Introduction

Background

The near-shore shallow seas are where humans first engaged with and began to understand the ocean, and they serve as the primary zones for marine development and utilization. In the future, at least 40% of the global population is expected to live within 100 kilometres of the coastline, within the economic influence zone of the near-shore shallow seas1. The near-shore shallow sea environment is complex, covering various features such as coastal mudflats, mixed sediments, underwater riverbeds, and swimming waters. This region also contains numerous artificial structures located in the seabed with highly variable topography and geological conditions2. For example, in Qingdao, China, marine development activities are concentrated in functional zones near the shoreline, particularly on mudflats and shallow waters closer to the land. These areas include many artificial structures such as ports, docks, cross-sea bridges, and tidal power stations3. During construction and maintenance, any instability or failure of these structures could lead to significant casualties, economic losses, and environmental damage4. Therefore, ensuring these structures’ safe construction and efficient maintenance under such complex conditions has become an urgent issue5. With the increasing human activities in near-shore shallow sea areas, using robots instead of humans to perform tasks such as patrolling, surveying, and construction in these amphibious environments has become an inevitable trend of the times6.

In recent years, amphibious robots have emerged as highly efficient transport systems for amphibious environments, demonstrating great potential in both scientific research7,8 and commercial applications9. These robots can perform various tasks in marine engineering, including maintenance, inspection, surveying, and monitoring, thereby enhancing operational efficiency, reducing risks, and decreasing reliance on human resources.

Related work

Current research in the field of amphibious mobile robots is mainly focused on two directions6,10. The first direction involves observing the movement characteristics of biological organisms and using biomimetic design to develop robots with amphibious adaptability. Baines et al. studied the streamlined flippers and columnar legs of sea turtles and tortoises, along with their respective gaits, to design an adaptive morphing robot11. Liu et al. used the California sea lion as a biomimetic model to develop a robotic sea lion flipper, which demonstrated superior thrust performance at a 15° pitch angle12. Lin et al. designed a robot capable of efficient and agile movement in terrestrial and aquatic environments by studying the mudskipper13. E. Milana et al. designed a biomimetic snake-like soft robot named EELWORM, composed of five inflatable bending and elongating actuator modules14. Inspired by sea turtles, Xing et al. designed a miniature biomimetic amphibious spherical robot called ASRobot, featuring a leg-like multi-vector water-jet composite drive mechanism (LMWCDM)15. Despite their remarkable locomotion performance and environmental adaptability, such biomimetic systems often exhibit relatively lower energy efficiency.

The second approach involves analyzing the different adaptation characteristics of wheeled, legged, and underwater propulsion mechanisms in terrestrial, uneven, and aquatic environments, leading to the innovative design and development of amphibious robots with diverse mobility capabilities. Amphihex-II is a robot equipped with flexible flippers and fan-shaped rigid legs, capable of moving over muddy and sandy terrains while swimming underwater16. RoboTerp uses a combination of paddles and legs to enable the transition between land and aquatic locomotion for amphibious robots17. SILVER2, a biomimetic underwater legged robot, can traverse irregular terrains and adapt its leg movements for tasks involving underwater exploration and seabed missions18. Wang et al. designed an underwater legged robot with active landing buffering capabilities, enabling fast and smooth landings. The feasibility and versatility of the designed soft-landing controller were successfully validated19. Epaddle integrates wheels, legs, and paddles, allowing it to conduct search and rescue operations in complex amphibious environments following earthquakes20. The RHex series of robots utilize six half-circle legs, enabling them to climb and perform oscillatory swimming in highly reliable amphibious environments. Y. Lin et al. improved the RHex leg design to create the RHex-T3, which can seamlessly switch between legged, wheeled, and RHex mobility modes, enhancing its movement flexibility. Additionally, RHex-T3 can use a hook mode to climb ladders21,22. X. Ma et al. designed a lightweight miniature amphibious robot based on a propeller-leg composite structure, which controls its mode by adjusting the body’s folding angle. This design endows the robot with amphibious mobility and outstanding performance in the field of lightweight miniature robotics23. Notwithstanding their exceptional locomotion efficiency and capabilities, these robots are often confined to specific application scenarios and struggle to adapt to a diverse range of mission requirements.

In recent years, research on amphibious mobile robots has gradually shifted from a primary focus on scientific analysis, such as exploring movement principles and characteristics, to more practical applications in engineering. The United States Naval Academy developed an amphibious robot named Sea-Dragon to support the Marine Corps in operations between the beachhead and inland positions24. The Wuhan University of Technology proposed an amphibious pipeline inspection robot that utilizes Mecanum wheels and propellers to achieve rapid movement within pipelines25. The Future Technologies Lab of Trivandrum developed a passive deformable wheel design that offers multi-mode (wheel-paddle-leg) capabilities to adapt to both land and water operations26.Bin Wang et al. designed a hull-cleaning robot using a combination of legs, wheels, vacuum suction, and permanent magnet adhesion, enabling it to operate across different media and adhere to curved surfaces with varying curvatures27.

However, the amphibious near-shore environment is complex, and amphibious mobile robots still need to improve in energy conversion efficiency, operational capability, and motion stability. Amphibious robots cannot independently enter their operational environment and typically require manual assistance for deployment. Additionally, they can only adapt to specific amphibious near-shore environments. This undoubtedly increases the burden on amphibious uncrewed and intelligent operations. Achieving robots with amphibious operational capabilities faces two significant challenges: a) the need to adapt to amphibious environments and b) the ability to maintain continuous movement across different amphibious settings. Additionally, the complex and dynamic nature of near-shore amphibious environments necessitates selecting different propulsion methods to meet the operational demands of varying environmental conditions.

The amphibious robot for near-shore operations studied in this paper is designed to meet the following criteria:

  1. 1.

    Strong environmental adaptability, capable of operating in complex amphibious environments.

  2. 2.

    Ability to maintain continuous movement across different amphibious environments to cope with the complexities of near-shore tasks.

  3. 3.

    Equipped with multiple modes of movement to address diverse environmental requirements.

  4. 4.

    Simple and efficient movement mechanisms ensure near-shore amphibious operations’ stability and safety.

  5. 5.

    Can independently enter the operational environment without requiring manual deployment, with the ability to complete tasks autonomously.

Contributions

This paper contributes by proposing a novel amphibious mobile robot design based on an integrated wheel-leg-paddle drive structure. The robot’s motion characteristics were explored through modeling, dynamic analysis, and simulation experiments. The effectiveness of the design was validated using a prototype robot. The amphibious robot developed in this study, named Amp-beetle, features an integrated wheel-leg-paddle structure that accounts for the distinct driving characteristics of wheels, legs, and propellers in various environments. By altering the driving modes, the robot can offer multiple motion modalities suitable for amphibious conditions and underwater vector thrust capabilities. Specifically, as shown in Fig. 1, the four integrated wheel-leg-paddle drive units are connected to the robot’s body through rotational joints. By adjusting the angle of these joints, the Amp-beetle can freely switch between a) wheel-leg locomotion mode and b) vector thrust mode. It enables the robot to operate effectively in various environments, including 1) coastal mudflats, 2) mixed sand and silt areas, 3) seabed riverbeds, 4) underwater, and 5) on the water surface.

