The pursuit of agile, adaptable, and efficient locomotion in robotics has long drawn inspiration from the natural world. Among biological models, the frog stands out for its remarkable swimming capability, achieved through a powerful, cyclic leg motion that generates substantial thrust while minimizing drag during the recovery phase. This bio-inspired approach offers a compelling blueprint for robotic systems, particularly for applications in underwater exploration, environmental monitoring, and search and rescue where silent, disturbance-minimizing propulsion is advantageous. Traditional attempts to replicate this form of locomotion have often relied on rigid actuators and complex linkage mechanisms, resulting in robots that are heavy, noisy, and limited in their environmental compliance.
The emergence of soft robotics, employing materials that can undergo large, reversible deformations, presents a paradigm shift. By utilizing soft actuators, a bionic robot can more closely mimic the fluid, high-strain movements of biological muscle. This not only enhances motion fidelity but also improves the system’s robustness and ability to interact safely with complex surroundings. For a frog-inspired bionic robot, the actuator must be capable of significant and rapid contraction/expansion, much like the leg muscles of its biological counterpart.
Dielectric Elastomer (DE) actuators have emerged as a leading candidate for such applications. These actuators, often called “artificial muscles,” consist of a soft, stretchable polymer film sandwiched between two compliant electrodes. When a high voltage is applied, electrostatic attraction between the electrodes generates a Maxwell stress, compressing the film in thickness and causing it to expand in area. This mechanism can produce strains exceeding 300%, with high energy density and fast response times, making DEs exceptionally well-suited for creating a high-performance swimming bionic robot. The design and optimization of a DE-driven, frog-inspired bionic robot involves a multidisciplinary synthesis of material science, mechanical design, bio-mechanics, and control theory.
The core working principle of the DE actuator is governed by electro-mechanical coupling. The applied electric field induces a pressure, known as Maxwell stress, on the elastomer film. For an ideal dielectric elastomer, this stress is given by:
$$
\sigma_z = \epsilon_0 \epsilon_r \left( \frac{U}{d} \right)^2
$$
where $\sigma_z$ is the Maxwell stress, $\epsilon_0$ is the vacuum permittivity, $\epsilon_r$ is the relative permittivity of the DE material, $U$ is the applied voltage, and $d$ is the instantaneous thickness of the film. This stress causes the film to deform, reducing its thickness and increasing its area. In a constrained configuration, such as a pre-stretched film bonded to a flexible but inextensible frame, this area expansion is converted into bending motion. This bending actuation is the fundamental mechanism that drives the leg stroke of the frog-inspired bionic robot.
The design of the bionic robot must translate this basic actuation into an effective swimming gait. The biological frog’s leg has multiple segments and a webbed foot that changes orientation during the power and recovery strokes. Emulating this, the robotic design incorporates a multi-segment leg structure. A primary DE actuator serves as the “thigh” joint, providing the main propulsive force. A passive, compliant secondary segment acts as the “shin,” and an adaptive foot pad completes the limb. This foot is designed to present a large surface area normal to the flow during the backward power stroke (maximizing thrust) and to reorient to a streamlined profile during the forward recovery stroke (minimizing drag). This adaptive geometry is crucial for achieving high net propulsion efficiency, a key goal for any underwater bionic robot.
The fabrication process of this bionic robot is meticulous and determines its final performance. The process can be summarized in the following key stages:
| Stage | Process Description | Key Materials |
|---|---|---|
| 1. Frame Fabrication | Laser-cutting the main leg frame, stiffening ribs, and adaptive foot components from PET films and acrylic sheets. | PET (0.188 mm, 0.25 mm), Acrylic |
| 2. DE Membrane Preparation | Biaxially pre-stretching a VHB elastomer film (e.g., 540% x 540%). Applying compliant carbon grease electrodes to defined areas on both sides. | VHB 4910, Carbon Grease |
| 3. Actuator Assembly | Sandwiching the pre-stretched, electroded DE membrane between the main frame and stiffening ribs. The pre-strain causes an initial curvature. | Assembled DE Actuator |
| 4. Leg Integration | Attaching the adaptive foot assembly to the actuator frame. Connecting high-voltage wires to the electrodes. | Completed Leg Module |
| 5. Robot Final Assembly | Mounting four identical leg modules onto a lightweight, streamlined body (e.g., 3D-printed PLA). | PLA Body, Four Legs |
The pre-stretch is a critical step. It not only thins the membrane, allowing for larger actuation strains at lower voltages, but also predisposes the actuator to a specific resting curvature. Upon application of voltage, the Maxwell stress works against the elastic restoring force of the pre-stretched membrane, causing the actuator to flatten. When the voltage is removed, the elastic force returns the leg to its bent state. This cyclic bending and recovery, powered by a square-wave high-voltage signal, replicates the frog’s kick cycle.
