2.1. Overall design of the bio-inspired flapping-wing robot

To conduct an in-depth study of the reciprocating flapping motion in flapping-wing flight, this paper appropriately simplifies the flapping patterns of birds and adopts the bio-inspired flapping-wing flying robot scheme shown in Figure 2. The prototype primarily consists of a fuselage frame, a transmission system with a flapping mechanism, wings, and a tail adjustment system.

The fuselage frame serves as the main structural component for mounting and securing other parts. The wings primarily provide lift and thrust for the flapping-wing robot, with their specific structure detailed in Figure 2 (C). The schematic of the transmission system and flapping mechanism is shown in Figure 2 (D). The transmission system comprises a brushless DC (Direct Current, DC) motor and a two-stage straight gear reduction mechanism. The flapping mechanism includes a crank, a ball-headed connecting rod, and a wing oscillating lever. The motor power is transmitted through the reduction gears to the crank, which drives the ball-headed connecting rod to actuate the wing oscillating lever in a reciprocating swing, thereby realizing the flapping motion of the wings.

Figure 2.

Bio-inspired flapping-wing robot system design. (A) Prototype system model diagram. (B) Exploded view of the prototype system model. (C) Wing assembly structure diagram. (D) Power and transmission assembly structure diagram. (E) Tail adjustment assembly structure diagram.

Figure 2. Long description

The diagram illustrates a bio-inspired flapping-wing robot system design. It includes five sub-diagrams labeled A through E. Sub-diagram A shows the prototype system model with flapping wings and a tail. Sub-diagram B presents an exploded view of the prototype system model, highlighting components such as the fuselage, flapping wing, tail, connecting rod, and transmission system and flapping mechanism. Sub-diagram C details the wing assembly structure, including elements like the rib, wing surface, leading edge rod, pendulum rod, swing rod base, swing rod, and diagonal beam. Sub-diagram D illustrates the power and transmission assembly structure, featuring a brushless motor, reduction gear set, pendulum rod, swing rod base, connecting rod, spindle, gear set, crank, and flapping angle. Sub-diagram E depicts the tail adjustment assembly structure, showing the tail surface, yaw servo, and pitch servo.

The tail adjustment system is mainly used for flight balancing and providing pitch, roll, and yaw moments for aircraft maneuvering (Figure 2 (E)). This system consists of two servos and corresponding linkage mechanisms. The pitch adjustment servo drives the entire tail to rotate around the pitch axis via a parallelogram mechanism, changing the control surface position to enable longitudinal pitching motion of the fuselage. The roll adjustment servo similarly drives the tail to achieve yaw motion around the yaw axis via a parallelogram mechanism. When the tail control surface rotates around the yaw axis, it provides roll and yaw moments to the fuselage. The assembled first-generation complete prototype is shown in Figure 2 (A) and (B).

2.2. Theoretical model of tail geometric parameters and pitch static stability

The contribution of tail area to the pitch stability of an aircraft primarily depends on the aerodynamic force it generates and the moment arm relative to the center of gravity. To quantify this contribution, the key dimensionless geometric parameter, the horizontal tail volume coefficient V _h , is introduced:

(1) \begin{equation} V_{h}=\frac{S_{h}\cdot l_{h}}{S\cdot \overline{c}} \end{equation}

where S _h is the horizontal tail area; l _h is the tail moment arm, i.e., the distance from the mean AC (aerodynamic center, AC) of the wing to the AC of the horizontal tail. S, $\overline{c}$ are the wing area and mean aerodynamic chord, respectively. V _h quantifies the “moment leverage” capability of the tail geometry relative to the wing. A larger value indicates a potentially stronger stabilizing effect.

Flapping-wing robots require sufficient flight stability during operation. Here, the design index “Pitch Static Margin (Static Margin, SM)” is introduced. The Pitch SM is a key metric for evaluating the longitudinal stability of an aircraft, defined as:

(2) \begin{equation} SM=\frac{x_{np}-x_{cg}}{\overline{c}} \end{equation}

where x _np is the AC position of the complete aircraft. x _cg is the center of gravity position of the complete aircraft. $\overline{c}$ is the wing mean aerodynamic chord. SM>0 indicates the aircraft is statically stable. A larger SM value indicates stronger stability, but may degrade maneuverability. For flapping-wing robot, SM is typically set between 5% and 15%.

According to longitudinal stability theory, the overall AC x _np is the superposition of contributions from the wing and the tail. The classic expression is:

(3) \begin{equation} x_{np}=x_{ac}+\frac{S_{h}}{S}\cdot \frac{a_{h}}{a}\cdot \left(1-\frac{\partial \epsilon }{\partial \alpha }\right)\cdot l_{h} \end{equation}

where x _ac is the AC position of the wing (typically located at about 25% chord); a and a _h are the lift curve slopes of the wing and tail, respectively; $\partial \epsilon/\partial \alpha$ is the rate of change of downwash angle induced by the wing wake.

Substituting the definition of the tail volume coefficient V _h into the above equation:

(4) \begin{equation} x_{np}=x_{ac}+\left(V_{h}\cdot \overline{c}\right)\cdot \left[\frac{a_{h}}{a}\cdot \left(1-\frac{\partial \epsilon }{\partial \alpha }\right)\right] \end{equation}

Let

(5) \begin{equation} \eta _{v}\equiv \eta \cdot \frac{a_{h}}{a}\cdot \left(1-\frac{\partial \epsilon }{\partial \alpha }\right) \end{equation}

Here, $\eta _{\textrm{v}}$ represents the tail efficiency factor, which denotes the ratio of the actual stabilizing moment effect produced by the tail to that under ideal, undisturbed conditions. It comprehensively accounts for the effects of tail lift efficiency, downwash, and dynamic pressure loss. $\eta$ : The ratio of dynamic pressure at the tail to free stream dynamic pressure (typically slightly less than 1). Thus, the formula (4) for the overall AC position can be simplified to:

(6) \begin{equation} x_{np}=x_{ac}+\eta _{v}\cdot V_{h}\cdot \overline{c} \end{equation}

Substituting this simplified relation into the definition of the SM:

(7) \begin{equation} SM=\frac{\left(x_{ac}+\eta _{v}\cdot V_{h}\cdot \overline{c}\right)-x_{cg}}{\overline{c}} \end{equation}

Rearranging and solving for the required tail volume coefficient V_h to meet a specified SM:

(8) \begin{equation} V_{h}=\frac{SM+\frac{x_{cg}-x_{ac}}{\overline{c}}}{\eta _{v}} \end{equation}

Here, the numerator (x _cg −x _ac )/ $\,\overline{c}$ represents the position of the center of gravity relative to the wing’s AC. If the center of gravity is behind the AC, this term is positive, which weakens the inherent stability, requiring a larger tail (larger V _h ) for compensation. The denominator $\eta _{\textrm{v}}$ : The lower the tail efficiency (smaller $\eta _{\textrm{v}}$ ), the larger the tail volume required to achieve the same stabilizing effect.

