Design and Experimental Study of a Robotic Tuna with Shell-like Tensegrity Joints

Abstract

We developed an untethered robotic tuna featuring tensegrity joints for the purposes of simplifying the design procedure, reserving enough internal space, reducing the frictional loss of structures and generating a relatively smooth fish body wave. To achieve these objectives, a novel shell-like tensegrity joint was introduced, paired with a single-motor multiple-joint driving mechanism. The morphology matching design method of the tensegrity joint was proposed to fit the streamlined fish body, where the deflection angles of each joint were predetermined to generate the specific body waveform. Stiffness analysis shows that the tensegrity joint could function equivalently to a traditional rotational joint, given certain geometric conditions. Based on the fabricated robotic tuna prototype, extensive free-swimming experiments were performed to optimize its swimming performance by varying key parameters, including the caudal fin‘s shape, flexibility and rotational stiffness and joint deflection angles. The results reveal that the robotic tuna achieved the highest swimming speed of 1.31 body lengths per second (BL/s) at a driving frequency of 2.4 Hz, and the maximum stride length increased to 0.81 BL/cycle at 1 Hz, demonstrating the effectiveness of the proposed design scheme. This study provides valuable insight for developing high-performance bio-inspired autonomous underwater vehicles.

1. Introduction

Fish are endowed with an elegant and agile beauty by long-term natural selection, which has attracted great attention from bionic scholars. Substantial efforts in the improvements of robotic fish have been made to reach the swimming capabilities of real fish since the 1990s [1,2,3,4], aiming to replicate the performance of their biological counterparts. Consequently, robotic fish are emerging as a promising biomimetic autonomous underwater vehicle, with potential applications in marine research, seabed exploration and other tasks [5,6].

Early robotic fish typically utilized a rigid serial chain mechanism for their backbone, with each joint independently driven by an actuator [7,8]. The body and/or caudal fin (BCF) undulation could be achieved by applying a time-varying prescribed shape control technique for all joints [9], which essentially is a kinematic biomimetic method. However, this method is hindered by low efficiency, high control complexity and a narrow driving frequency bandwidth [10,11]. With the advent of the soft robot era, many robotic fish have employed a single actuator to drive the entire soft body [1,12,13,14]. This approach excites inherent vibration modes, generating oscillating motions and is referred to as a dynamic biomimetic method. While such robotic fishes are easy to control and simple to design, there exist two key drawbacks to this method. First, the efficiency is considerably reduced when the driving frequency deviates from the natural frequency of a soft body. Second, oscillatory swimming is less efficient than undulatory swimming [15,16].

Nowadays, a single-actuator multiple-joint mechanism, which uses a single motor to drive discreet serial body segments, is proposed to combine the advantages of the above two methods [10,17]. However, when traditional rigid joints are used to connect these segments, a friction-loss problem arises, which inevitably leads to low efficiency. For instance, the eight-joint robotic tuna developed by Barrett et al. experienced a transmission loss of up to 10% [18]. Moreover, as the number of joints and driving frequency increase, the efficiency loss grows rapidly. To address this issue, a tensegrity joint was introduced [19]. A tensegrity structure is composed of compressed rods and tensioned cables, where the rods are suspended within a network of tensioned cables, resulting in minimal direct contact between the rods. Therefore, tensegrity joints constructed by tensegrity structures not only nearly eliminate friction but also have lightweight, modular and compliant properties [20,21]. The first application of tensegrity structures in robotic fish occurred in 2012 when Bliss et al. developed a tensegrity swimmer [22]. However, in this design, the tensegrity structure was fixed in holders and only used to drive the foil to oscillate. A more advanced implementation came in 2019 when Bingxing Chen and Hongzhou Jiang developed a wire-driven tensegrity robotic fish [19]. This bio-inspired fish skeleton, composed of six tensegrity joints and covered with elastic skin to displace external fluid, achieved a swimming speed of 0.7 body lengths per second (BL/s), demonstrating the promising potential of tensegrity joints in the robotic fish field.

