuous, calculating the trajectory of Point B is equivalent to obtaining the expression for the $\Gamma$ curve.

Based on the analysis in the previous section, the motion trajectories of Points A and C are known to be functions of $\alpha$. By using infinitesimal analysis, $\alpha$ can be divided into n segments, and the motion trajectories of Points A and C can be represented by n + 1 points, which can be expressed as Equation (20).

$$\begin{array}{cc}\hfill & \left(x_{Ai},y_{Ai}\right)=\left(x_a\left(\alpha \left(i\right)\right),y_a\left(\alpha \left(i\right)\right)\right),i\in [0,n]\hfill \\ \hfill & \left(x_{Ci},y_{Ci}\right)=\left(x_c\left(\alpha \left(i\right)\right),y_c\left(\alpha \left(i\right)\right)\right),i\in [0,n]\hfill \end{array}$$

(20)

At the same time, there were n + 1 tangent points on the $\Gamma$ trajectory, which can be denoted as Equation (21).

$$\left(x_{Bi},y_{Bi}\right)=\left(x_b\left(\alpha \left(i\right)\right),y_b\left(\alpha \left(i\right)\right)\right),i\in [0,n]$$

(21)

It should be noted that Point B is a tangent point on the $\Gamma$ trajectory, which is attached to Point A, and that Point A is a point on the circle. Therefore, it is convenient to convert each Point B into a point in polar coordinates with Point D as the center and DA as the positive direction, i.e., to perform a conversion from Cartesian coordinates to the polar coordinates of the $\Gamma$ trajectory as system of Equations (22).

$$\begin{array}{cc}\hfill {\rho }_{Bi}& =\sqrt{{\left(x_{Bi}+d\right)}^2+{y_{Bi}}^2}\hfill \\ \hfill {\theta }_{Bi}& =\theta -arctan\frac{y_{Bi}}{x_{Bi}+d}\hfill \end{array}$$

(22)

When considering the infinitesimal method, the constraint condition of the unchanged string length was no longer expressed as a continuous integral, but rather represented as the sum of lengths between the different points remaining invariant. Thus, the following constraint condition applies:

$$m_0=BC+\widehat{AB},$$

(23)

which can be transformed into Equation (24).

$$m_0=C{B}_k+\sum _{i=1}^k{B}_{i-1}{B}_i,i\le k\le n,$$

(24)

where k represents the position wherein Points A and C have moved to the kth position, and the tangent point at this moment is located at $B_k$. $B_i$ indicates the position of the previous tangent point on the $\Gamma$ trajectory in the current state. Transforming it into a point in polar coordinates for calculation has two advantages. Firstly, because the $\Gamma$ trajectory itself was moving, converting the tangent point $B_i$ to a point attached to its own polar coordinate system facilitates the elimination of this rotation. Secondly, when representing the tangent point in Cartesian rectangular coordinates, its value in the x direction was one that was non-increasing. Meanwhile, in polar coordinates, its angle was found to be increasing. Furthermore, using polar coordinates for representation can prevent the “tangent point retreat” phenomenon and facilitate the calculation of the distance between tangent points.

In an infinitesimal analysis, the constraint condition of the equal tangent angles was no longer applicable as the curved term of the cam trajectory was replaced by extremely small line segments. It should be noted that, after k iterations, the value of the $k-1$ steps is a known value in the formula, and the formula becomes an elliptic Equation (25).

$$\mathbf{s}=m_0-\sum _{i=1}^{k-1}{B}_{i-1}{B}_i=B_{k}C+B_{k-1}{B}_k,1\le k\le n$$

(25)

where the tangent point $B_k$ lies on an ellipse with $B_{k-1}$, and C is its foci and $\mathbf{s}$ is its conjugate length (this holds for all k). This results in infinitely many solutions to the equation. To address this issue, we introduced the constraint that the angle of the tangent point at the current moment was equal to the angle of the tangent point that was obtained by solving the circular trajectory mechanism in the Cartesian coordinates. The angle of the tangent point obtained by solving the circular trajectory, i.e., the angle $\angle B_{k}DO$ in Cartesian coordinates, can be expressed in terms of a second-order Fourier series expansion. As such, it can be represented as Equations (26).

$$\begin{array}{cc}\hfill {\vartheta }_B& =84.25-1.793cos\left(\omega \alpha \right)+8.917sin\left(\omega \alpha \right)+\hfill \\ \hfill & 0.6591cos\left(2\omega \alpha \right)-0.1282sin\left(2\omega \alpha \right),\omega =0.0201\hfill \end{array}$$

(26)

Based on the above analysis, numerical solutions were obtained for all tangent points. Moreover, the solutions were plotted in polar coordinates, as shown in the comparison with the circular trajectory shape in Figure 18. This was also an illustration of the $\Gamma$ trajectory of the cam.

From Figure 18a, it can be seen that due to the “tangent point retreat” phenomenon, there was a backward iteration in the tangent points, thus resulting in a “enlarged” phenomenon at the root of the cam mechanism. Similarly, the slight tightening of the string due to this factor was also unavoidable but can be mitigated by a cam mechanism.

For convenience in the structural design, the “enlarged” area was smoothed, and the final cam mechanism based on the parameters designed in the table was obtained, as shown in Figure 18b. Figure 19a depicts a complete comparison between the smoothed cam trajectory and the circular trajectory, and Figure 19b shows the practical design of a cam mechanism using the parameters detailed above from Solidwork.

This section describes the computational method for calculating the trajectory of a cam mechanism that is based on the various parameters of circular rollers. In addition, the trajectory of the circular rollers are also compared with it. In the next section, the designed cam trajectory parameters will be validated through physical experiments.