From the definition of V _h , the required horizontal tail area can be derived:

(9) \begin{equation} S_{h}=V_{h}\cdot \frac{S\cdot \overline{c}}{l_{h}}=\frac{S\cdot \overline{c}}{l_{h}}\cdot \frac{SM+\frac{x_{cg}-x_{ac}}{\overline{c}}}{\eta _{v}} \end{equation}

Based on design experience for flapping-wing robot, the static margin SM is set to 10%. The typical range for the tail efficiency factor $\eta _{\textrm{v}}$ is 0.80 to 0.95. Considering the significant reduction in equivalent $\eta _{\textrm{v}}$ value due to intense fluctuations in the downwash field caused by the unsteady wing wake vortices, low Reynolds number effects, and possible dynamic pressure ratio losses in flapping-wing robot, a value of 0.80 is adopted here. Finally, by substituting parameters such as the wing area and mean aerodynamic chord of the flapping-wing robot into the formula (9), the required tail area for the flapping-wing robot can be calculated, providing a theoretical basis for the subsequent parametric design of multi-configuration tails.

2.3. Application of the theoretical model in tail design

The above theoretical model provides a fundamental area constraint for the tail design in this study. Substituting the parameters of the aircraft in this paper (wing area S = 0.234 m², mean aerodynamic chord l = 0.156 m, tail arm length l _h = 0.45 m, relative center-of-gravity position x _cg/ $\,\overline{c}$ = 0.35, relative wing aerodynamic center position x _ac/ $\,\overline{c}$ = 0.25, desired static stability margin SM = 10%, tail efficiency factor $\eta _{v} = 0.80$ ), the required horizontal tail area is calculated to be S _h≈102,440 mm². This area value is used as a uniform constraint in the parametric design of all six tail configurations in Chapter 3, ensuring a fair baseline for performance comparison among different configurations.

It should be noted that this model primarily addresses longitudinal static stability. For lateral-directional control effectiveness and channel coupling characteristics, there is currently no analytical theory that directly provides optimal geometric parameters. Therefore, in Chapter 3, we adopted a control-variable method to design a series of parametric specimens (varying characteristic width, opening angle, control surface area, tail arm angle, etc.), and quantitatively evaluated the influence of each parameter through wind tunnel experiments in Chapter 4, thereby obtaining optimization recommendations for specific configurations. Theory and experiment complement each other, together forming a complete tail design process.

2.4. Scope and limitations of the theoretical model

The theoretical model established in this paper is mainly intended for the preliminary parametric design of flapping-wing robot tails. Its core function is to establish a quantitative relationship between tail geometric dimensions and pitch static stability margin requirements, providing a unified basis for selecting the area and scale of multi-configuration tails. It should be noted that this model is a simplified theoretical analysis oriented toward design, focusing primarily on the dominant influence of the tail on longitudinal static stability. For the detailed response laws of different tail configurations in terms of lateral-directional aerodynamic forces, control coupling, and unsteady wake interference, quantitative evaluation still needs to be conducted through subsequent wind tunnel experiments. Based on this, Chapter 3 further carries out the parametric design of multi-configuration tails, and Chapter 4 systematically compares the three-axis aerodynamic moment characteristics of each configuration through wind tunnel experiments.

2.5. Theoretical innovations of this research

Compared with existing research on flapping wing robot tails, the theoretical innovations of this research are mainly reflected in the following aspects. First, to address the long standing problem in preliminary tail design for flapping wing robot that “parameter selection relies mainly on experience and lacks a unified theoretical basis,” this paper, based on the tail volume coefficient and longitudinal static stability theory, introduces the pitch static stability margin as a core design indicator and establishes a quantitative relationship between tail size parameters and longitudinal static stability, thereby directly linking tail geometric design to flight stability requirements.

Second, building upon the traditional static stability analysis framework for fixed-wing tails, this paper further incorporates the unsteady aerodynamic environment of flapping-wing robot by including the effects of flapping-wing wake, downwash, and dynamic pressure loss in the form of an equivalent tail efficiency factor. This allows the model to more reasonably reflect the actual variation in tail static effectiveness under flapping conditions.

Finally, based on the above model, this paper further derives an expression for the required tail area to satisfy a given static stability margin, providing a unified theoretical basis for the parametric design of multi-configuration tails. Thus, the theoretical framework established in this paper can not only be used to explain the coupling relationships among tail area, tail moment arm, and stability, but also provides a calculable and generalizable analytical basis for preliminary size design and configuration comparison of flapping wing robot tails.

Therefore, the theoretical contribution of this paper is not limited to a simple application of existing tail stability concepts, but rather lies in constructing a unified analytical framework tailored to flapping wing aircraft application scenarios that can serve the parametric design of multi-configuration tails.

3.1. Arc-shaped tail

In nature, arc-shaped tails are widely found in various bird species. Their morphological characteristics are primarily manifested as a typical fan shape when fully deployed, featuring a relatively high aspect ratio and large area, which can provide significant lift or drag to the body. When folded, the projected area is substantially reduced, contributing to lower aerodynamic drag. This morphological adaptability enables such tails to operate across a wide range of flight speeds.

Figure 3.

Whole-machine assembly diagram and structural parameter diagram of the arc-shaped tail.

Figure 3. Long description

The image consists of two diagrams. The left diagram illustrates the whole-machine assembly of an arc-shaped tail attached to a mechanical structure. The right diagram shows the structural parameters of the arc-shaped tail, including the distance d, radius R, angle theta_out, and the arc length s. The diagrams highlight the mechanical design and key dimensions of the tail, which are crucial for understanding its functionality and integration into the overall system.