Fig. 1
figure 1

Typical movement patterns of the Amp-beetle, its amphibious operational environments, and associated tasks. (Keyshot11, https://www.keyshot.com/blog/luxion-releases-keyshot-11/WPS, https://www.wps.cn/, Figs. 2, 3, 4, 5, 6, 7 also uses the same software).

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A key feature is that the position of the integrated wheel-leg-paddle drive structure varies in different motion modes, altering the thrust generated and the relative positions of the robot’s center of gravity and buoyancy. As a result, the robot exhibits excellent balance or maneuverability in different motion modes. Additionally, the propellers are affected by the rotational joint angles, creating a vector thrust effect. Adjusting the thrust direction minimizes redundant thrust, enhancing the robot’s adaptability and efficiency across various environments.

Paper structure

The remainder of this paper is structured as follows: Section 2 presents a configuration analysis, motion mechanism design, and motion mode analysis of the amphibious robot featuring an integrated wheel–leg–paddle structure. Section 3 provides kinematic and dynamic modeling of the Amp-beetle, along with an analysis of the influence of leg tilt angles on its dynamic performance. Section 4 establishes a simulation environment for the Amp-beetle and verifies its motion characteristics through simulated experiments. Section 5 describes the evaluation of the robot’s motion performance and functional capabilities via experiments conducted both in a test pool and in a coastal environment. Section 6 discusses the core contributions and limitations of this work. Finally, Section 7 concludes the paper.

Configuration analysis and dimension design of the wheel-leg-paddle integrated structure and amp-beetle robot

Design of the wheel-leg-paddle integrated structure

The integrated wheel-leg-paddle structure is illustrated in Fig. 2. It comprises four main components: A) the wheel mechanism, B) the paddle mechanism, C) the suspension mechanism, and D) the leg mechanism. The wheel and paddle mechanisms are among the Amp-beetle’s primary power output components.

Fig. 2
figure 2

Key components and functional divisions of the integrated Wheel-Leg-Paddle structure.

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The design of the propulsion system was driven by the need to achieve stable and efficient motion in low-velocity marine currents. Each propeller was selected to provide a thrust of approximately 10 kg, resulting in a total thrust output of 40 kg, ensuring sufficient force for propulsion and maneuverability. To accommodate the integrated wheel-propeller structure and structural efficiency, a tire diameter of 32 cm was chosen. This dimension was derived based on the required thrust output and spatial constraints. The overall dimensions of the robot—900 mm in length, 750 mm in width, and 870 mm in height—were determined to ensure proportional balance among all functional components while maximizing stability and mobility.

Material selection prioritized lightweight and high-performance properties. The primary structures, such as the wheels and parts of the leg mechanisms, were manufactured using glass-fiber-reinforced nylon to reduce weight and enhance corrosion resistance. Key load-bearing components, including sections of the leg and suspension systems, were made from aluminum alloy to ensure structural integrity under dynamic conditions. The remaining parts were produced using polyoxymethylene (POM) and buoyancy materials, achieving an optimal balance of structural strength, weight, and durability. These material choices collectively result in a system that is significantly lighter than all-metal designs while maintaining superior performance in amphibious environments.

The wheel mechanism provides the necessary rolling motion for the Amp-beetle. The tire features a chevron-pattern tread designed to improve traction and maneuverability on varied terrains, including loose sand and muddy surfaces. The tire surface generates friction with the ground to propel the robot. The inner and outer wheel hubs connect with the tire’s inner grooves to form a mortise-and-tenon structure, securing the tire in place. Synchronous pulley A is connected to the inner hub via a threaded structure. An underwater reduction motor powers the wheel mechanism, transmitting power through synchronous pulley B and a timing belt. The pulley set alters the power source position required for the wheel mechanism’s movement while increasing its torque. A gap is left in the center of the wheel mechanism to accommodate and operate the paddle mechanism.

The paddle mechanism provides underwater propulsion for the Amp-beetle. A propeller is positioned at the center of the fixed hub to form the underwater propulsion unit. The propeller is secured using a fixed collar and is concentric with the wheel mechanism. The wheel mechanism connects to the fixed hub of the paddle mechanism via thin-walled bearings A and B. The wheel rotates around the fixed hub without affecting the internal propeller mechanism. Since the propeller and wheel mechanism share the same axis, the thrust has a more extended lever arm, resulting in a more excellent thrust moment and improved control over the robot’s body posture changes.

The fixed hub in the paddle mechanism is secured to the elbow suspension housing using a threaded structure, ensuring stability between the wheel and paddle mechanisms. The elbow suspension housings A and B encase the timing pulley set and, together with the support legs, form a structure similar to a mammalian elbow, providing cushioning while protecting the internal timing pulleys. The support legs and elbow suspension housing are connected via suspension springs to absorb impacts.

The suspension mechanism plays a crucial role in securing, connecting, and protecting the various components, reducing the impact of shocks and vibrations during the robot’s movement. It also ensures that the wheel mechanism remains in contact with the ground, enhancing traction.

The leg mechanism is responsible for altering the wheel mechanism’s contact point and the paddle’s thrust vector direction. A servomotor connects to the support leg via a tilting flange and tilting bearing, providing the power to rotate the entire leg mechanism. The support structure and tilting flange secure the support leg, preventing the servomotor from being subjected to radial forces. The support structure uses a threaded connection to attach the integrated wheel-leg-paddle assembly to the Amp-beetle’s side mounting plate.

Overall structural design of the robot

The Amp-beetle consists of the robot body and four integrated wheel-leg-paddle drive mechanisms, as illustrated in Fig. 3. Its main components include a) the integrated wheel-leg-paddle mechanism, b) composite frame plates, c) the remote communication module, d) cameras and lights, e) flotation materials, f) a pressure-resistant waterproof control cabin, and g) auxiliary propulsion mechanisms. Four composite frame plates are secured together using a mortise and tenon structure and threaded connections to form the main frame of the robot, supported by metal rods as the internal skeleton. The integrated wheel-leg-paddle mechanisms are symmetrically attached to the front and rear frame plates using fixed flanges and screws. The robot can change its motion modes in different environments, allowing continuous movement across various amphibious settings by adapting its locomotion form.

Fig. 3
figure 3

The typical movement patterns of the Amp-beetle, its amphibious working environment, and related tasks.