Despite the good development prospect of tensegrity robotic fish, its design methodology remains to be perfected. At present, one of the most critical challenges is achieving an optimal match between tensegrity joints and fish morphology. Morphology matching refers to retaining the streamlined shape of a fish to the greatest extent possible while incorporating tensegrity joints, a key factor that impacts the fish-like propulsion performance. In this paper, a novel shell-like tensegrity joint is proposed, building upon the work of [19]. To overcome the limitations of the wire-driven method, namely, its unsuitability for high-frequency swimming and inability to precisely control the deflection angle of individual body segments, a single-actuator multiple-joint mechanism is adopted, and the maximum rotation angle of the shell-like tensegrity joint is predetermined through its structure design, enabling the generation of specific robotic fish body waves by tuning these rotation angles. Given that tuna, known for their exceptional high-speed swimming and endurance, are the quintessential representatives of thunniform swimmers and have long been an ideal model for biomimetic research, we took tuna as the bionic subject for this study. A novel robotic tuna featuring shell-like tensegrity joints with predefined deflection angles was designed and fabricated. The influence of the deflection angles on swimming performance is examined via experiments. Furthermore, since the lunate-shaped caudal fins are the primary source of thrust for tuna, we also took caudal fin characteristics as the focus of swimming performance optimization.

The remainder of this paper is organized as follows: In Section 2, the overall design of robotic tuna, including its structure design based on a shell-like tensegrity joint and control system, is presented. Section 3 gives detailed discussions about the swimming performance results obtained from the experiments. In Section 4, a brief conclusion and future research directions are drawn.

2. Design of Robotic Tuna

2.1. Evolutionary Development of Tensegrity Joint

2.1.1. Previous Work

The tensegrity saddle joint is a commonly used structure in biomimetic robots, and it mimics the biomechanics of the human spine structure, as shown in Figure 1a [23,24]. In this configuration, the tension elements keep the rigid elements out of contact with each other, and the joint’s rotational motion is entirely governed by the tension networks formed by tension elements. This results in a universal joint with a large range of motion. However, the lack of a fixed center of rotation and its difficulty in controlling stiffness led Bingxing Chen and Hongzhou Jiang to propose some improvements and introduce a novel variant for use in the robotic fish TenFiBot-I [19]. The main modifications involved redesigning the structure of the rigid elements into the plate with a Y-shaped foot and dividing the tension network into an axial and a horizontal tension network (Figure 1b). The axial tension network primarily provides translational stiffness, while the horizontal tension network is responsible for rotational stiffness.

2.1.2. Improvement Work

The TenFiBot-I’s design faced a critical limitation: inadequate internal space. For an underwater mobile platform, sufficient internal space is essential for housing functional modules required to perform underwater tasks. To address this issue, we re-engineered the Y-shaped foot into a Π-shaped foot that stands perpendicular to the plate and redesigned the plate into a hollow structure (Figure 2a) while retaining the original tension network.

Another challenge with the TenFiBot-I is the need for an elastic skin to cover the skeleton. The elastic skin generates tension as the robotic fish bends its body, which limits the joints’ range of motion. Moreover, the skin tends to wrinkle during undulatory swimming, disrupting the smoothness of the streamlined surface and negatively affecting hydrodynamics [17]. It is still hard to quantitatively evaluate the influence of skin on swimming performance. In addition, forming a watertight seal with an elastic skin poses a challenge. To overcome these limitations, we extended the length of the hollow plate and shortened the Π-shaped foot, forming a novel shell-like tensegrity joint, as shown in Figure 2b. This new design eliminates the need for elastic skin, significantly simplifying the robotic fish’s construction. It should be noted that despite the substantial structural changes between Figure 1a and Figure 2b, the fundamental topology of the tension elements was consistent throughout the design evolution.