To focus on studying the aerodynamic characteristics of the arc-shaped tail itself, this paper designs an arc-shaped tail configuration as shown in Figure 3, without considering its folding mechanism. The left part of the figure shows a schematic of the bio-inspired flapping-wing robot assembled with this tail, and the right part provides a detailed view of the tail assembly. The key characteristic parameters of this design include: tail area (S), characteristic width (d), wing surface opening angle (θ _out), and arc profile radius (R). Based on this, this paper systematically varies the characteristic width d and the opening angle θ _out to design three tail specimens (#11, #12, #13) to investigate the influence of geometric parameters on aerodynamic performance. Specific parameters are listed in Table I, where #11 is the baseline design, #12 reduces the characteristic width while maintaining the same area, and #13 increases the characteristic width.

Table I.

Parameters of three arc-shaped tails.

3.2. Triangular-shaped tail

In nature, certain birds possess tails with nearly straight trailing edges, which can be morphologically approximated as triangular profiles, such as some raptors (e.g., the Common Buzzard). Compared to arc-shaped tails, this type is often associated with specific flight modes.

To investigate the aerodynamic characteristics of straight-edged tails and to conduct a comparative analysis as an intermediate form between arc-shaped and swallow-shaped configurations, this study designed a tail configuration with a standard triangular profile. The overall schematic and detailed structural diagram are shown in Figure 4.

The geometric characteristics of the triangular tail are defined by the following key parameters: the included angle of the outer contour (θ _out), the outer contour edge length (L), the characteristic width between wingtips (d), and the effective wing area (S). Based on these parameters, this study desgned and fabricated three triangular tail specimens (#21, #22, #23) with different characteristic widths (d). Specific parameters are listed in Table II, where #21 is the baseline design, #22 reduces the characteristic width while maintaining the same area, and #23 increases the characteristic width.

Table II.

Parameters of three triangle-shaped tails.

Figure 4.

Whole-machine assembly diagram and structural parameter diagram of the triangular-shaped tail.

Figure 4. Long description

The image shows a detailed diagram of a triangular-shaped tail assembly for an aircraft. The left side of the image displays the whole-machine assembly, highlighting the tail’s integration with the aircraft’s body. The right side of the image presents a structural parameter diagram, illustrating key measurements such as the length (L), the distance (d), and the angle (theta out). The diagram also includes labels for the x-axis and y-axis, indicating the orientation of the tail structure. The tail’s triangular shape is clearly depicted, with specific attention to its aerodynamic design and structural components.

3.3. Swallow-shaped tail

Swallow-shaped tails are commonly found in nature among species such as swifts, which are known for high-speed, highly maneuverable flight. Their morphological characteristic is a concave central trailing edge, forming a forked structure. This configuration can effectively maintain aerodynamic moments while significantly reducing drag, particularly facilitating rapid roll and yaw maneuvers.

To study the aerodynamic characteristics and control effectiveness of this type of tail on a flapping-wing platform, this paper designed a swallow-shaped configuration based on its forked contour features (Figure 5). Its structural characteristics are primarily reflected in parameters such as wing area, characteristic width, inner and outer contour angles, and outer contour edge length. This paper uses the following parameters to define its geometric shape: outer contour angle (θ _out), inner contour angle (θ ^′), outer contour edge length (L), characteristic width between wingtips (d), and effective wing area (S).

Table III.

Parameters of three swallow-shaped tails.

Figure 5.

Whole-machine assembly diagram and structural parameter diagram of the swallow-shaped tail.

Figure 5. Long description

The image features a detailed diagram of a swallow-shaped tail, showcasing its biomechanical structure and parameters. The left side of the image displays the whole-machine assembly, highlighting the tail’s connection to the main body and its articulated segments. The right side presents a structural parameter diagram, illustrating angles such as theta prime and theta out, along with dimensions like distance d and length L. The tail’s design appears to be inspired by bird anatomy, emphasizing its role in flight dynamics. Annotations indicate the tail’s influence on lift, drag, and stability, with references to biomechanical studies and fluid-structure interactions. The diagram includes labels for axes and key measurements, providing a comprehensive view of the tail’s functional components and their interactions.

Based on the above parameters, this paper designed and fabricated three swallow-shaped tail specimens (#31, #32, #33) with different characteristic widths (d) to investigate the influence of characteristic width on aerodynamic control forces. Specific geometric parameters are listed in Table III, where #31 is the baseline configuration, #32 reduces the characteristic width while maintaining the same area, and #33 increases the characteristic width.

3.4. Webbed tail

The design inspiration for the webbed tail comes from the phenomenon observed in certain water birds using their webbed feet to assist in attitude adjustment during flight. This configuration features two independent control surfaces, enabling aerodynamic moment control in three channels — pitch, roll, and yaw — through symmetric or differential pitching motions of the two wing surfaces (Figure 6).

The tail consists of two symmetric and independently controllable wing surfaces. Its geometric characteristics are defined by the following parameters: triangular contour angle of a single wing surface (θ _web), dihedral angle between the tail support arms ( ${{\alpha}}_{\textrm{web}}$ ), angle between the centerlines of the two tail surfaces ( $\beta_{\textrm{web}}$ ), which value is jointly determined by the tail arm opening angle ${{\alpha}}_{\textrm{web}}$ and the triangular profile of a single tail surface. The specific correspondence is shown in Table IV, outer contour edge length of a single wing surface (L), distance from the root of a single wing surface to the centerline of the tail module (L ^′), characteristic width between the wingtips of a single wing surface (d), and effective area of a single wing surface (S).

To investigate the influence of the tail-arm dihedral angle ( ${{\alpha}}_{\textrm{web}}$ ) on control effectiveness, this paper designed and fabricated two basic configuration webbed tail specimens (#41, #42) (Table IV), with the core difference being the value of ${{\alpha}}_{\textrm{web}}$ .

Table IV.

Parameters of two webbed tails.

Figure 6.

Whole-machine assembly diagram and structural parameter diagram of the webbed tail.

Figure 6. Long description

The image shows a detailed diagram of a webbed tail assembly, including its structural parameters. The left side of the image depicts the whole-machine assembly, highlighting the interconnected components and their spatial arrangement. The right side of the image focuses on the structural parameters, illustrating various angles, lengths, and distances with labeled lines and arrows. Key elements include the tail structure, support mechanisms, and geometric relationships between different parts. The diagram provides a comprehensive view of how the webbed tail is assembled and how its structural parameters are defined.