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To ensure operational reliability in aquatic environments, comprehensive waterproofing measures are implemented across all critical subsystems. The core electronic components, including the main control board, battery, and sensors, are housed within custom-designed pressure-resistant dry compartments fabricated from aluminum alloy and sealed with O-rings. These chambers provide robust protection against hydrostatic pressure at the intended operating depth. Electromechanical actuators, such as the underwater gearmotors, are enclosed in hard-anodized aluminum casings and employ a dual-layer dynamic sealing strategy at rotational shaft outputs, using two professional-grade rotary lip seals in series for enhanced redundancy and reliability. All external cable penetrations are routed through industry-standard waterproof connectors and cable glands rated for the target depth. The waterproof integrity of each compartment was validated through rigorous pressure tests—sealed chambers were internally pressurized to 8 atmospheres (~0.8 MPa) for 8 hours, exceeding the maximum operational hydrostatic pressure by a safety factor of 1.5. No leakage or pressure drop was detected, confirming the effectiveness of the sealing design and structural integrity. The central pressure-resistant waterproof cabin serves as the power and control hub, distributing signals to regulate the movement of the integrated wheel-leg-paddle structures. This protected internal system enables reliable mode switching, allowing the robot to perform complex amphibious tasks and maintain adaptive mobility across diverse environments.

Flotation materials are attached above the composite frame plates to provide buoyancy for the Amp-beetle. A remote communication device is placed in the center of the flotation materials for remote communication and GPS positioning when the robot is on the water surface or land. The pressure-resistant waterproof control cabin is fixed at the center of the frame plates using threaded connections, and watertight connectors link the internal electronics of the control cabin to each port, providing power and signals to various sensors and drive components. In the prototype design, auxiliary thrusters are installed on both sides of the flotation material and inside the frame plates to provide return power if any issues arise during testing.

Multi-modal motion design

From coastal mudflats to seabed floors, from crawling through pipelines to underwater hovering, the Amp-beetle can independently traverse various amphibious environments through multiple locomotion mode transitions. The Amp-beetle combines leg, wheel, and paddle mechanisms, forming two primary modes: the wheel-leg mode and the vector thrust mode. As shown in Fig. 4, these two modes can be further subdivided into sub-modes based on the tilt angle of the leg mechanism, with each sub-mode suited to specific amphibious environments, as indicated on the right side of the figure.

Fig. 4
figure 4

Mobility modes of the Amp-beetle.

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The first category is the wheel-leg mode, which uses a coordinated control strategy between the wheel and leg mechanisms. By leveraging the friction generated between the tires and the contact surface, the Amp-beetle can move across flat terrains in amphibious environments where swimming is not required. Since the robot needs to exert pressure on the ground even underwater, it is designed with negative buoyancy—ensuring that when fully submerged, its weight exceeds its buoyancy, allowing it to maintain ground contact for bottom operations.

When the Amp-beetle’s integrated wheel-leg-paddle structure is perpendicular to the ground, it operates in the wheel-leg mode, as shown in the top-left panel of Fig. 4. The Amp-beetle can move quickly and flexibly on flat surfaces. When encountering obstacles, the tilt angle of the leg structure is adjusted to reposition the wheel mechanism for obstacle clearance. Using this combined wheel-leg gait, the robot can overcome obstacles that typical bottom-crawling underwater robots cannot and efficiently navigate complex, unstructured terrain. Amphibious environments are complex, often requiring the robot to adjust height, especially in the pipeline or rocky areas. The Amp-beetle can simultaneously adjust the tilt angles of its four integrated wheel-leg-paddle structures, widening its track and lowering its height to meet the demands of more challenging amphibious conditions.

The second category is the vector thrust mode, which combines the paddle and leg mechanisms to create a vector thrust effect. This mode relies on the rotation of the paddles to generate thrust underwater, combined with the tilt of the leg mechanism, to form a vector propulsion structure. The vector thrust strategy allows the Amp-beetle to maneuver flexibly and stably underwater, meeting the demands of complex underwater navigation.

When the leg mechanism is parallel to the body and remains fixed, the thrust generated by the paddle mechanism is entirely used to counteract negative buoyancy. In this configuration, only forces perpendicular to the robot’s body can be produced. This mode of operation is referred to as quadrotor mode, the Amp-beetle controls the rotational speed of the paddle mechanism to create thrust differentials, altering its body posture and thereby controlling its movement direction. Furthermore, reactive torque from the propellers is utilized to induce a yaw moment, enabling precise heading adjustments. In this mode, the integrated wheel-leg-paddle structure can only actively control four degrees of freedom: heave, roll, pitch, and yaw. However, underwater operations often involve various complex terrains, with different underwater environments exerting varying degrees of influence on the robot’s control, imposing significant pressure in a single direction. The vector thrust mode, as illustrated in the bottom-right panel of Fig. 4, is achieved by coordinating the tilt angle of the leg mechanism with the thrust output of the paddle mechanism. The Amp-beetle has more versatile motion capabilities and excellent current resistance when traversing various unstructured terrains. The Amp-beetle’s vector thrust mode features five actively controlled degrees of freedom (sway, heave, roll, pitch, yaw), offering enhanced environmental adaptability and maneuverability.

Modeling of the Amp-beetle

A dynamic model analysis of its structure is conducted to facilitate the control of the Amp-beetle robot. We assume that all internal structures of the Amp-beetle are rigid and that the robot is perfectly symmetrical along its xoz plane. Furthermore, we assume that no sliding or adhesive forces affect its movement. The specific body parameters of the robot are shown in Fig. 5, while Table 1 summarizes the relevant parameters and their abbreviations as used in the subsequent descriptions. The variable represents the identifier of each integrated wheel-leg-propeller structure, starting with the structure located at the front-right of the robot as number 1 and increasing clockwise. Additionally, the leg joint tilt angle, denoted as \(\theta_{ci}\), is defined as 0° when the leg mechanism is perpendicular to the body and 90° when it is parallel to the body.

Fig. 5
figure 5

Schematic of the Amp-beetle’s body parameters (A. Front View,B. Side View,C. Top View).

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Table 1 Amp-beetle parameter definition.
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Kinematic analysis of the wheel-leg mode

As shown in Fig. 6 the centre of mass (COM) is the origin of the body coordinate system, which coincides with the centre of mass of the Amp-beetle when all leg joint tilt angles are set to 0°. In this coordinate system, the positive x-axis points in the direction of the front camera, while the positive z-axis points vertically upward from the ground. In the wheel-leg mode, the robot employs a four-wheel differential drive system. The motion model is derived by establishing the relationship between the turning radius, velocity, and angular velocity, as shown in Eq. (1).

$$r_{c} = \frac{{v_{c} }}{{\omega_{c} }} = \frac{{\left( {v_{l} + v_{r} } \right)d_{LR} }}{{2\left( {v_{r} - v_{l} } \right)}}$$
(1)
Fig. 6
figure 6

Diagram illustrating motion in the world coordinate system for the wheel-leg mode.