To understand and optimize the swimming dynamics of this bionic robot, a detailed analysis of its kinematics and hydrodynamics is essential. The motion of key points on the leg can be tracked and modeled. If we establish a coordinate system on the leg’s primary segment, the trajectory of points like the “knee” and “ankle” can be described as functions of time and actuation parameters. For a point on the leg, its position $\vec{r}(t)$ during actuation can be approximated by:
$$
\vec{r}(t) = \vec{r}_0 + \theta(t) \cdot \hat{k} \times (\vec{r}_p – \vec{r}_0)
$$
where $\vec{r}_0$ is the pivot point position, $\vec{r}_p$ is the point’s initial position, $\theta(t)$ is the time-varying bending angle of the DE actuator, and $\hat{k}$ is the unit vector along the axis of rotation. The function $\theta(t)$ is directly driven by the applied voltage waveform $U(t)$. The second segment (shin) and foot then move under the combined influence of this primary actuation and hydrodynamic forces, leading to the complex, adaptive stroke profile. The propulsive thrust $F_T$ generated is related to the leg’s velocity $\vec{v}_{leg}$ relative to the water and its effective area $A_{eff}$:
$$
F_T \propto \frac{1}{2} \rho_w C_D A_{eff} |\vec{v}_{leg}|^2
$$
where $\rho_w$ is the density of water and $C_D$ is the drag coefficient, which is high during the power stroke due to the foot orientation and low during the recovery stroke.
Experimental characterization of the bionic robot is necessary to validate its performance. Key metrics include swimming speed, thrust force, and power consumption under varying electrical inputs (voltage $U$, frequency $f$, and duty cycle $K$). A typical test setup involves placing the robot in a water tank, applying controlled high-voltage signals, and using cameras for motion capture and force sensors for direct thrust measurement. The data reveals fundamental relationships. For instance, swimming speed generally increases with voltage (due to larger leg stroke amplitude) but has an optimal point with respect to frequency, as too high a frequency prevents the leg from completing its full stroke.
| Voltage (kV) | Frequency (Hz) | Avg. Swimming Speed (mm/s) | Peak Thrust (mN) | Leg Stroke Amplitude (deg) |
|---|---|---|---|---|
| 3 | 1.0 | 45 | 12.5 | 45 |
| 4 | 1.0 | 78 | 18.2 | 62 |
| 5 | 0.5 | 67 | 22.1 | 85 |
| 5 | 1.0 | 105 | 20.5 | 78 |
| 5 | 2.0 | 132 | 15.8 | 65 |
The performance of a swimming bionic robot is often evaluated using dimensionless numbers that allow comparison across different scales and with biological organisms. The most relevant are the Reynolds number ($Re$), the Strouhal number ($St$), and the Swimming number ($Sw$).
- Reynolds Number ($Re$): Represents the ratio of inertial forces to viscous forces. $$ Re = \frac{v L \rho_f}{\mu} $$ where $v$ is swimming speed, $L$ is body length, $\rho_f$ is fluid density, and $\mu$ is dynamic viscosity.
- Strouhal Number ($St$): Correlated with propulsive efficiency in oscillatory propulsion. $$ St = \frac{f A}{v} $$ where $f$ is tail/leg beat frequency and $A$ is the peak-to-peak amplitude of the trailing edge. Many efficient biological swimmers operate in the range $0.2 < St < 0.4$.