3.5. T-shaped tail

The T-shaped tail is one of the most common tail configurations in fixed-wing aircraft, featuring a “T”-shaped layout with a horizontal tail and a vertical tail. Its main advantage is the ability to achieve aerodynamic decoupling between the pitch and yaw channels, thereby simplifying flight-control logic. In flapping-wing vehicle research, some prototypes have also adopted this layout to improve the independence and predictability of attitude control.

Figure 7.

Whole-machine assembly diagram and structural parameter diagram of the T-shaped tail.

Figure 7. Long description

The image shows a detailed diagram of the whole-machine assembly and structural parameters of a T-shaped tail. The diagram includes a side view of the assembly, highlighting the main components such as the wings, tail, and fuselage. The structural parameter diagram on the right side illustrates the geometric relationships and dimensions of the T-shaped tail. Key elements include the horizontal stabilizer (S_H), vertical stabilizer (S_V), rudder (S_r), elevator (S_e), and various angles and distances such as theta_r, d, and L. The z-axis and y-axis are marked to indicate the orientation of the components.

To systematically evaluate the aerodynamic performance of the T-shaped tail on a flapping-wing platform, this paper designed a T-shaped tail configuration as shown in Figure 7. Its geometric characteristics are defined by the following parameters: triangular contour angle of the horizontal wing surface (θ _T), which is fixed at 55.4° in this study, outer contour edge length of the horizontal wing surface (L), characteristic width between the wingtips of the horizontal wing surface (d), area of the horizontal stabilizer (S _H ), area of the elevator control surface (S _e ), area of the vertical stabilizer (S _V ), and area of the rudder control surface (S _r ).

To investigate the influence of control-surface area on handling effectiveness, three sets of T-shaped tail specimens (#51, #52, #53) were designed by varying the elevator area (S _e ) while keeping the total horizontal tail area (S _H + S _e ) constant. Their specific geometric parameters are listed in Table V, where #51 is the baseline design, #52 reduces the elevator area, and #53 increases it.

Table V.

Parameters of three conventional T-shaped tails.

3.6. V-shaped tail

The V-shaped tail is a tail configuration widely used in fixed-wing aircraft. Its structural feature is two stabilizers arranged symmetrically at a given dihedral angle in a “V” shape, with independently controllable surfaces usually attached to their trailing edges. Compared with the T-shaped tail, the V-shaped tail theoretically offers lower overall drag and structural weight, but its control logic is more complex, requiring control allocation as symmetric deflection mainly generates pitch moments while differential deflection primarily produces coupled roll and yaw moments.

Figure 8.

Whole-machine assembly diagram and structural parameter diagram of the V-shaped tail. Overall assembly of the flapping-wing robot equipped with the V-shaped tail (left). Structural parameter definition of the V-shaped tail configuration (right).

Figure 8. Long description

The image shows a detailed diagram of a V-shaped tail assembly for a machine. The diagram includes a side view of the entire assembly, highlighting the tail structure and its connection to the main body. Key components such as the tail surfaces, support structures, and attachment points are labeled. Additionally, a close-up view illustrates the structural parameters, including angles, lengths, and distances, which are crucial for understanding the tail’s design and functionality. Arrows and lines indicate the relationships and measurements between different parts of the assembly.

To evaluate the applicability of the V-shaped tail on flapping-wing vehicle platforms and its unique control-coupling characteristics, this paper designed a V-shaped tail configuration as shown in Figure 8. In this paper, the V-shaped tail is defined using the following parameters according to the structural form illustrated: the dihedral angle between the two stabilizers (θ _V), the triangular contour angle of a single stabilizer ( ${{\alpha}}_{\textrm{V}}$ ), which is fixed at 30° in this study, the total contour edge length of a single wing surface (L), the characteristic width between the two wingtip points of the control surface on a single wing (d), the total area of the V-shaped tail (S), and S ₀ is the total control surface area of the V-shaped tail, i.e., the sum of the movable surface areas on the left and right sides. In this study, it is fixed at 31,155.52 mm².

To investigate the influence of the stabilizer dihedral angle θ _V on aerodynamic performance and control characteristics, three sets of V-shaped tail specimens (#61, #62, #63) were designed by varying the angle θ _V while keeping the tail area constant. Their specific geometric parameters are listed in Table VI.

Table VI.

Parameters of three tails with V-shaped tails.

3.7. Innovations in parametric design of multi-configuration tails

The innovations in tail design in this chapter are mainly reflected in the following three aspects.

First, this paper does not conduct local optimization for a single tail configuration, but rather systematically proposes six representative tail configurations for flapping-wing robot tail design, including the arc-shaped tail, triangular-shaped tail, swallow-shaped tail, webbed tail, and two typical fixed-wing tail forms (T-shaped tail and V-shaped tail), completing the design of a total of 17 parametric specimens. Compared with existing research that simplifies the tail into a single, rigid, fixed-wing configuration, this paper establishes a unified comparison system covering both bio-inspired configurations and conventional engineering configurations, which is conducive to more comprehensively revealing the influence patterns of different tail layouts on the aerodynamic control capabilities of flapping-wing robot.

Second, this paper adopts a parametric design approach of “maintaining area constraints while highlighting key variables.” For different tail configurations, key geometric parameters such as characteristic width, tail aspect ratio, control surface area, and stabilizer dihedral angle are extracted, and parameter series are constructed while keeping the total area substantially constant. This design method effectively reduces interference caused by area differences, enabling the experimental results to better reflect the influence of the configuration itself and key geometric parameters on the three-axis aerodynamic moments, thereby improving the fairness of horizontal comparisons among different tails and the reliability of the conclusions.

Third, based on the structural characteristics of different tail configurations, this paper designs corresponding differentiated actuation modes and testing logics. For the arc-shaped tail, triangular-shaped tail, and swallow-shaped tails, the focus is on examining the influence of global pitch and lateral deflection actuation on the three-axis aerodynamic moments. For the webbed tail, combined with its dual control surface independent control feature, symmetric/differential actuation modes are designed. For the T-shaped and V-shaped tails, corresponding actuation states are set according to their elevator-rudder separate control and symmetric/differential coupling control characteristics, respectively. This “configuration-feature-based actuation mode definition” allows each type of tail to be tested and compared under conditions that better match its actual control approach, thereby improving the applicability of the experimental design.