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Here,\(r_{c}\) represents the turning radius of the robot, \(v_{c}\) is its linear velocity, \(\omega_{c}\) is its angular velocity, \({\text{d}}_{{{\text{LR}}}}\) denotes the distance between the left and right wheels, \(v_{l}\) and \(v_{r}\) represent the linear velocities of the left and right wheel mechanisms, respectively. The simplified forward kinematic model calculates the velocity of the geometric centre of mass (COM) based on the velocities of the virtual left and right driving wheels, which can be expressed as:

$$\left[ {\begin{array}{*{20}l} {v_{c} } \hfill \\ {\omega_{c} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{{v_{r} + v_{l} }}{2}} \hfill \\ {\frac{{v_{r} - v_{l} }}{{d_{LR} }}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1/2} & {1/2} \\ {1/d_{LR} } & { - 1/d_{LR} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {v_{r} } \hfill \\ {v_{l} } \hfill \\ \end{array} } \right]$$
(2)

However, unlike conventional four-wheeled differential robots, the Amp-beetle possesses leg transformation capabilities. The change in the leg joint’s tilt angle affects (\(\theta_{ci}\)) the wheelbase (\({\text{d}}_{{{\text{LR}}}}\)) of the Amp-beetle.

$$d_{LR} = 2L_{{\text{ry }}} + L_{{\text{leg }}} *\sin \left( {\theta_{{\text{leg }}} + \theta_{{\text{cl }}} } \right) + L_{{\text{leg }}} *\sin \left( {\theta_{{\text{leg }}} + \theta_{{\text{cr }}} } \right)$$
(3)

Here, \(\theta_{cr}\) and \(\theta_{cl}\) represent the tilt angles of the right and left sides, respectively. By controlling \(\theta_{cr}\),\(\theta_{cl}\),\(v_{l}\) and \(v_{r}\), the Amp-beetle can achieve wheel-leg mode movement on both land and underwater surfaces.

Kinematic and dynamic analysis of the vector thrust mode

This paper uses a six-degree-of-freedom (DOF) parameterization to describe the motion characteristics of the Amp-beetle in vector propulsion mode. The Amp-beetle is an underactuated underwater robot with variable thrust vectoring, where the tilt angles of its integrated wheel-leg-propeller structures do not contribute to thrust in the surge direction. In vector motion mode, the Amp-beetle has five active degrees of freedom and one passive degree of freedom.

When the tilt angles are at the specific value of 90°, the Amp-beetle loses its ability to move horizontally in the X and Y directions within the body-fixed frame. The design of the variable thrust vectoring allows the robot to achieve omnidirectional movement using only four integrated wheel-leg-propeller structures. However, such omnidirectional motion is constrained by its limited degrees of freedom due to the underactuated nature of the robot.

The inertial frame and the body-fixed frame of the Amp-beetle during underwater swimming are shown in Fig. 7. Here, \(\eta_{1} = \left[ {\begin{array}{*{20}c} x & y & z \\ \end{array} } \right]^{T}\) and​ \(\eta_{2} = \left[ {\begin{array}{*{20}c} \varphi & \theta & \psi \\ \end{array} } \right]^{T}\) represent the robot’s spatial position and attitude information in the global coordinate system. \(\phi\), \(\theta\) and \(\psi\) are the robot’s roll, pitch, and yaw angles. The body-fixed frame defines the robot’s linear velocities \(v_{1} = \left[ {\begin{array}{*{20}c} u & v & w \\ \end{array} } \right]^{T}\) and angular velocities \(v_{2} = \left[ {\begin{array}{*{20}c} p & q & r \\ \end{array} } \right]^{T}\). Here \(u\),\(v\),and \(w\) denote the surge, sway, and heave velocities of the robot. \(p\),\(q\),and \(r\) represent the robot’s roll, pitch, and yaw angular velocities, respectively. It is important to note that the robot’s position and attitude information are defined in the global coordinate system. In contrast, the velocity and angular velocity information are defined in the body-fixed frame.

Fig. 7
figure 7

In the vector thrust mode, the situation of Amp-beetle in both the inertial and body reference frames may vary. All parameters are uniquely defined for the vector thrust mode, differing in meaning from the symbols used in the previous section.

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Using an IMU, attitude information in the global coordinate system is obtained, and a rotation matrix must be applied to convert this data into the body-fixed frame for control purposes. The linear velocity can be expressed as:

$$\dot{\eta }_{1} = J_{1} \left( {\eta_{1} } \right)v_{1}$$
(4)

Here, \(J_{1} (\eta_{1} )\) is the transformation matrix for converting linear velocity from the body-fixed frame to the global coordinate system:

$$J_{1} \left( {\eta_{1} } \right) = \left[ {\begin{array}{*{20}c} {c\theta c\psi } & {c\psi s\phi s\theta - s\psi c\phi } & {c\psi s\theta c\phi + s\psi s\phi } \\ {c\theta s\psi } & {s\psi s\phi s\theta + c\psi c\phi } & {s\psi s\theta c\phi - c\psi s\phi } \\ { - s\theta } & {s\phi c\theta } & {c\phi c\theta } \\ \end{array} } \right]$$
(5)

Here, \(s( * )\) and \(c( * )\) are simplified forms of \(\sin (\rm{ * })\) and \(\cos (\rm{ * })\). The angular velocity can be expressed as:

$$\dot{\eta }_{2} = J_{2} \left( {\eta_{2} } \right)v_{2}$$
(6)

Here, \(J_{2} (\eta_{2} )\) is the transformation matrix that converts angular velocity from the body-fixed frame to the global coordinate system:

$$J_{2} \left( {\eta_{2} } \right) = \left[ {\begin{array}{*{20}c} 1 & {\tan \theta {\text{s}}\varphi } & {\tan \theta {\text{c}}\varphi } \\ 0 & {{\text{c}}\varphi } & { - {\text{s}}\varphi } \\ 0 & { {\text{s}}\varphi /{\text{c}}\theta } & {{\text{c}}\varphi /{\text{c}}\theta } \\ \end{array} } \right]$$
(7)

Therefore, the kinematic model of the Amp-beetle is:

$$\dot{\eta } = \left[ {\begin{array}{*{20}c} {J_{1} \left( {\eta_{1} } \right)} & {0_{3 \times 3} } \\ {0_{3 \times 3} } & {J_{2} \left( {\eta_{2} } \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {v_{1} } \hfill \\ {v_{2} } \hfill \\ \end{array} } \right] = J(\eta )v$$
(8)

The dynamic model during underwater swimming mode can be expressed as:

$$\left( {M_{A} + M_{R} } \right)v + (C_{A} + C_{R} )v + D_{{({\mathbf{v}})}} v + g(\eta ) = Bu$$
(9)

Here\(M_{A}\) is the added mass matrix, \(M_{R}\) is the rigid body mass matrix, \(C_{A}\) and \(C_{R}\) are the Coriolis. Due to the robot’s low cruising speed, the values of \(C_{A}\) and \(C_{R}\) are relatively small compared to other terms and are typically neglected. \(D_{(V)}\) is the damping matrix, which counteracts the resistance caused by velocity and the surrounding water. \(g(\eta )\) represents the combined forces and moments generated by the robot’s weight and buoyancy. \(B\) is the thrust coefficient matrix of the Amp-beetle.