- Swimming Number ($Sw$): A comprehensive metric combining scale, speed, and kinematics. $$ Sw = \frac{2 A \omega L \rho_f}{\mu} $$ where $\omega = 2\pi f$ is the angular frequency.
Calculating these numbers for the DE-driven bionic robot at its peak performance (e.g., $v=0.132$ m/s, $L=0.1$ m, $f=2$ Hz, $A=0.026$ m) yields $Re \approx 1.32 \times 10^4$, $St \approx 0.39$, and $Sw \approx 3.04 \times 10^4$. The $St$ number falling within the biological efficiency range and the $Re$/$Sw$ numbers aligning with those of small amphibians confirm that this bionic robot successfully operates in a biologically relevant hydrodynamic regime, validating its bio-inspired design principles.
A critical figure of merit for any robotic system, especially an untethered bionic robot, is its propulsion efficiency $\eta$. This is defined as the ratio of useful mechanical power output ($P_{out}$) to the electrical power input ($P_{in}$): $\eta = P_{out} / P_{in}$.
The mechanical output power is the product of steady-state swimming speed $v$ and the thrust force $F_T$ required to overcome drag at that speed: $P_{out} = v \cdot F_T$. The input electrical power can be modeled by considering the DE actuator as a variable capacitor. The energy consumed per cycle is approximately the energy stored in the capacitive film at maximum actuation. For a square-wave signal with duty cycle $K$, the average input power is:
$$
P_{in} = C_{act} U^2 f K
$$
where $C_{act}$ is the capacitance of the actuator at its maximum deformed state (maximum area, minimum thickness). The capacitance is given by $C_{act} = \epsilon_0 \epsilon_r A_{act}/d_{min}$, where $A_{act}$ and $d_{min}$ are the area and thickness of the DE film under full actuation. Therefore, the propulsion efficiency model becomes:
$$
\eta = \frac{v \cdot F_T}{ \left( \epsilon_0 \epsilon_r \frac{A_{act}}{d_{min}} \right) U^2 f K }
$$
This model highlights the complex trade-offs in optimizing a DE-driven bionic robot. Efficiency depends not only on hydrodynamic performance ($v$, $F_T$) but also on the electro-mechanical coupling ($A_{act}/d_{min}$) and electrical driving parameters ($U$, $f$, $K$). Experimental data fitted to this model allows for identifying optimal operating points.
| Operating Condition (U, f) | Calculated $P_{in}$ (mW) | Measured $P_{out}$ (mW) | Estimated Efficiency $\eta$ (%) |
|---|---|---|---|
| 5 kV, 0.5 Hz | 8.2 | 1.48 | 18.0 |
| 5 kV, 1.0 Hz | 16.1 | 2.16 | 13.4 |
| 5 kV, 2.0 Hz | 31.5 | 2.09 | 6.6 |
| 4 kV, 1.0 Hz | 10.3 | 1.43 | 13.9 |
| 3 kV, 1.0 Hz | 5.8 | 0.56 | 9.7 |
The analysis shows that maximum speed and maximum efficiency do not necessarily coincide for this bionic robot. Lower frequencies often allow for a more complete leg stroke and longer glide phase, leading to higher efficiency despite a lower top speed. This underscores the importance of mission-specific optimization for a practical bionic robot.
In conclusion, the development of a dielectric elastomer-driven, frog-inspired bionic robot demonstrates a highly effective synergy between soft actuator technology and bio-inspired design. The DE actuators provide the large-strain, muscle-like motion essential for replicating the frog’s powerful kick. The robot’s segmented leg with an adaptive foot successfully mimics the drag-minimizing recovery stroke observed in nature. Performance characterization confirms that this bionic robot can achieve swimming speeds comparable to its biological scale and operates within hydrodynamic regimes indicative of efficient propulsion. The constructed efficiency model provides a framework for future optimization of power consumption and endurance. Future work on such a bionic robot will focus on integrating onboard power and control systems for full autonomy, further refining the leg geometry and actuation pattern using machine learning, and exploring the use of advanced DE materials for lower voltage operation and improved cycle life. This research pathway solidifies the role of soft, bio-inspired robotics in creating a new generation of agile, efficient, and environmentally interactive machines.