In summary, the design-level innovation of this paper is not merely the proposal of multiple tail shapes, but more importantly, the construction of a systematic design framework consisting of “multi-configuration selection – key variable extraction – equal-constraint parametric design – differentiated actuation definition.” This framework provides a repeatable and extensible research path for the scheme selection, preliminary design, and subsequent optimization of flapping-wing robot tails.

3.8. Indirect guidance of the theoretical model for parametric design

Although the theoretical model in Section 2.2 mainly determines the total tail area, the physical principles behind it still provide indirect guidance for the selection of internal geometric parameters:

1. 1. Tail arm length: The tail arm length l _h in the theoretical model directly affects the tail volume coefficient and should be made as large as structurally permissible. All configurations in this study adopt the same tail length (450 mm) to ensure fair comparison.

2. 2. Aspect ratio and lift curve slope: The lift curve slope a _h in the theoretical model is positively correlated with the tail aspect ratio. Therefore, under a fixed area, appropriately increasing the characteristic width (i.e., increasing the aspect ratio) is beneficial for improving tail aerodynamic efficiency. This provides a theoretical basis for the variation range of the characteristic width d in the arc-shaped tail, triangular-shaped tail, and swallow-shaped tail.

3. 3. Control surface area allocation: For the T-shaped tail, the theoretical model indicates that the pitch control moment is proportional to the control surface area on the horizontal tail. Therefore, this paper designs three different elevator areas to quantitatively verify this linear relationship.

4. 4. Tail efficiency factor and wake interference: The efficiency factor $\eta _{\textrm{v}}$ in the theoretical model is affected by the position of the tail relative to the wing. For the V-shaped tail and the webbed tail, changes in the tail surface configuration alter the effective moment arm of the control surfaces and the degree of downwash interference, thereby affecting $\eta _{\textrm{v}}$ . This paper investigates these effects by varying $\alpha _{\textrm{web}}$ (60°/90°) and $\alpha_{\textrm{v}}$ (90°/120°/150°).

In summary, although the theoretical model does not directly give the optimal value for each parameter, it provides a physical basis and initial estimates for the selection ranges of parameters and the design of experiments.

4.1 Wind tunnel experimental platform and testing

To accurately evaluate the aerodynamic performance of the six tail configurations under realistic airflow conditions, this chapter conducted systematic quantitative tests based on a wind-tunnel experimental platform. To precisely distinguish the respective aerodynamic contributions of the flapping-wing body and the tail and accurately measure the aerodynamic moments of the tail at different positions and actuation states, the experiment adopted a modular separation-measurement approach. The free stream velocity was set to 10 m/s, and the flapping frequency was fixed at 2.5 Hz, corresponding to the typical cruise speed and flapping frequency of the flapping-wing robot, respectively. The experiments were conducted in a recirculating/open-return wind tunnel at the Harbin Institute of Technology (Shenzhen) campus. The dimensions of the wind tunnel test section are 2.4 m in length, 6 m in width, and 3.6 m in height, with a maximum controllable wind speed of 35 m/s. The flow quality in the test section is excellent, with a turbulence intensity below 1%, meeting the requirements for high-precision aerodynamic testing. The single-side flapping stroke angle ranges from –5° to + 40°. All force and moment data were acquired via a six-axis force/torque sensor at a sampling frequency of 1,000 Hz and were appropriately filtered to ensure signal quality.

Figure 9.

Wind tunnel experimental platform and measurement system. (A) Force analysis diagram of the prototype system. (B) Prototype measurement system diagram. (C) Physical diagram of the prototype system. (D) Wind tunnel experimental measurement diagram of the prototype.

Figure 9. Long description

The image consists of four panels illustrating a flapping wing robot and its experimental setup. Panel A shows a force analysis diagram of the prototype system, highlighting various forces such as lift, tail force, and gravitational force acting on the robot. Key components like the wing measurement module, tail wing component, and tail measurement module are labeled. Panel B provides a detailed diagram of the prototype measurement system, including six-axis force sensors, linear position sensors, lead screws, and stepper motors. Panel C displays a physical photograph of the prototype system, showcasing the robot with its wings and tail. Panel D depicts the wind tunnel experimental setup where the prototype is being tested. The image collectively illustrates the design, measurement system, and experimental process of the flapping wing robot.

The wind-tunnel experimental platform is shown in Figure 9, mainly comprising a lower support platform and an upper test device. The test system consists of a flapping-wing performance measurement module (orange dashed box in Figure 9 (B)) and a tail performance measurement module (red dashed box), which can operate independently or synchronously. The flapping-wing performance measurement module is used to measure the aerodynamic forces/moments generated by the flapping mechanism itself. The tail performance measurement module is used to independently and accurately measure the aerodynamic forces/moments acting on the tail assembly in the airflow while enabling controllable adjustment of its position relative to the fuselage. It includes: a two-dimensional translation slide system (consisting of precision lead-screw rails driven by two stepper motors) for controlling the longitudinal (X-axis) and vertical (Z-axis) positions of the tail assembly; linear displacement sensors and connecting rods integrated into the slide, providing real-time feedback on the precise coordinates of the tail assembly; and a high-precision six-axis force/torque sensor mounted on the translation slide, directly measuring the aerodynamic forces and moments on the tail assembly. The data acquisition and control system is used to synchronously collect output signals from all sensors and to control the actuation of motors and servos.

The experimental test content includes tail performance test: With the flapping wing operating normally, measure the time-average and periodic aerodynamic forces/moments acting on the tail under different actuation modes.

The actual installation and test scenarios of the experimental prototype in the wind tunnel are shown in Figure 9 (C) and (D). Through the above separation-measurement design, the aerodynamic contributions of the flapping-wing body and the tail can be effectively decoupled, providing a high-precision experimental data foundation for subsequent analysis of the independent aerodynamic effectiveness and control-coupling characteristics of various tails.

4.2 Analysis and discussion of aerodynamic moment results

Based on wind-tunnel experimental data, this section systematically analyzes the aerodynamic performance of the six types of tails. For clarity, the definition of actuation states is grouped according to the control logic of the tails: arc-shaped, triangular-shaped, and swallow-shaped tails, which share the same control logic, use a common set of standard state definitions (provided in Section 4.2.1); while webbed, T-shaped, and V-shaped tails, which have unique control logic, have their states defined independently at the beginning of their respective subsections.