Center of gravity, buoyancy, and righting moment

The multiple motion modes of the Amp-beetle are achieved through the coordinated combination of tilting joints, wheel, and propeller mechanisms. The integrated wheel-leg-propeller structures undergo tilting motion during operation. When the integrated wheel-leg-propeller structures tilt, the robot’s center of gravity and center of buoyancy change accordingly. The positions and magnitudes of the center of gravity and center of buoyancy affect the robot’s ability to regain balance. The Amp-beetle exhibits different balance recovery capabilities under different modes. In the wheel-leg mode, the Amp-beetle should have a strong balance recovery ability, allowing the robot to remain stable without external interference. However, an underactuated robot must adjust its attitude to control movement in vector propulsion mode. It means that poor balance recovery due to the misalignment of the center of gravity and center of buoyancy can hinder the robot from achieving the desired attitude, leading to inefficiency in propeller thrust.

Calculations of the positions of the center of gravity and center of buoyancy, performed using SolidWorks, reveal that the tilting angle of the integrated wheel-leg-propeller structure has a more significant impact on vertical changes than on planar position changes. The impact of the tilting angle of a single integrated wheel-leg-propeller structure on the vertical position of the robot’s overall center of gravity and center of buoyancy is shown in Fig. 8. As the tilting angle of the integrated structure increases, the center of gravity and center of buoyancy rise in a curved trajectory. Since the density of the integrated structure is much greater than that of water, the tilting angle affects the rate of change of the center of gravity more than that of the center of buoyancy. Additionally, the greater the distance between the center of gravity and the center of buoyancy, the stronger the robot’s balance recovery capability underwater. It causes the Amp-beetle’s center of buoyancy to be much higher than its center of gravity in wheel-leg mode, resulting in superior static stability. However, in vector propulsion mode, as the tilting angle of the integrated structure increases, the distance between the center of gravity and the center of buoyancy rapidly decreases, making the robot more prone to attitude changes and more accessible to control.

Fig. 8
figure 8

The shift in the center of gravity’s position (Keyshot11, https://www.keyshot.com/blog/luxion-releases-keyshot-11/ Matlab2022b,https://ww2.mathworks.cn/?s_tid=gn_logo WPS,https://www.wps.cn/).

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The tilting angle of the integrated wheel-leg-propeller structure alters the Amp-beetle’s centre of gravity positions \(\left[ {\begin{array}{*{20}c} {x_{w} } & {y_{w} } & {z_{w} } \\ \end{array} } \right]\) and centre of buoyancy positions \(\left[ {\begin{array}{*{20}c} {x_{B} } & {y_{B} } & {z_{B} } \\ \end{array} } \right]\).\(F_{B}\) and \(F_{W}\) represent the net buoyant force and gravitational force, respectively. The restoring moment of the Amp-beetle can be derived from the calculation of the gravitational and buoyant forces along with their respective moment arms, as shown below:

$${\mathbf{g}}({{\varvec{\upeta}}}) = \left[ {\begin{array}{*{20}c} {\left( {F_{B} - F_{W} } \right)s_{\theta } } \\ {\left( {F_{W} - F_{B} } \right)c_{\theta } s_{\phi } } \\ {\left( {F_{W} - F_{B} } \right)c_{\theta } c_{\phi } } \\ {\left( {y_{W} F_{W} - y_{B} F_{B} } \right)c_{\theta } c_{\phi } - \left( {z_{W} F_{W} - z_{B} F_{B} } \right)s_{\phi } c_{\theta } } \\ {\left( {z_{B} F_{B} - z_{W} F_{W} } \right)s_{\theta } - \left( {x_{W} F_{W} - x_{B} F_{B} } \right)c_{\phi } c_{\theta } } \\ {\left( {x_{W} F_{W} - x_{B} F_{B} } \right)s_{\phi } c_{\theta } + \left( {y_{W} F_{W} - y_{B} F_{B} } \right)s_{\theta } } \\ \end{array} } \right]$$
(10)

Since the changes in the centre of gravity and centre of buoyancy along the z-axis of the Amp-beetle are much greater than those along the y-axis, the center of gravity and buoyancy can lie on the same vertical line. Eq. (10) can be rewritten as:

$${\mathbf{g}}({{\varvec{\upeta}}}) = \left[ {\begin{array}{*{20}c} {(B - W)s_{\theta } } \\ {(W - B)c_{\theta } s_{\phi } } \\ {(W - B)c_{\theta } c_{\phi } } \\ {z_{B} Bs_{\phi } c_{\theta } } \\ {z_{B} Bs_{\theta } } \\ 0 \\ \end{array} } \right]$$
(11)

Thrust vector matrix

The vector propulsion mode is the primary motion mode for the Amp-beetle’s underwater swimming. The tilting angle of the integrated wheel-leg-propeller structure influences the change in vector direction. By modeling the positions and thrust of the body propellers, we can derive the thrust coefficient matrix :

$$B = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ {c\theta_{c1} } & {c\theta_{c2} } & { - c\theta_{c3} } & { - c\theta_{c4} } \\ {s\theta_{c1} } & {s\theta_{c2} } & {s\theta_{c3} } & {s\theta_{c4} } \\ { - L_{{\text{xarm }}} } & { - L_{{\text{xarm }}} } & {L_{{\text{xarm }}} } & {L_{{\text{xarm }}} } \\ {L_{{\text{yarm }}} *s\theta_{c1} } & { - L_{{\text{yarm }}} *s\theta_{c2} } & { - L_{{\text{yarm }}} *s\theta_{c3} } & {L_{{\text{yarm }}} *s\theta_{c4} } \\ { - L_{{\text{zarm }}} *c\theta_{c1} } & {L_{{\text{zarm }}} *c\theta_{c2} } & { - L_{{\text{zarm }}} *c\theta_{c3} } & {L_{{\text{zarm }}} *c\theta_{c4} } \\ \end{array} } \right]$$
(12)

The underwater thrust coefficient matrix of the robot can be obtained by multiplying the moment arms with the thrust of the underwater propellers. Here,\(L_{xarm}\), \(L_{yarm}\) and \(L_{zarm}\) are variable parameters related to the robot’s leg-raising angle \(\theta_{ci}\). Eq.(13), (14) and (15) can be obtained from Fig. 5:

$$L_{xarm} = L_{ry} *\sin \left( {\theta_{ci} } \right) + L_{{\text{leg }}} *\cos \left( {\theta_{{\text{leg }}} } \right) + L_{rz} *\cos \left( {\theta_{ci} } \right)$$
(13)
$$l_{yarm} = l_{rx}$$
(14)
$$l_{zarm} = l_{rx}$$
(15)

Simulation of the wheel-leg-paddle integrated amphibious robot

Based on the hydrodynamic parameters identified through high-fidelity numerical simulations using ANSYS Fluent CFD software, a dynamic model of the amphibious robot was constructed in MATLAB/Simulink environment. The obtained hydrodynamic coefficients, including the rigid-body mass matrix and added mass matrix, were integrated into the model to accurately represent the robot’s hydrodynamic characteristics. Where rigid-body mass matrix is obtained as:

$$M_{R} = {\text{diag}} \left\{ {\left( {\begin{array}{*{20}l} {112.380} \hfill & {112.380} \hfill & {112.380} \hfill & {16.067} \hfill & {8.944} \hfill & {23.844} \hfill \\ \end{array} } \right)} \right\}.$$
(16)

