The pursuit of mobile robots capable of operating in complex, unstructured environments has driven significant research in bio-inspired engineering. Among these, bionic robots that emulate the locomotion of animals offer superior mobility and adaptability compared to traditional wheeled or tracked platforms. This article presents the design, control, and optimization of a novel crocodile-inspired bionic robot. The crocodile, a master of amphibious locomotion, exhibits remarkable versatility, seamlessly transitioning between high-walking, sprawling, and even rolling motions during predation. To capture this multi-modal capability, we introduce a metamorphic leg mechanism, enabling structural reconfiguration between a sprawling posture for crawling and a locked, arched posture for rolling. A Central Pattern Generator (CPG) network based on nonlinear Hopf oscillators generates the rhythmic joint control signals for stable walking gaits. Furthermore, we analyze the dynamic stability using the Zero Moment Point (ZMP) principle and propose an active tail-swinging strategy to mitigate undesired yaw during high-speed trotting. Comprehensive co-simulations using MATLAB and ADAMS validate the robot’s stable performance across different terrestrial gaits and its rolling locomotion capability.
1. Bio-inspired Design with a Metamorphic Mechanism
The core innovation of our bionic robot lies in its metamorphic leg structure, directly inspired by the crocodile’s ability to perform a “death roll.” This drastic motion requires a rigid, arched body configuration, contrasting with the articulated legs used for walking. Our design goal is a single robotic platform that can morph between these two distinct functional states.
The robot’s overall architecture consists of a main torso, four independently driven metamorphic legs, and a multi-segmented active tail. Each leg is a closed-chain three-link mechanism, with its joint constraints alterable to change the limb’s degrees of freedom (DoF). In Configuration 1 (Crawling Mode), the leg operates as a standard 2-DoF limb (hip and knee pitch), allowing for cyclic lifting and placing motions. In Configuration 2 (Rolling Mode), specific joints are locked via latching mechanisms (conceptually implemented with electromagnets in the design phase), transforming the leg into a rigid, C-shaped spoke. This transforms the entire robot into a segmented, wheel-like structure capable of sidewise rolling.
The metamorphic transition is governed by changes in kinematic constraints. Let the leg links be denoted as A (thigh), B (shin), C (foot), D, and E, with revolute joints a (hip), b, c (knee), and d. In Configuration 1, all joints are active. The transition to Configuration 2 involves creating a rigid connection between links A and E. This can be represented as a change in their kinematic pair from a non-rigid connection to a fixed joint:
$$ C_{A,E}^{(1)} = R \rightarrow C_{A,E}^{(2)} = 0 $$
where $C$ denotes the constraint, $R$ is a revolute joint, and $0$ indicates a rigid fixation. Concurrently, the composite link formed by D and E changes its connection to link A from a revolute joint to a fixed connection:
$$ \begin{aligned}
C_{D,E}^{(1)} &= R \\
C_{D,E,A}^{(2)} &= D \oplus E \rightarrow 0 \text{ (fixed to A)}
\end{aligned} $$
Other joints (a, b, c) may also be locked depending on the specific design to achieve the desired rigid spoke shape. This morphing ability is central to the robot’s multifunctionality, allowing the same bionic robot platform to adopt the optimal configuration for crawling over rough terrain or rolling rapidly over smoother ground. The key dimensional parameters of the designed platform are summarized in Table 1.
| Component | Dimension (m) |
|---|---|
| Total Body Length | 1.364 |
| Body Height (standing) | 0.272 |
| Body Width | 0.302 |
| Multi-segment Tail Length | 0.675 |
| Upper Leg Link | 0.095 |
| Lower Leg Link | 0.105 |
2. Gait Generation Using a Central Pattern Generator Network
To endow our bionic robot with lifelike, adaptive, and stable locomotion, we employ a bio-inspired control strategy based on Central Pattern Generators (CPGs). CPGs are neural networks capable of producing coordinated rhythmic signals without requiring high-level feedback for each cycle, making them ideal for generating the periodic joint motions of legged locomotion.
2.1. Biological Gait Analysis and Target Gaits
Crocodiles employ different gait strategies depending on speed and substrate. For terrestrial locomotion, two primary patterns are observed: the high walk and the low sprawl. The high walk, used for faster movement, is characterized by the body being held clear of the ground with diagonal limbs moving in phase, closely resembling a trot gait. The low sprawl, used for slower movement or transition, keeps the body close to the ground with three legs in support at any time, akin to a walk gait.