It should be noted that the aforementioned theoretical model provides a basis for selecting tail area and scale parameters from the perspective of longitudinal static stability design, whereas the specific aerodynamic responses of different tail configurations in terms of yaw, roll, and multi-channel coupling still need to be revealed through experimental data. Therefore, the focus of the analysis in this section is not simply to repeat the validation of the theoretical formulas in Chapter 2, but rather, under unified design constraints and comparable test conditions, to further quantitatively compare the effects of different tail configurations and their key geometric parameters on three-axis aerodynamic moment outputs and control coupling characteristics, thereby supplementing the theoretical model’s limitations in describing complex configuration effects.

4.2.2 Influence of triangular-shaped tail structural parameters on maneuverability

To investigate the influence of the characteristic width (d) of the triangular-shaped tail on its attitude control effectiveness, this study designed and fabricated three tail specimens with different d values according to Table II (Figure 11 (A)). During wind tunnel experiments, the three-axis aerodynamic moments of each tail under four standard actuation states were systematically measured, with the resulting curves presented in Figure 11 (B)–(D). The analysis of the triangular-shaped tail in this section follows the four standard actuation states (State 1–4) defined in Section 4.2.1 to maintain consistency for comparison.

Figure 10.

Arc-shaped tail and its moment curves. (A) Physical images of three arc-shaped tails with the same area but different characteristic widths. (B) Pitch moment curves of the three tails under wind tunnel conditions. (C) Yaw moment curves of the three tails under wind tunnel conditions. (D) Roll moment curves of the three tails under wind tunnel conditions.

Figure 10. Long description

The image contains three photos of red arc-shaped tails with different widths labeled as eleven, twelve, and thirteen, each with a specified width in millimeters. The tails are shown side by side. Additionally, there are three line graphs displaying the pitching, yawing, and rolling moments of the tails under wind tunnel conditions. Each graph has three lines representing the different tails, with the pitching moment graph showing variations in pitching moment across four actuation states, the yawing moment graph showing variations in yawing moment across four actuation states, and the rolling moment graph showing variations in rolling moment across four actuation states.

Figure 11.

Triangular tail and its moment curves. (A) Physical images of three triangular tails with the same area but different characteristic widths. (B) Pitch moment curves of the three tails under wind tunnel conditions. (C) Yaw moment curves of the three tails under wind tunnel conditions. (D) Roll moment curves of the three tails under wind tunnel conditions.

Figure 11. Long description

The image contains one photo and three graphs. The photo shows three triangular-shaped tails with different widths. The tails are labeled as twenty-one, twenty-two, and twenty-three, with widths of five hundred ninety-one point thirty-one millimeters, five hundred forty-one point thirty-one millimeters, and six hundred forty-one point thirty-one millimeters, respectively. The graphs display pitch, yaw, and roll moment curves for the tails under wind tunnel conditions. The pitch moment graph shows the pitching moment in newton meters against actuation states for the three tails. The yaw moment graph shows the yawing moment in newton meters against actuation states for the three tails. The roll moment graph shows the rolling moment in newton meters against actuation states for the three tails. Each graph uses different colors to represent the data for each tail.

Longitudinal control characteristics: As shown in Figure 11 (B), during pure pitch actuation (State 2), the pitch moment (M _p ) of all configurations showed a significant jump (#21: 0.113 N·m; #22: 0.125 N·m; #23: 0.122 N·m), while changes in yaw and roll moments were minimal. This indicates that the triangular tail exhibits weak coupling interference to the lateral-directional channels during pitch actuation, demonstrating good longitudinal control purity.

Lateral-directional control and channel coupling: Figure 11 (C) and (D) reveal that under pure sideslip actuation (State 3), the yaw moment (M _y ) and roll moment (M _r ) of all configurations increased substantially, accompanied by noticeable changes in M _p , confirming that sideslip operation introduces significant longitudinal cross-coupling. Among them, tail #23 with the largest characteristic width demonstrated the most outstanding lateral-directional control capability, with its M _y and M _r outputs in State 3 being the highest among the three (M _y ≈ 0.086 N·m, M _r ≈ 0.094 N·m). Compared to the baseline configuration #21, the roll moment of #23 increased by approximately 91.8%, representing an extremely significant enhancement.

Combined actuation performance analysis: In State 4 (combined actuation), the lateral-directional moments of the #23 tail remained leading (M _y ≈ 0.101 N·m, M _r ≈ 0.097 N·m), reflecting its strong lateral-directional control potential and disturbance rejection capability. It can also be observed that the M _p of each configuration generally decreased in State 4 compared to State 2, indicating the presence of nonlinear interference and moment redistribution between control channels.

Comprehensive evaluation and configuration optimization: The results show that increasing the characteristic width significantly enhances lateral-directional control, particularly roll moment. Configuration #23 provides the best overall performance for high-maneuverability applications due to its superior lateral-directional control margin during sideslip and combined actuation.

4.2.3 Influence of swallow-shaped tail structural parameters on maneuverability

To study the effect of characteristic width (d) on aerodynamic moments and coupling, three swallow-shaped tails with varying d (Table III) were tested in a wind tunnel (Figure 12(A)). Time-averaged three-axis moments under four actuation states were measured (Figure 12 (B)–(D)). The analysis of the swallow-shaped tail also adopts the standard actuation states (State 1–4) defined in Section 4.2.1.

Figure 12.

Swallow-shaped tail and its moment curves. (A) Physical images of three swallow-shaped tails with the same area but different characteristic widths. (B) Pitch moment curves of the three tails under wind tunnel conditions. (C) Yaw moment curves of the three tails under wind tunnel conditions. (D) Roll moment curves of the three tails under wind tunnel conditions.

Figure 12. Long description

The image displays three swallow-shaped tails with identical areas but varying characteristic widths. The top tail, labeled 31, measures 709.33 millimeters in width. The middle tail, labeled 32, measures 659.33 millimeters in width. The bottom tail, labeled 33, measures 759.33 millimeters in width. The graphs on the right show the moment curves for these tails under wind tunnel conditions. Graph B illustrates the pitch moment curves, Graph C shows the yaw moment curves, and Graph D presents the roll moment curves. Each graph compares the performance of the three tails across different actuation states.