6. Experiment and Analyses

For the purpose of this project, the details of the physical mechanism were omitted. To verify the accuracy of the string drive mechanism simulation, high-speed photography was used to capture the flapping trajectory of the physical mechanism, which was then compared with the simulation results. Since the validation target of this part was the accuracy of the drive mechanism simulation and to eliminate the influence of inertia under high-speed flapping conditions, the drive mechanism was set to operate at a lower voltage. This was captured under high-speed photography (5000 fps) at a voltage of 1.5 V. As shown in the schematic diagram of the testing environment (Figure 20), two high-speed photography captures of the flapping trajectory from different angles were conducted, where the blue dots represent the marker points that were marked on the carbon fiber rods used for installing wings with a white oil-based ink. This was performed to facilitate a spatial position calibration, which was achieved with three-dimensional video processing software.

Through high-speed photography captures of the actual mechanism, as well as a reproduction of the three-dimensional space position, the motion trajectory of the marker points in three-dimensional space could be obtained. Figure 21 shows the three-dimensional spatial motion trajectory of the four marker points on the two carbon fibers. When compared with the string-based mechanism simulation trajectory with the same design parameters, it can be seen from Figure 22 that, under the condition of low flapping speed and minimal inertia influence, the actual flapping trajectory of the cam mechanism was approximate to the simulation flapping trajectory, which proves the simulation accuracy of the string drive flapping mechanism; thus, it can be used as a guide for physical design.

Also, a phase difference between the left and right flapping was observed. The reason for this was because the string was trying to keep an equal phase, specifically due to the elongation error of the string and the error of manual installation. In the subsequent design, a device for adjusting the tightness of the string was added to keep the tension of the string, which will thus decrease the error in the manual installation.

To verify the superiority of the circle roller over the cam roller, we designed an experiment to compare the energy consumption difference between the different roller mechanisms. We added the same load (8 cm long and a 0.75 mm radius carbon fiber rod) to the circular roller and the cam roller, as well as drove them with the same DC motor at the same driving voltage (which were 2 V and 2.5 V). The current of motor was tested using a hall-effect high-sensitivity current probe, and the force torque fluctuations of the mechanism could be analyzed by comparing the currents of the DC motor to drive the circular roller and cam mechanism. The reason for this was due to the fact that the output power of a DC motor is positively correlated with the current at the same voltage. The test result is shown in Figure 23, where Figure 23a,b are the current of the DC motor when driving the circle roller mechanism at voltage of 2 V and 2.5 V. Figure 23c,d are the current of the DC motor when driving the cam roller mechanism at a voltage of 2 V and 2.5 V. From an intuitive perspective, Figure 23a,b, show that the current of the motor driving circular roller has greater fluctuation, while Figure 23c,d show that the current of the motor driving cam roller is smoother (which can also be proved by data analysis).

Table 5. Effective current of the motors when the circle mechanism and cam mechanism are being driven.

Mechanism	Effective Current at 2 V	Effective Current at 2.5 V
Circle mechanism	223.9 mA	230.4 mA
Cam mechanism	158.7 mA	173.2 mA

shows the effective values of the current passed through the DC motor driving circular roller and the cam roller under different voltages, and the calculation equation of effective value is Equation (27).

$$I_e=\sqrt[2]{\frac{{\displaystyle \sum _{i=1}^n}{\left(I_i\right)}^2}{n}}$$

(27)

where $I_e$ is the effective current, and $I_i$ is the transient current that is measured by an oscilloscope with a total data count of n.

Compared to a circular roller, a motor driving cam mechanism has a smaller effective current and lower energy consumption under the same voltage. At the same time, the current waveform was smoother, which proves that the cam mechanism reduced the phenomenon of the motor torque increase, which was caused by the excessive tension of the string. This also proved, from an experimental perspective, that the cam mechanism that was designed according to parameter analysis had more advantages than the circular roller mechanism.

It is worth mentioning that our experiments were conducted at a relatively low flapping speed with a load of carbon fiber rods of which the mass was almost negligible. At that point, the measured flapping trajectory and maximum amplitude showed a good similarity with the simulation data, which validated the correctness of our kinematic design. However, as the flapping frequency increased and the wings were installed, the aerodynamic and inertial effects gradually increased. Due to the elasticity of the rope and the tensile properties of the structural components, the measured flapping amplitude would be higher than the designed amplitude. Factors such as these are difficult to consider in the design of structural morphology, so they are not within the scope of experimental verification in this article. Studies related to this aspect can be included in future research plans.

7. Conclusions

This article presents an analysis of the simulation and analysis methods for the design parameters in a string-based mechanism. Compared to previous studies on string-based mechanisms, this article conducts a detailed analysis of the various parameters in a string drive mechanism, thereby identifying the core design parameter, namely the ratio of the pulley radius to the installation radius of the center gear (which has the greatest impact on the flapping amplitude and trajectory). Additionally, this article analyzes the causes of over tension and the relaxation issues in circular roller mechanisms, as well as proposes a corresponding cam roller design method that is based on the relevant parameters in circular roller mechanisms, while also providing simulation results for it. Based on the parameters for the cam roller, a 22 mm miniature flapping mechanism was designed, assembled, and tested to verify the consistency between the simulated flapping results of the cam mechanism and the actual measured flapping results. The experimental results showed that, under the condition of minimal inertial influence, the cam mechanism that was designed based on the parameters obtained from the simulation method described above was found to have a trajectory that closely matched with the designed target, and it was also found to be able to effectively reduce the phenomenon of stagnation during the flapping motion. Compared to a circular roller, the motor driving cam mechanism exhibited a smaller effective current and lower energy consumption under the same voltage, and the smoother current waveform indicated that the cam mechanism mitigated the increase in the motor torque due to string tension. In conclusion, the cam mechanism that was designed by parameters analysis can serve as an optimization method for string-based flapping mechanisms.