We define the gait duty factor $\beta$ as the fraction of a cycle a leg spends in the stance phase. For a walk gait, $\beta = 0.75$, ensuring a stable tripod support polygon. For a trot gait, $\beta = 0.5$, where diagonal leg pairs (Left-Front & Right-Hind, LF-RH; Right-Front & Left-Hind, RF-LH) move in synchrony, with each pair alternating between swing and stance phases.
2.2. Hopf Oscillator-Based CPG Model
We utilize a Hopf nonlinear oscillator as the fundamental unit of our CPG network due to its stable limit cycle and the ease with which its frequency, amplitude, and waveform can be modulated. The dynamics of a single, uncoupled Hopf oscillator are given by:
$$ \begin{aligned}
\dot{u} &= \alpha (\mu – r^2) u – \omega v \\
\dot{v} &= \alpha (\mu – r^2) v + \omega u \\
r &= \sqrt{u^2 + v^2} \\
\omega &= \omega_{st} + \frac{\omega_{sw} – \omega_{st}}{1 + e^{-a(\beta – \phi)}}
\end{aligned} $$
Here, $u$ and $v$ are the state variables; $\alpha$ governs the convergence rate to the limit cycle; $\mu$ determines the amplitude of oscillation $A = \sqrt{\mu}$; $r$ is the instantaneous amplitude; $\omega$ is the modulated frequency, which smoothly transitions between the stance frequency $\omega_{st}$ and the swing frequency $\omega_{sw}$ based on the oscillator’s phase $\phi$ relative to the duty factor $\beta$; $a$ is a shaping constant.
To control the four-legged bionic robot, we construct a network of four such oscillators, one assigned to each leg (i = LF, RF, LH, RH). The oscillators are coupled to enforce the desired inter-limb phase relationships (e.g., 0.5$\pi$ phase difference for walk, 0$\pi$/$\pi$ for trot between diagonal pairs). The coupled network dynamics are:
$$ \begin{aligned}
\begin{bmatrix} \dot{u}_i \\ \dot{v}_i \end{bmatrix} &=
\begin{bmatrix} \alpha(\mu – r_i^2) & -\omega_i \\ \omega_i & \alpha(\mu – r_i^2) \end{bmatrix}
\begin{bmatrix} u_i \\ v_i \end{bmatrix} +
\sum_{j=1, j\neq i}^{4} \mathbf{R}(\theta_i^j)
\begin{bmatrix} u_j \\ v_j \end{bmatrix} \\
r_i &= \sqrt{u_i^2 + v_i^2} \\
\mathbf{R}(\theta_i^j) &=
\begin{bmatrix}
\cos \theta_i^j & -\sin \theta_i^j \\
\sin \theta_i^j & \cos \theta_i^j
\end{bmatrix}
\end{aligned} $$
where $\mathbf{R}(\theta_i^j)$ is the rotation matrix representing the desired phase offset $\theta_i^j$ of oscillator $j$ relative to oscillator $i$.
2.3. Signal Mapping to Joint Angles
The CPG output signals are mapped to the physical joint angles of the bionic robot. For each leg $i$, the hip pitch angle $\theta_{h,i}$ is directly mapped from the $u_i$ state variable, while the knee pitch angle $\theta_{k,i}$ is mapped from the $v_i$ variable, often with a piecewise function to ensure the knee only flexes during the swing phase:
$$ \theta_{h,i} = G_h \cdot u_i $$
$$ \theta_{k,i} = \begin{cases}
G_k \cdot ( -v_i ), & \text{if } v_i \leq 0 \text{ (swing phase)} \\
0, & \text{if } v_i > 0 \text{ (stance phase)}
\end{cases} $$
where $G_h$ and $G_k$ are gain factors translating the oscillator output to meaningful joint angles in radians. This mapping, driven by the CPG network, produces the rhythmic, coordinated joint trajectories necessary for walk and trot gaits in the crawling configuration of the bionic robot.