Where added mass matrix is obtained as:

$$M_{A} = {\text{diag}} \{ [44.5695,75.1587,248.4437,2.4245,3.4101,2.5106]\} .$$
(17)

And the damping matrix is:

$$D_{(v)} = \left[ {\begin{array}{*{20}c} {153.3843u^{2} - 12.036u + 0.9813} \\ {225.2857v^{2} - 23.6494v + 2.2503} \\ {215.0329w^{2} + 157.4679w - 15.1667} \\ {1.4414p^{2} + 0.7773p - 0.3260} \\ {2.1568q^{2} - 0.6593q - 0.3127} \\ {0.8680r^{2} + 0.0157r + 0.0114} \\ \end{array} } \right]$$
(18)

Furthermore, it is noted that during its initial operational mode, the Amp-beetle typically does not engage in long-distance, high-speed, or prolonged duration missions. Consequently, the Coriolis and centripetal forces, which are velocity-dependent and significant under high-speed conditions, are considered negligible for this specific analysis and are thus omitted from the dynamic model. This simplification is justified by the expected operational envelope and enhances computational efficiency without sacrificing model fidelity for the intended use case.

To rigorously validate the dynamic model and comprehensively evaluate the closed-loop performance, two distinct control strategies—Nonlinear Model Predictive Control (NMPC) and a dual-loop cascaded PID architecture—were designed, implemented, and compared through simulations. The parameter configuration of the cascaded dual-loop PID controller is as follows:

$$\left\{ {\begin{array}{*{20}l} {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{p}}}} = {\text{diag}} ([2.1,\;2.1,\;2.6]),\quad {\mathbf{K}}_{{{\mathbf{i}},{\mathbf{v}}}} = {\text{diag}} ([0.01,\;0.01,\;0.03])} \hfill \\ {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{v}}}} = {\text{diag}} ([50,\;50,\;50]),\quad {\mathbf{K}}_{{{\mathbf{i}},{\mathbf{v}}}} = {\text{diag}} ([1,\;1,\;0.7]),\quad {\mathbf{K}}_{{{\mathbf{d}},{\mathbf{v}}}} = {\text{diag}} ([0.001,\;0.001,\;0])} \hfill \\ {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{angle}}}} = {\text{diag}} ([1.55,\;1.55,\;1]),\quad {\mathbf{K}}_{{{\mathbf{i}},{\mathbf{v}}}} = {\text{diag}} ([0.0002,\;0.0002,\;0.0.0001])} \hfill \\ {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{anglev}}}} = {\text{diag}} ([4.2,\;4.2,\;3.45]),\quad {\mathbf{K}}_{{{\mathbf{i}},{\mathbf{anglev}}}} = {\text{diag}} ([5.5,\;5.5,\;10]),\quad {\mathbf{K}}_{{{\mathbf{d}},{\mathbf{anglev}}}} = {\text{diag}} ([0,\;0,\;0])} \hfill \\ \end{array} } \right.$$
(19)

The NMPC strategy was formulated with the following cost function to minimize tracking error and control effort:

$$J = \sum\limits_{k = 1}^{{N_{p} }} {\left\| {{\mathbf{x}}(k) - {\mathbf{x}}_{{{\text{ref}}}} (k)} \right\|^{2} } {\mathbf{Q}} + \sum\limits_{k = 0}^{{N_{c} - 1}} {\left\| {{\mathbf{u}}\left( k \right)} \right\|^{2} {\text{R}}}$$
(20)

where \({\text{x}}(k)\) denotes the system state,\(x_{{{\text{ref}}}} (k)\) is the reference trajectory, and \(u(k)\) represents the control input. The weighting matrices \(Q\rm{ \succcurlyeq }0\) and \(R \succ 0\) penalize tracking errors and control efforts, respectively. In this simulation the prediction horizon and control horizon were set to \(N_{P} = 18\) and \(N_{c} = 3\), respectively. The controller is subject to the following thrust magnitude and direction constraints:

$$\left\{ {\begin{array}{*{20}l} { - 70N\rm{}u_{i} \rm{}100N,} \hfill & {i \in 1,2,3,4} \hfill \\ {0\deg \rm{}u_{i} \rm{}150\deg ,} \hfill & {i \in 5,6,7,8} \hfill \\ { - 10N\rm{}u_{i} \rm{}10N,} \hfill & {i \in 1,2,3,4} \hfill \\ { - 5\deg \rm{}u_{i} \rm{}5\deg ,} \hfill & {i \in 5,6,7,8} \hfill \\ \end{array} } \right.$$
(21)

where \(u_{i}\) represents the thrust magnitude and thrust angle of each propeller, and \(\vartriangle u_{\theta i}\) represents their rates of change. These constraints ensure that the control commands remain within the physical actuation limits of the platform.

As illustrated in Fig. 9, the simulation results demonstrate that both control algorithms are capable of effectively maneuvering the Amp-beetle. The dual-loop cascaded PID control architecture offers computational efficiency and ease of implementation on embedded platforms, its performance remains limited in terms of tracking precision, disturbance rejection, and constraint handling. In contrast, the nonlinear model predictive control (NMPC) strategy exhibits significantly superior overall performance, demonstrating higher trajectory tracking accuracy, enhanced stability, and smoother motion control. The NMPC controller achieves dynamic responses with reduced overshoot and lower steady-state error, while maintaining robustness even under challenging motion scenarios. These outcomes confirm the controllability and stability of the developed dynamic model.

Fig. 9
figure 9

Amp-beetle position and posture tracking curve (Matlab2022b, https://ww2.mathworks.cn/?s_tid=gn_logo), Figs. 10, 13, 14, 15, 16 also uses the same software.

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The actuator outputs, depicted in Fig. 10, further illustrate the thrust vectoring behavior during maneuvering. The smooth and coordinated variation in thrust magnitude and direction across the propellers validates the effectiveness of the vector control methodology. It can be observed that the physical constraints of the actuators are satisfactorily maintained throughout the simulation, demonstrating the practical applicability of the proposed control strategy.

Fig. 10
figure 10

Amp-beetle nonlinear model predictive control for thrust vectoring and angle adjustment.

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Experimental trials of the wheel-leg-paddle integrated amphibious robot

To verify the motion characteristics of the Amp-beetle, as well as its environmental adaptability and continuous cross-environmental mobility in near-shore amphibious environments, we conducted experiments with the Amp-beetle in pool environment. The specific parameters of the Amp-beetle prototype are shown in Table 2.