2.2. Morphology Matching Design Method

The flowchart of the morphology matching design method is illustrated in Figure 3. The method, developed for shell-like tensegrity joints, can be applied to all robotic fish based on the BCF propulsion mode, including those with anguilliform, carangiform and thunniform modes. We used tuna as a case study to describe the method.

The first step involves constructing a three-dimensional model of robotic tuna. Extensive studies of real tuna morphology using micro-computed tomography have generated a substantial amount of profile data [25]. Hence, processes such as data extraction, curve fitting and length scaling can be performed sequentially according to empirical data. During this process, only the outlines of the tuna’s body and caudal fin are retained, while the outlines of the pectoral fin, dorsal fin and anal fin, which could complicate later discretization, are excluded. Thus, the overall dimension of robotic fish can be obtained.

The second step is to determine the number of fish body segments and the location of bending points according to different propulsion modes. Akanyeti et al. suggested that fish swimming kinematics can be accurately captured by dividing the fish body into six or fewer segments [26]. Lauder et al. demonstrated that although increasing the number of body segments enhances the flexibility and swimming performance of robotic fish, the performance gain decreases, and the design complexity increases at the same time [17]. Based on these findings, we segmented the robotic tuna into six parts: one for the head, one for the caudal fin and four for the posterior body. After that, the bending point distribution along the body becomes crucial for multi-joint robotic fish. Biological data indicate that for thunniform swimmers, the head and tail segments account for about 47% ± 7% and 20% ± 5% of the total body length, respectively. Akanyeti et al. also held that shorter posterior body segments allow easier control of the tail’s amplitude and deflection angle [26]. Taking these insights into account, we set the length proportions of the six segments in our robotic tuna, from head to tail, as 42.7%, 9.5%, 9.5%, 9.5%, 6.4% and 22.4%, respectively.

In the third step, a V-shaped segmentation method is introduced to create the modular skeleton unit. A shell-like tensegrity joint consists of a front segment, a posterior segment and an associated tension network. For clarity, the tension network is omitted in Figure 4. The cross-section of the robotic fish body at the bending point serves as the datum plane, denoted as plane A. On one side of the neutral plane (as shown in Figure 4), parting plane B defines the posterior boundary of the front segment. The dihedral angle α₁ between planes A and B is referred to as the offset angle, which determines the tension degree of the tensegrity joint. Parting plane C defines the front boundary of the posterior segment, and the dihedral angle α₂ between planes B and C is termed the deflection angle, which controls the maximum rotational angle of the joint and affects the oscillation amplitude of the robotic fishtail. The same process is mirrored along the neutral plane, forming a V-shaped structure at both ends of the segments, hence the name ‘V-shaped segmentation method’.

The fourth step focuses on designing the tension network. While the horizontal tension networks shown in Figure 2b are theoretically feasible, they lack sufficient stability in practice. To enhance joint rotation stability and prevent unwanted movement in other directions, we developed an improved tension network, as shown in Figure 5. In Figure 5a, holes are punched near the datum plane on both sides of the adjacent segments, allowing the tension element, such as an elastic string, to pass through them in an orderly fashion. Figure 5b details the specific connection method, with the ends of the tension element secured by a collet. The key difference between Figure 2b and Figure 5 is the transformation of two discrete horizontal tension networks into a unified circumferential tension network.

Figure 6 presents some new structures added to the body segment. The segment features a three-tier structure, separated by two spacer plates. Foam material fills the upper section, while lead weights occupy the bottom section, both of which are used to fine-tune the buoyancy balance of the robotic tuna. Moreover, the weight distribution on both the left and right sides of the robotic fish should also be carefully calibrated to prevent rollover during undulatory swimming. The middle section allows for the passage of a rigid rod that serves as the backbone of the robotic fish. Stop blocks have been incorporated into the segment structure to prevent kinematic interference between adjacent segments, which could otherwise cause structural failure. Additionally, the finlet structure mimics the finlet of real tuna, which is beneficial to swimming performance [27].