Longitudinal control channel characteristics: As shown in Figure 12(B), under pure pitch actuation, the pitch moment (M _p ) of all swallow-shaped tail configurations showed a significant jump (#31: 0.135 N·m; #32: 0.149 N·m; #33: 0.145 N·m), indicating efficient longitudinal control capability. At this time, changes in lateral-directional moments were minimal, suggesting good static decoupling between the longitudinal and lateral-directional channels under this actuation mode.

Lateral-directional control capability and coupling effects: During pure sideslip actuation, Figure 12 (C) and (D) show that the lateral-directional moments increased sharply and became the dominant response. Simultaneously, the pitch moment also changed significantly (e.g., #31 jumped from ≈0 N·m in the baseline state to 0.108 N·m), revealing significant longitudinal cross-coupling during sideslip actuation of the swallow-shaped tail. It is noteworthy that configuration #33 (largest d) exhibited the strongest lateral-directional control output in State 3 (M _y ≈ 0.109 N·m, M _r ≈ 0.123 N·m). Compared to baseline #31, the roll moment of #33 increased by approximately 68%, indicating that increasing d can effectively enhance the lateral-directional, and especially the roll control gain.

Nonlinear interaction under combined actuation: Under combined pitch and sideslip actuation, the lateral-directional moment output of #33 further reached its peak. However, compared to State 2, the pitch moment of all configurations in State 4 decreased to varying degrees, confirming the existence of nonlinear interaction between the longitudinal and lateral-directional control channels during multi-channel combined actuation.

Comprehensive configuration evaluation and optimization: Experiments show that, under the constraints of constant area (S) and inner contour angle (θ ^′), increasing the characteristic width effectively enhances the lateral-directional control moments of the swallow-shaped tail. Although #32 exhibits the highest peak pitch moment under pure pitch actuation, #33 demonstrates better lateral-directional control margin and overall attitude control potential during sideslip and combined actuation, making it the optimal configuration choice for high-maneuverability missions.

4.2.4 Influence of webbed tail structural parameters on maneuverability

To evaluate the influence of the key structural parameter of the webbed tail with two independent control surfaces — the tail-arm opening angle ( ${{\alpha}}_{\textrm{web}}$ ) — on multi-channel control effectiveness and coupling characteristics, this study fabricated two basic configuration tail specimens according to Table IV (Figure 13(A)). Given its unique dual-control-surface configuration, three characteristic actuation states were defined to systematically assess its control logic and performance: State 1 (baseline, both control surface deflection angles at 0°); State 2 (symmetric pitch actuation, both control surfaces deflected upward by 20°); State 3 (differential pitch actuation, one control surface deflected upward by 20°, the other downward by 20°). The time-averaged three-axis aerodynamic moment curves measured in the wind tunnel are shown in Figure 13 (B)–(D).

Longitudinal control performance under symmetric pitch actuation: In State 2, the pitch moment (M _p ) of both configurations reached their peak values (#41: M _p ≈ 0.153 N·m; #42: M _p ≈ 0.132 N·m), while the yaw moment (M _y ) and roll moment (M _r ) showed only minor changes. This confirms that symmetric pitch actuation can efficiently and nearly decoupledly excite longitudinal control moments. Notably, configuration #41 with the smaller opening angle ( ${{\alpha}}_{\textrm{web}}$ = 60°) exhibited a pitch moment peak approximately 16% higher than that of configuration #42, indicating that a smaller ${{\alpha}}_{\textrm{web}}$ can generate greater pitch control efficiency at the same deflection angle.

Lateral-directional control and coupling under differential pitch actuation: In State 3, the aerodynamic response pattern undergoes a fundamental shift. The lateral-directional moments (M _y , M _r ) increase sharply (#41: M _y ≈ 0.115 N·m, M _r ≈ 0.125 N·m; #42: M _y ≈ 0.076 N·m, M _r ≈ 0.106 N·m) and become the dominant output. Concurrently, the pitch moment (M _p ) significantly decreases compared to State 2, clearly demonstrating the functional logic: “symmetric actuation controls pitch, differential actuation controls yaw/roll.”

Influence mechanism and trade-off of the tail-arm opening angle ( ${\boldsymbol{\alpha}}_{\textrm{web}}$ ): Comparing the two configurations, #41 demonstrates stronger moment output capabilities across all performance metrics. This primarily stems from a smaller ${{\alpha}}_{\textrm{web}}$ increasing the effective moment arms of the two control surfaces in both the longitudinal and lateral directions, thereby enhancing the aerodynamic control efficiency of each channel. However, stronger control capability is accompanied by more pronounced channel coupling: in State 3, the residual pitch moment for #41 is much higher than that for #42, indicating stronger parasitic “differential-to-pitch” coupling, which requires more decoupling compensation in the control law design.

Figure 13.

Webbed tail and its moment curves. (A) Physical images of two webbed tails with the same area but different tail-arm opening angles. (B) Pitch moment curves of the two tails under wind tunnel conditions. (C) Yaw moment curves of the two tails under wind tunnel conditions. (D) Roll moment curves of the two tails under wind tunnel conditions.

Figure 13. Long description

Two webbed tails with different tail-arm opening angles are shown. The first tail has an angle of 60 degrees, while the second has an angle of 90 degrees. The moment curves for pitch, yaw, and roll are displayed in separate graphs. The pitch moment curves show the highest moment at actuation state 2 for both tails. The yaw moment curves indicate a steady increase in moment with actuation state, with tail 41 showing higher values. The roll moment curves also show an increase in moment with actuation state, with tail 41 again showing higher values. All values are approximated.

Configuration selection strategy: The experimental results indicate that the tail-arm opening angle ${{\alpha}}_{\textrm{web}}$ is a key parameter for adjusting the trade-off between “control authority” and “control purity” in the webbed tail. If the mission requirement prioritizes maximizing the three-axis control moment margin, especially emphasizing rapid lateral-directional response in differential mode, then configuration #41 exhibits superior comprehensive performance.

Figure 14.

T-shaped tails and its moment curves. (A) Physical images of three T-shaped tails with the same area but different elevator areas. (B) Pitch moment curves of the three tails under wind tunnel conditions. (C) Yaw moment curves of the three tails under wind tunnel conditions. (D) Roll moment curves of the three tails under wind tunnel conditions.