Table 2 Amp-beetle chassis specifications.
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The electrical control system of the Amp-beetle is shown in Fig. 11. The Amp-beetle utilizes a wireless TTL UART downloader, a Wi-Fi module, and a wired Ethernet port for control and debugging. The wireless TTL UART downloader allows rapid modifications to the low-level control program without disassembling the entire robot structure. The Wi-Fi module is used for wireless control on land. The wired Ethernet module is connected via a buoyancy cable, controlling the robot’s underwater movement and transmitting body data and underwater imagery. Real-time depth status is captured by an integrated depth sensor fixed outside the main control cabin. The Raspberry Pi 4B, integrated within the waterproof control cabin, is the Amp-beetle’s onboard computer, handling extensive matrix calculations in complex underwater scenarios. The STM32F407 functions as the motion controller, responsible for transmitting key robotic data signals—such as speed, attitude, and depth—to the Raspberry Pi. The Raspberry Pi receives and processes these data signals, using the control algorithms designed in this paper to manage the robot’s movements. The battery is connected to a DC voltage regulator module, which provides a stable 24 V for the underwater actuators and servos and a stable 48 V for the waterproof gear motor. A switch circuit composed of a waterproof switch and relay prevents the need to repeatedly disassemble and reassemble the waterproof structure when powering the robot on or off. All external components are sealed using epoxy adhesive and sealing rings.

Fig. 11
figure 11

Schematic of hardware control connections for the Amp-beetle. (Visio,https://www.microsoft.com/zh-cn/microsoft-365/visio/).

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Motion characteristic experiments

Wheel-leg mode

As shown in Fig. 12, the Amp-beetle was tested in various environments, including hard ground, slopes, grass, and the bottom of a pool (10m). The Amp-beetle could operate quickly and smoothly in all environments while also demonstrating a certain degree of obstacle-crossing capability. The movement speed is related to the motor rotation speed and the roughness of the ground, with a maximum speed exceeding 1.2 m/s. During land operation, once a single integrated wheel-leg-propeller structure tilts to the designated angle and stabilizes, the servo current stabilizes around 0.1A. The Amp-beetle can climb slopes with angles greater than 30° and stop and start on slopes. Additionally, the Amp-beetle can perform in-place spins through the differential motion of its four wheels, achieving zero-radius turns. The experimental results demonstrate that the robot is capable of operating effectively in land environments, amphibious transition zones, and along the bottom of underwater environments.

Fig. 12
figure 12

Amp-beetle’s movement across various amphibious environments, including hard surfaces, grasslands, dry sand, wet sand, and underwater conditions.

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Vector propulsion mode

To evaluate its maneuverability, the Amp-Beetle was subjected to testing in a 10 m × 30 m × 10 m water tank under quadcopter mode, employing a double closed-loop PID control algorithm for stability and precision, with specific parameters configured as follows:

$$\left\{ {\begin{array}{*{20}l} {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{p}}}} = {\text{diag}} ([6,6,2]),\quad } \hfill \\ {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{v}}}} = {\text{diag}} ([4,\;4,\;3]),\quad {\mathbf{K}}_{{{\mathbf{i}},{\mathbf{v}}}} = {\text{diag}} ([2,\;2,\;1.4]),\quad {\mathbf{K}}_{{{\mathbf{d}},{\mathbf{v}}}} = {\text{diag}} ([0.0001,\;0.0001,\;0])} \hfill \\ {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{angle}}}} = {\text{diag}} ([3,\;3,\;0.8]),\quad } \hfill \\ {{\mathbf{K}}_{{{\mathbf{p}},{\mathbf{anglev}}}} = {\text{diag}} ([4.2,\;4.2,\;3.45]),\quad {\mathbf{K}}_{{{\mathbf{i}},{\mathbf{anglev}}}} = {\text{diag}} ([2.2,\;2.2,\;1.7]),\quad {\mathbf{K}}_{{{\mathbf{d}},{\mathbf{anglev}}}} = {\text{diag}} ([0,\;0,\;0])} \hfill \\ \end{array} } \right.$$
(22)

The depth and attitude information of the Amp-beetle were measured by the depth sensor and IMU onboard the robot. Figures 13, 14, and 15 show the body attitude angles and the prescribed thrust values of the propeller blades during depth-holding surge, yaw, and depth-holding maneuvers of the Amp-beetle. The RF PWM, RB PWM, LB PWM, and LF PWM correspond to the rotational speeds of the propeller blades on wheel-leg-propeller structures 1 through 4, respectively. The quadcopter mode is selected for detailed analysis here as it represents a canonical and highly constrained mode of Amp-beetle vector-based propulsion with limited attitude variation.

Fig. 13
figure 13

Changes in the rotational speed of the four Wheel-Leg-Paddle structures and the pitch angle of the Amp-beetle during depth-hold forward motion.

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Fig. 14
figure 14

Changes in the rotational speed of the four Wheel-Leg-Paddle structures and the pitch angle of the Amp-beetle during depth-hold yaw motion.

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Fig. 15
figure 15

Changes in depth, pitch angle, and roll angle of the Amp-beetle, as well as the rotational speed of the four Wheel-Leg-Paddle structures, during depth-hold hover motion.

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Fig. 13 illustrates the Amp-beetle performing a depth-holding surge motion underwater. In the quadcopter mode, surge and sway are controlled through similar mechanisms by modulating the pitch and roll angles via differential thrust from the propellers. Therefore, only the surge motion is presented here as a representative case. As an underactuated robot, the Amp-beetle adjusts its pitch and roll attitudes by regulating the rotational speeds of the propellers, thereby generating the desired horizontal motions. As shown in the figure, after an initial stabilization period, the robot initiates surge motion at the 18 second mark by applying appropriate control commands. The pitch angle rapidly changes to tilt the robot forward, while the propeller speeds are adjusted accordingly to achieve and maintain the motion. By 30 seconds, the robot reaches a steady surge velocity with constant propeller speeds, demonstrating effective attitude and depth control during the maneuver.

Fig. 14 shows the Amp-beetle performing depth-holding yaw motion underwater. The robot generates a yawing moment through the inherent counter-torque effects of the propellers, while depth is maintained through balanced thrust allocation. Due to hydrodynamic disturbances and combined with minor water flow variations in the pool environment, contribute to the observed variations in yaw rate despite stable overall depth maintenance.

The results of both the surge and yaw maneuvers demonstrate that the quadcopter mode provides effective motion control in multiple degrees of freedom while reliably maintaining depth stability. As an underactuated system, the Amp-beetle successfully achieves complex underwater locomotion through coupled attitude and thrust modulation, despite limitations in independent control of all degrees of freedom. The slight fluctuations observed during the yaw maneuver highlight the influence of hydrodynamic effects and propeller-induced torques, which are inherent challenges in underwater robotic systems. Nevertheless, the overall performance confirms the effectiveness of the mechanical design and control strategy. This integrated approach reduces mechanical complexity, system weight and power consumption, proving advantageous for amphibious operations. The consistent performance in different motion modes indicates that the Amp-beetle is well-suited for applications requiring precise underwater positioning and navigation.

The hovering experiment demonstrated that the Amp-beetle can adjust its posture to maintain a fixed depth underwater. Data from the underwater thrusters, IMU, and depth gauge are shown in Fig. 15. It can be seen that when the attitude angles change, each propeller leg adjusts its respective speed to correct the body’s error angles, ensuring that the roll angle remains below 3° during the entire hovering process. The results indicate that after adjustments during swimming, the Amp-beetle can achieve stable hovering with a depth-holding error of less than 0.1m.