Finally, all the hollow body segments, functioning as skeletal units, are connected via tension networks to form the complete structure of the tensegrity robotic fish. It should be checked whether the resulting fish body wave is smooth. If not, go back to the second step to redesign and adjust the relevant parameters, then proceed through the remaining steps until the design requirements are fully satisfied.

2.3. Stiffness Analysis of Shell-like Tensegrity Joint

The front and side views of the shell-like tensegrity joint are presented in Figure 7. The joint connects two segments using three tension elements. Specifically, two tension elements, A₁A₂ and B₁B₂ (depicted as purple lines), lie in the xOz plane, while the third tension element (shown as green lines) lies in the xOy plane. This study primarily focuses on the stiffness in the y, z and φ directions, which influence the robotic tuna’s planar swimming in the yOz plane. Assuming point O is the virtual rotation center of the joint, the planar stiffness matrix can be derived based on the stiffness theory of parallel mechanisms when the joint is in its neutral position (φ = 0) [28].

$$K=\left[\begin{array}{ccc}{K}_y& 0& K_{\mathrm{y}\phi }\\ 0& K_z& 0\\ K_{\mathrm{y}\phi }& 0& K_{\phi }\end{array}\right]$$

(1)

where K_y, K_z and K_φ represent the stiffness in the y, z and φ directions, respectively, while K_yφ represents the coupling stiffness between the y and φ directions. The expressions for these stiffnesses are given by Equations (2)–(4).

$$K_y=4k{c}^2{\cos }^2\beta /{l_2}^2$$

(2)

$$K_z=2k{(H-H_0)}^2/{l_1}^2+4k{H}_0^2/{l_2}^2$$

(3)

$$K_{\phi }=K_{y\phi }=4k{c}^2{(H-h)}^{2}co{s}^2\beta /{l_2}^2$$

(4)

where k denotes the stiffness of the tension elements, and c is the radius of the circle defined by the four points C₂, C₅, D₂ and D₅. For simplicity, the same symbols are used for both the holes and the center points of the corresponding holes. Β is the acute angle between the radius vector OC₂/OC₅/OD₂/OD₅ and the y-axis. H and H₀ are the distances from point C₃/C₄/D₃/D₄ to lines A₂B₂ and A₁B₁, respectively, while h is the distance from point O to line A₂B₂. l₁ and l₂ represent the lengths of the axial tension elements (A₁A₂/B₁B₂) and the partial length of circumferential tension element (C₂C/C₅C/D₂D/D₅D), respectively.

$$K_y=4k{c}^2{\cos }^2\beta /{l_2}^2$$

(5)

$$K_z=2k{(H-H_0)}^2/{l_1}^2+4k{H}_0^2/{l_2}^2$$

(6)

where a, b and d are the distances from points A₂, B₁ and D₃ to the symmetry plane, respectively.

Three stiffness characteristics can be concluded from Equations (2)–(4). First, when h equals H, both K_φ and K_yφ are zero, allowing the joint to rotate freely around point O in the φ direction. Second, since K_y and K_z are independent of h, the stiffnesses in the y and z directions remain unaffected by the virtual rotation center. Third, the shell-like tensegrity joint demonstrates stiffness anisotropy. By adjusting the axial tension elements so that point A₁ (BS₁) is as close as possible to point A₂ (B₂), K_φ can be minimized to nearly zero. The offset angle influences the arrangement of the circumferential tension elements, thereby affecting the values of K_y and K_z. These values are positively correlated with the offset angles. Consequently, fine-tuning the tension networks can result in high K_y and K_z values, while K_φ remains near zero. Thus, the tensegrity joint can effectively function as a traditional rigid rotational joint. When the joint is away from the neutral position, the virtual rotation center has some drift. In our experiments, with the joint’s rotation angle limited to 14°, the dimensionless drift was less than 5% and can be considered negligible [29].