Figure 14. Long description

The image contains three photos of T-shaped tails labeled as fifty-one, fifty-two, and fifty-three, each with the same area but different elevator areas. The first photo shows a T-shaped tail with an elevator area of thirty thousand ninety-three point ninety-two millimeters. The second photo shows a T-shaped tail with an elevator area of twenty-two thousand three hundred fifty-eight point zero one millimeters. The third photo shows a T-shaped tail with an elevator area of thirty-seven thousand seven hundred ninety-seven point zero one millimeters. Additionally, the image includes three line graphs. The first graph displays the pitch moment curves of the three tails under wind tunnel conditions, with the pitch moment on the y-axis and the actuation state on the x-axis. The second graph shows the yaw moment curves of the three tails under wind tunnel conditions, with the yaw moment on the y-axis and the actuation state on the x-axis. The third graph presents the roll moment curves of the three tails under wind tunnel conditions, with the roll moment on the y-axis and the actuation state on the x-axis. Each graph includes three lines representing the data for the three different T-shaped tails.

4.2.6 Influence of V-shaped tail structural parameters on maneuverability

To investigate the influence of the stabilizer dihedral angle ( ${{\alpha}}_{\textrm{V}}$ ) of the V-shaped tail on its control effectiveness and channel-coupling characteristics, this study designed and fabricated three V-shaped tail specimens with different ${{\alpha}}_{\textrm{V}}$ values based on Table VI (Figure 15(A)). Based on the symmetric–differential control logic of the V-shaped tail, three key actuation states were defined: State 1 (baseline, control surfaces neutral); State 2 (symmetric pitch actuation, both control surfaces deflected upward simultaneously by 45°); State 3 (differential pitch actuation, one control surface deflected upward by 45°, the other downward by 45°). The time-averaged three-axis aerodynamic moments obtained from wind-tunnel experiments are shown in Figure 15 (B)–(D).

Figure 15.

V-shaped tails and its moment curves. (A) Physical images of three V-shaped tails with the same area but different stabilizer dihedral angles. (B) Pitch moment curves of the three tails under wind tunnel conditions. (C) Yaw moment curves of the three tails under wind tunnel conditions. (D) Roll moment curves of the three tails under wind tunnel conditions.

Figure 15. Long description

The image displays three V-shaped tails with identical areas but varying stabilizer dihedral angles. The tails are labeled as sixty-one, sixty-two, and sixty-three, with angles of one hundred twenty degrees, ninety degrees, and one hundred fifty degrees, respectively. The accompanying graphs illustrate the pitching, yawing, and rolling moments of these tails under wind tunnel conditions. Each graph compares the three tails across different actuation states, showing how the dihedral angle affects the aerodynamic moments.

Longitudinal control and coupling under symmetric actuation: In State 2, the pitch moment (M _p ) of all configurations increased significantly, reaching or approaching their peak values (#61: 0.146 N·m; #62: 0.142 N·m; #63: 0.150 N·m), confirming that symmetric deflection is an efficient method for longitudinal control. Configuration #62 ( ${{\alpha}}_{\textrm{V}}$ = 90°) exhibited stronger lateral-directional coupling in State 2, with its M _y and M _r increments being noticeably higher than the others, suggesting that a smaller stabilizer dihedral angle may intensify inherent pitch–lateral-directional coupling.

Lateral-directional dominant response under differential actuation: In State 3, the aerodynamic response pattern shifts fundamentally. The lateral-directional moments (M _y , M _r ) increase sharply and become the dominant output, with configuration #62 performing most prominently. A smaller dihedral angle (#62, ${{\alpha}}_{\textrm{V}}$ = 90°) results in stronger lateral-directional moment output under differential actuation. Compared to #61, M _y and M _r of #62 increase by approximately 27 % and 23 %, respectively, with even more significant improvements compared to #63. Concurrently, the pitch moment of all configurations decreases to a low level, clearly demonstrating the functional division: “symmetric actuation controls pitch, differential actuation controls yaw/roll.”

Influence patterns and trade-offs of the stabilizer dihedral angle: Comparing configurations with different ${{\alpha}}_{\textrm{V}}$ values reveals that a smaller dihedral angle (#62, ${{\alpha}}_{\textrm{V}}$ = 90°) yields stronger lateral-directional moment output under differential actuation, while a larger angle (#63, ${{\alpha}}_{\textrm{V}}$ = 150°) improves longitudinal control purity but significantly weakens lateral-directional effectiveness in differential mode. Essentially, reducing ${{\alpha}}_{\textrm{V}}$ enhances lateral-directional potential at the cost of stronger coupling; increasing ${{\alpha}}_{\textrm{V}}$ favors channel decoupling but reduces differential control effectiveness.

Configuration selection recommendations: If the mission prioritizes lateral-directional maneuverability and rapid roll/yaw response — especially during flight phases where differential control dominates — configuration #62 offers superior comprehensive performance and the largest lateral-directional control margin.

Figure 16.

Six types of optimal-performance tails and their moment curves. (A) Physical images of six different tail shapes with the same total area. (B) Pitch moment curves of the six tails under wind tunnel conditions. (C) Yaw moment curves of the six tails under wind tunnel conditions. (D) Roll moment curves of the six tails under wind tunnel conditions.(;Note: For the webbed tail (#41) and V-shaped tail (#62), due to differences in control logic, only three actuation states (State 1–3) are included, and state 4 is not applicable).

Figure 16. Long description

The image contains one photo and three graphs. The photo shows six different tail shapes with the same total area. The first tail has a diameter of 438.50 millimeters and is labeled as number 13. The second tail has a diameter of 641.31 millimeters and is labeled as number 23. The third tail has a diameter of 759.33 millimeters and is labeled as number 33. The fourth tail, labeled as number 41, has an angle of 60 degrees. The fifth tail, labeled as number 53, has an area of 3779.01 square millimeters. The sixth tail, labeled as number 62, has an angle of 90 degrees. The first graph displays the pitch moment curves of the six tails under wind tunnel conditions. The second graph displays the yaw moment curves of the six tails under wind tunnel conditions. The third graph displays the roll moment curves of the six tails under wind tunnel conditions. For the webbed tail (number 41) and V-shaped tail (number 62), due to differences in control logic, only three actuation states (State 13) are included, and state 4 is not applicable.