From the above experiments, it can be observed that the Amp-beetle controls its posture by creating thrust differentials by varying the thrust of the four underwater thruster groups, thereby controlling its movement direction and speed. The experiment proves that, despite its underactuated design lacking surge thrust and an additional degree of freedom, it can still adjust its posture to modify its displacement characteristics. The experiment demonstrates its capability to fulfill complex task movement requirements in three-dimensional space.

Amphibious adaptability and cross-environment continuous motion experiments

The Amp-beetle possesses autonomous deployment capabilities and the ability to move continuously across amphibious environments, which means it must be able to traverse land independently before transitioning to underwater operation. The Amp-beetle’s transition process in water involves five stages. The first stage is wheel-leg motion, where it performs bottom-following movement at a depth of 8 meters. The second stage occurs when the Amp-beetle encounters an obstacle it cannot traverse, prompting it to switch to vector propulsion mode at 13 seconds. The third stage involves vector-propulsion ascent, forward movement, and hovering. The Amp-beetle rapidly ascends 5 meters within 10 seconds and proceeds with a depth-holding surge toward the designated underwater central platform. Upon reaching the underwater central platform, it enters the fourth stage, transitioning back to wheel-leg mode. In the fifth stage, it descends to the bottom and moves using the wheel-leg mode, executing turns and finally coming to rest at the edge of the experimental pool. The entire process is illustrated in Fig. 16, with sensor data recorded by the depth sensor, calibrated before the experiment began. The Amp-beetle underwent complex transitions and motion adaptations throughout these stages to suit various underwater environments. The experiments showed that the robot does not disturb the sediment during bottom-following movement, thereby minimizing environmental impact. A video of the experimental process is provided in Supplementary Video S1. The entire experimental process demonstrates that the Amp-beetle is capable of autonomous deployment and can continuously operate across complex amphibious environments, exhibiting excellent mobility and environmental adaptability.

Fig. 16
figure 16

Five-stage motion transition of the Amp-beetle in water.

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Discussion

This study presents the development and experimental validation of the Amp-beetle, an amphibious robot featuring a novel integrated wheel-leg-propeller architecture that enables multi-modal locomotion across terrestrial and aquatic environments. The compact structural design and adaptive control strategies demonstrate significant advantages over conventional amphibious systems, particularly in terms of mechanical simplicity, environmental adaptability, and energy efficiency.

The primary innovation lies in the integrated actuation mechanism that combines three locomotion modalities into a unified structure. This design eliminates the need for additional propulsion mechanisms or complex transformation systems typically required in conventional amphibious robots. The compact configuration not only reduces overall system weight and inertia but also minimizes potential leakage points through reduced sealing requirements, thereby enhancing operational reliability and maintainability. Experimental results confirm that this integrated structure delivers high-performance mobility on land while providing sufficient underwater thrust, effectively balancing the demands of terrestrial mobility and aquatic propulsion.

The robot’s multi-modal adaptive capability represents another significant advancement. Through seamless transitions between wheeled, legged, and vector propulsion modes, the system autonomously adapts to diverse environments ranging from hard ground, slopes, and grassy terrain to underwater substrates. Particularly noteworthy is the robot’s minimal environmental disturbance during bottom-contacting locomotion in wheel-leg mode, coupled with precise three-dimensional moveing capability in vector propulsion mode. This multi-environment adaptability enables operations that are challenging for conventional amphibious robots, particularly in applications requiring minimal environmental impact such as fragile ecosystem monitoring and underwater infrastructure inspection.

Through independent regulation of the tilting angles of the four leg structures and the rotational speeds of the four propellers, the system achieves integrated attitude and displacement control. Simulation results indicate that in vectored thrust mode, the robot exhibits motion characteristics with five independent degrees of freedom, wherein forward-backward translation is accomplished solely through pitch angle modulation. Experimental data corroborate that in quadcopter mode, the platform maintains stable hovering with roll angles constrained within 3°, while surge motion is achieved via pitch orientation adjustments, albeit operating with only four active degrees of freedom in three-dimensional space. Although this strategy imposes certain limitations on the decoupled control along specific degrees of freedom, it markedly reduces system complexity and power consumption—a feature particularly beneficial for extended amphibious missions.

However, the study also identifies several limitations worthy of consideration. The underactuated nature requires horizontal motion to be achieved indirectly through attitude adjustments, which may affect control precision in strongly perturbed environments. Additionally, the computational capabilities of the current platform (STM32F407 + Raspberry Pi 4B) constrain the implementation of more advanced control algorithms, such as NMPC, in real-time applications. Future improvements could incorporate dedicated computing hardware and optimize control allocation algorithms to enhance performance.

In conclusion, the Amp-beetle’s integrated wheel-leg-propeller architecture successfully unifies structural compactness with functional diversity, offering a novel approach to amphibious robot design. Its exceptional environmental adaptability and low-disturbance characteristics make it particularly suitable for marine research, ecological monitoring, and coastal exploration applications.

Conclusions

This paper contributes by proposing a novel amphibious mobile robot design based on an integrated wheel-leg-propeller driving structure. Utilizing its integrated wheel-leg-propeller mechanism, the Amp-beetle possesses specific obstacle-crossing capabilities and multiple locomotion modes. The Amp-beetle can navigate complex, unstructured amphibious terrains with agility and achieve six-degree-of-freedom movement underwater in any direction. Through modeling, dynamic analysis, and simulation experiments, it has been demonstrated that the robot, despite its underactuated nature, possesses the capability for six degrees of freedom underwater movement.

Subsequently, we developed a prototype of the amphibious robot. We validated its running and swimming capabilities in simulated experimental environments, demonstrating its locomotion characteristics, amphibious adaptability, and ability to move continuously across complex amphibious environments. Experimental results show that in the wheel-leg mode, the robot achieves a maximum speed of approximately 1.2 m/s, is capable of adapting to hard surfaces, slopes, grass, and pool bottoms (8m deep), and can carry a load of 45 kg while moving at 0.8 m/s. Tests conducted in a pool demonstrated that the Amp-beetle can operate at depths of up to 10 meters (The current experimental environment depth) with a negative buoyancy of −100N. It possesses six-degree-of-freedom movement capabilities underwater, including but not limited to surge, sway, diagonal movement, and depth-holding. The Amp-beetle demonstrates excellent environmental adaptability and the ability to move continuously across complex amphibious environments. Compared to other robots, it eliminates the need for additional manual deployment, extends the continuous operational range, and reduces environmental impact.

In future work, the integrated wheel-leg-propeller mechanism will be powered by a single motor to reduce system complexity and redundancy. Further exploration of alternative configurations for this integrated structure will be conducted to enhance the robot’s amphibious adaptability. Vector control algorithms will be incorporated to improve propulsion efficiency and minimize thrust loss. For extreme soft terrains such as mudflats and sandy beaches, advanced locomotion mechanisms including reconfigurable triangular tracks will be introduced and evaluated in comparison with the current wheel-leg design, alongside further in-depth study of the dynamics of movement across various terrains. Meanwhile, optimal vector propulsion control strategies for the Amp-beetle robot under varying center-of-gravity and buoyancy conditions will be thoroughly investigated.