2.4. Mechatronics Design

A schematic of the robotic tuna is shown in Figure 8a. Its total length, maximum height and maximum width are 410 mm, 110 mm and 68 mm, respectively. During the V-shaped segmentation process, the offset angle α₁ is set to zero for simplicity. θ₁, θ₂, θ₃ and θ₄ in Figure 8a represent different deflection angles of the tensegrity joints. The robotic tuna’s driving mechanism operates as follows: the steering motor directly drives the last segment of the posterior body via a rod, causing the four tensegrity joints to rotate around their respective virtual rotation centers in sequence. As a result, the rod’s straight-line configuration transitions into a folded-line arrangement of body segments, approximating the smooth curvature of a fish body wave.

The head section of the robotic tuna contains a sealed cabin that houses the control system (shown in Figure 8b). The microcontroller unit (MCU) generates a pulse-width modulation signal to control the steering motor, producing a sinusoidal motion at frequencies below 3 Hz. Communication between the transmitter and receiver is achieved via pulse position modulation. The transmitter allows adjustment of the MCU’s control parameters. A lithium battery, regulated by a voltage regulator, provides stable power to both the MCU and the receiver.

By changing the rotation mode of the servos, such as increasing the maximum rotation angle on one side while simultaneously decreasing the rotation angle range on the opposite side, the direction of the robotic fish can be effectively adjusted. On the other hand, as the primary focus of the study is to steady the swimming speed, the robotic fish does not feature an additional ascent and descent module, which typically requires the design of a pair of pectoral fins driven by steering motors [30].

3. Results and Discussion with Swimming Experiments

A series of free-swimming experiments were conducted in a water tank measuring 3 m × 1.2 m × 0.5 m after fabricating the robotic tuna prototype. The swimming sequences were recorded using an industrial digital camera operating at 60 frames per second with a resolution of 800 × 600 pixels. To ensure accurate results, the image data underwent preprocessing. The RGB images were initially converted to grayscale to eliminate color information and subsequently transformed into monochrome images using a threshold to remove noise. The Canny edge detection algorithm was applied to outline the robotic tuna’s body, allowing for the calculation of its midline coordinates. Preliminary body waveforms were extracted through batch image processing based on these steps. The undulatory reconfiguration technique was then employed to refine the raw data and achieve accurate swimming kinematics [31]. According to the rules of this technique, data identified as unsatisfactory—such as cases where roll and pitch movements disrupt straight-line swimming performance—can be filtered out and excluded from analysis.

3.1. Swimming Optimization with Predetermined Joint Angles

Drawing from empirical knowledge and the relevant reference [17], we assigned different deflection angles to each tensegrity joint to generate three cases, as listed in

Table 1. Predetermined joint angle settings for three cases.

Case	θ1 (°)	θ2 (°)	θ3 (°)	θ4 (°)	θ (°)
1	6	8	10	14	15.4
2	6	6	8	12	13.4
3	25	25	25	25	13.4

. The angles θ₁, θ₂, θ₃ and θ₄ represent the maximum deflection angles of the first to fourth tensegrity joints, respectively (illustrated in Figure 8a). θ, the maximum rotational angle of the steering motor, can be calculated inversely from the four joint angles in Cases 1 and 2, based on the structure parameters of the robotic tuna. Compared to Case 2, the maximum deflection angles in Case 1 are 2° larger for all joints, except for the first joint. Case 3 served as the control group, with θ set the same as in Case 2, at 13.4°. Consequently, despite the maximum deflection angles for all four joints being set to 25°, the actual deflection angles were significantly lower, never reaching the preset values. As a result, the predefined deflection angles are ineffective in this case, which can be regarded as operating in an unconstrained mode. In contrast, Cases 1 and 2 are referred to as constrained modes.

Three prototypes were developed based on the three cases to conduct swimming experiments with eight driving frequencies (1 Hz, 1.3 Hz, 1.5 Hz, 1.8 Hz, 2.1 Hz, 2.4 Hz, 2.7 Hz and 3 Hz, respectively). During fabrication, all parameters remained constant except for the deflection angles. According to the morphology matching design method, these small variations in joint angles result in small structural differences among the prototypes. The swimming performance of the robotic tuna, including the relative steady forward speed U, stride length U* and Strouhal number St at different frequencies f for the three cases, is shown in Figure 9.

U = U_e/L, U* = U_e/(fL), St = fA/U_e

(7)

where U_e is the actual steady forward speed, and L represents the total body length (BL) of the robotic tuna. A denotes the tail-beat amplitude. U* indicates the distance traveled during one tail-beat cycle. U and U* are measured in the BL/s and BL/cycles, respectively. The observations of real fish suggest that their stride length remains constant at approximately 0.7 BL/s, regardless of frequency. The Strouhal number (St) is a critical parameter that characterizes vortex shedding behind the tail and serves as a qualitative indicator of swimming efficiency [32]. Studies indicate that optimal propulsion performance is achieved when the St number is around 0.4 [33].

Figure 9a shows that, for all cases, the forward speed U initially increases and then decreases as the frequency rises. This trend is attributed to momentum exchange near the tail. The peak speeds of 1.18 BL/s (0.48 m/s), 0.94 BL/s (0.39 m/s) and 0.64 BL/s (0.26 m/s) occur at 2.4 Hz, 2.4 Hz and 1.3 Hz, respectively, for Cases 1, 2 and 3. The peak speeds in Cases 1 and 2 are 84% and 47% higher, respectively, than in Case 3. Figure 9b illustrates that stride length U* generally decreases with increasing frequency in all cases, but the rate of decline narrows from Case 3 to Case 1. The maximum U*, approximately 0.7 BL/cycle, is reached at 1 Hz in both Cases 1 and 2. Figure 9c indicates that the St value increases with frequency in Case 3, while in Cases 1 and 2, St remains mainly within the 0.4–0.5 range, suggesting that swimming efficiency is generally higher in Cases 1 and 2 compared to Case 3. Notably, the St values at 1 Hz for Case 3 and 2.4 Hz for Cases 1 and 2 approach 0.4, indicating good propulsion performance. These findings demonstrate that optimizing the deflection angles of each joint can significantly enhance the swimming performance of the robotic fish.

The body waves of the robotic tuna in air and water are depicted in Figure 10 and Figure 11, respectively. The differences in body waves between the two environments arise from three factors: the deflection angles, driving frequency, and hydrodynamic effects. In Cases 1 and 2, the body waves in air are similar. However, slight adjustments to the deflection angles lead to significant changes in the wave envelopes in water, highlighting the high sensitivity of the swimming kinematics to deflection angles. While frequency has a limited effect on the wave envelope, it significantly impacts the wave amplitude.

When comparing the three cases, Case 3 exhibits the poorest swimming performance. In this scenario, the posterior body primarily rotates around the virtual rotation center of the first tensegrity joint, resembling a single-degree-of-freedom swinging motion, as seen in the nearly straight wave envelopes in Figure 10c and Figure 11c. Consequently, Case 3 fails to fully utilize the advantages of the single-actuator multiple-joint mechanism. Although the swimming speeds in Cases 1 and 2 are similar at lower frequencies, Case 1 shows a more pronounced improvement at higher frequencies, indicating that deflection angle optimization is more effective at high frequencies. A video of the swimming performance for Case 1 is provided in the Supplementary Materials.

According to the properties of the shell-like tensegrity joint, the joint’s rotational stiffness approaches zero when the rotation angle is below the deflection angle. Theoretically, rotational stiffness becomes infinite once the rotation angle reaches the deflection angle, resulting in a step change during joint rotation. This represents a unique form of time-varying stiffness, which has the potential to significantly improve robotic fish performance. While most recent studies have focused on spatial stiffness, our findings underscore the importance of exploring stiffness variations over time [34].

It should be noted that the deflection angles for the three cases were set based on our empirical knowledge. A detailed theoretical understanding of the effect of deflection angles on swimming performance requires further in-depth research, incorporating computational fluid dynamics.

3.2. Swimming Optimization with Caudal Fin

We continued the stiffness optimization experiments on the caudal fin to enhance swimming performance, focusing on four key factors: the rotational stiffness at the caudal peduncle, material flexibility, caudal fin height and the trailing edge shape. The deflection angles were set to those of Case 1 for the following controlled experiments.

The rotational stiffness was adjusted by replacing five different torsional springs, with corresponding stiffness values (K_tail) of 30 Nmm/rad, 51 Nmm/rad, 77 Nmm/rad, 132 Nmm/rad and 197 Nmm/rad. The original caudal fin of the robotic tuna was made from resin. To investigate the effect of material flexibility, three caudal fins with Shore hardness values of 30 A, 60 A and 90 A (where lower values indicate greater flexibility) were fabricated using thermoplastic polyurethane (TPU). Additionally, four caudal fins with varying heights of 110 mm, 138 mm, 166 mm and 192 mm (corresponding to 1×, 1.25×, 1.5× and 1.75× the maximum body height of the robotic tuna) were also produced. Minor structural modifications were made to ensure consistency in shape across the four fins (Figure 12a). Xin ZhiQiang et al. identified that the primary variation in fishtails lies in the trailing edge shape [35]. Therefore, we changed the trailing edge shape to study its effect. As shown in Figure 12b, L_c denotes the chord length, while L refers to the total horizontal length of the caudal fin. By adjusting the L_c/L ratio (four ratios—0.4, 0.5, 0.6 and 0.7—were tested), the trailing edge shape was modified. The experimental results, shown in Figure 13 and Figure 14, demonstrate the effects on swimming speed and stride length. In general, swimming speed initially increases with frequency, reaching a maximum at 2.4 Hz, and then declines. Meanwhile, stride length first decreases, then fluctuates slightly, and eventually declines again as frequency rises. The effects of these four control variables will be discussed in detail below.

3.2.1. Rotational Stiffness

Both swimming speed and stride length decrease as rotational stiffness increases within the frequency range of 1 Hz to 2.4 Hz, but they rise with increasing stiffness once the frequency exceeds 2.4 Hz, as shown in Figure 13a and Figure 14a. This suggests an important matching mechanism of rotational stiffness to the frequency: the rotational stiffness at the caudal peduncle should increase in response to higher frequencies. In the controlled experiment, the maximum speed of the robotic fish increased from 1.13 BL/s to 1.2 BL/s, a 6.2% improvement, while the maximum stride length increased from 0.66 BL/cycle to 0.74 BL/cycle, marking a 12.1% gain.

3.2.2. Material Flexibility

Swimming performances with a material hardness of 30 A and 60 A are similar, as shown in Figure 13b and Figure 14b. Compared to the swimming performance with resin material, the performances with hardness of 30 A and 60 A show significant improvement at frequencies below 2.4 Hz, while the performance with 90 A hardness shows a notable improvement at frequencies above 2.4 Hz. This reveals another matching mechanism: the caudal fin’s flexibility should decrease as frequency rises. In this controlled experiment, the robotic fish’s maximum speed increased from 1.18 BL/s to 1.31 BL/s, an increase of 11.2%, while the maximum stride length grew from 0.7 BL/cycle to 0.81 BL/cycle, a 15.7% increase.

3.2.3. Caudal Fin Height

In this series of controlled experiments, the maximum speed of the robotic fish increased from 1.12 BL/s to 1.2 BL/s, a 7.1% improvement, while the maximum stride length increased from 0.67 BL/cycle to 0.7 BL/cycle, a 4.5% gain, as shown in Figure 13c and Figure 14c. Both the maximum swimming speed and stride length were achieved with a caudal fin height of 166 mm, highlighting that a larger caudal fin height does not necessarily result in better performance.

3.2.4. Trailing Edge Shape

As shown in Figure 13d and Figure 14d, both swimming speed and stride length increase with a higher L_c/L ratio in the frequency range of 1 Hz to 2.4 Hz. This is likely due to the increased surface area of the caudal fin, which enhances the thrust generated by its interaction with the fluid. In the controlled experiment, the robotic fish’s maximum speed increased from 1.14 BL/s to 1.25 BL/s, a 9.6% improvement, while the maximum stride length grew from 0.63 BL/cycle to 0.81 BL/cycle, a 28.6% gain.

3.3. Discussion

The above controlled experiments reveal that material flexibility leads to the most notable improvement in swimming speed, with an 11.2% increase, while the trailing edge shape results in the greatest enhancement in stride length, with a 28.6% increase. Overall, swimming performance was further optimized, achieving a maximum speed of 1.31 BL/s and a stride length of 0.81 BL/cycle. This demonstrates that precise matching of the deflection angles of tensegrity joints angle and caudal fin can significantly improve performance. Compared to the tensegrity robotic fish developed by Bingxing Chen and Hongzhou Jiang in 2019 [19], our robotic tuna shows remarkable gains, with swimming speed increasing by 87% and stride length by 62%, highlighting the advantages of the shell-like tensegrity joints and the morphology matching design method. Notably, since the stride length of thunniform fish is typically around 0.7 BL/cycle, this study’s achievement of 0.81 BL/cycle suggests the potential to exceed the swimming performance of real fish.

Some swimming performance data for robotic fish operating at low frequencies (below 3 Hz) from the literature are presented for comparison. The tendon-driven robotic fish made by Changlin Qiu et al. achieved a maximum speed of 1.04 BL/s [36]. The wire-driven elastic robotic fish developed by Xiaocun Liao et al. reached a top speed of 0.74 BL/s, with a stride length of 0.25 BL/cycle [37]. The wire-driven robotic fish fabricated by Fengran Xie et al. [38] reached a maximum speed of 0.84 BL/s and a stride length of 0.78 BL/cycle. In comparison, our shell-like tensegrity robotic tuna demonstrates a strong performance and a clear advantage in mechanism configuration.

4. Conclusions

In this paper, we present significant improvements based on the tensegrity robotic fish, TenFiBot-I, developed by Bingxing Chen and Hongzhou Jiang in 2019. Our innovations include the development of a novel shell-like tensegrity joint featuring a predetermined deflection angle, the corresponding morphology matching method, and a single-motor multiple-joint driving mechanism. These advancements expand internal space for functional modules, maintain the streamlined shape of real fish, eliminate the need for elastic skin, and simplify the overall design process. This approach is applicable to all robotic fish utilizing the BCF swimming mode. We designed a cost-effective robotic tuna and conducted swimming optimization experiments. The results show that the proper deflection angle setting has a strong improvement effect on the swimming performance of robotic tuna. Changing these angles is an optimization for the fish body wave. From a stiffness perspective, these angles are essentially a special form of time-varying stiffness. Additionally, four controlled experiment series reveal that material hardness has the greatest influence on swimming speed, while the trailing edge shape has the most impact on stride length. A key finding is that both rotational stiffness at the caudal peduncle and the caudal fin’s material hardness should increase with frequency to achieve better swimming performance. By matching deflection angles and caudal fin parameters, the swimming performance was further optimized and greatly surpassed that of TenFiBot-I, highlighting the superiority of our approach.

In the future, we will focus on theoretical and simulation studies to explore the optimization rule for deflection angles. Furthermore, we will aim to develop a new robotic fish specifically designed for high-frequency swimming.