3. Stability Analysis and Gait Optimization via Active Tail
Dynamic stability is paramount for legged bionic robots. We employ the Zero Moment Point (ZMP) criterion as a primary stability measure. The ZMP is the point on the ground where the net moment of the inertial and gravitational forces has no horizontal component. For stable locomotion, the ZMP must remain within the support polygon formed by the contact points of the stance legs and the tail tip.
The coordinates of the ZMP projected on the ground plane can be approximated, neglecting angular dynamics, as:
$$ x_{zmp} \approx x_{com} – \frac{z_{com}}{g} \ddot{x}_{com}, \quad y_{zmp} \approx y_{com} – \frac{z_{com}}{g} \ddot{y}_{com} $$
where $(x_{com}, y_{com}, z_{com})$ is the robot’s center of mass (COM) position and $g$ is gravity. The stability margin $S$ is defined as the shortest distance from the ZMP to the boundary of the support polygon. For the trot gait (two diagonal legs in stance plus tail tip), the polygon is a triangle. For the walk gait (three legs in stance plus tail tip), it is a quadrilateral. Our simulations confirmed positive stability margins for both gaits, with walk naturally providing a larger margin than trot.
3.1. Tail Swing for Yaw Compensation in Trot Gait
Although stable, the trot gait of a sprawling bionic robot like ours can induce an undesired yaw rotation due to asymmetrical friction forces between the diagonal stance legs. During the first half of the trot cycle (LF and RH stance), the forward propulsion forces $F_{1y}$ and $F_{3y}$ at the foot-ground contact points create a net yaw moment $M_{B1}$ about the body’s vertical axis:
$$ M_{B1} = F_{1y} l_1 – F_{3y} l_3 $$
where $l_1$ and $l_3$ are the moment arms from the COM to the respective contact points. This moment, if unbalanced, causes the robot to yaw. To counteract this, we propose using the multi-segmented active tail. By swinging the tail in the horizontal plane with an appropriate phase, an inertial reaction torque is imparted on the main body.
Modeling the robot as a two-body system (Body B: torso and legs; Body A: tail), the inertial torque from the tail’s motion opposes the friction-induced yaw moment. The equation of motion for the body’s yaw during a stance phase becomes:
$$ I_B \dot{\omega}_B = (F_{iy} l_i – F_{jy} l_j) – \mathbf{r}_{A/B} \times \mathbf{F}_{A, inert} $$
where $I_B$ is the body’s yaw inertia, $\omega_B$ is the yaw rate, $\mathbf{r}_{A/B}$ is the vector from body B’s COM to the tail’s average swing axis, and $\mathbf{F}_{A, inert}$ is the inertial force due to the tail’s swing acceleration. By synchronizing the tail swing (with a specific phase lead/lag) to the leg movements, the net yaw moment over a full stride can be minimized. The reduction factor $\eta$ for the yaw rate achieved with active tail control compared to without it is:
$$ \eta = \frac{\omega_{B,\text{with tail}}}{\omega_{B,\text{no tail}}} < 1 $$
This active tail control strategy is a key optimization, enhancing the straight-line walking performance of the bionic robot during high-speed trotting.
3.2. Rolling Gait via Tail and Body Actuation
When the bionic robot morphs into Configuration 2 (rolling mode), the legs are locked. Locomotion is achieved by generating a traveling wave along the segmented body and tail, similar to the motion of a sidewinder or a crocodile performing a roll. The actuation focuses on the pitch joints of the torso segments and the yaw joints of the tail segments. A coordinated sinusoidal actuation profile creates the rolling motion:
$$ \phi_i(t) = A_i \sin(\omega_r t + \lambda_i + \delta \cdot i) $$
where $\phi_i$ is the angular displacement of the i-th actuated joint in the rolling chain, $A_i$ is the joint amplitude, $\omega_r$ is the rolling frequency, $\lambda_i$ is a joint-specific phase offset, and $\delta$ is the inter-joint phase difference that creates the traveling wave. This transforms the rigid, arched bionic robot into a actively driven, rolling wheel.
4. Simulation Results and Analysis
The proposed design and control schemes were validated through co-simulation using ADAMS for multi-body dynamics and MATLAB/Simulink for implementing the CPG controller. The bionic robot model included contact forces between the feet/tail and the ground.
4.1. Crawling Gaits: Walk and Trot
The CPG parameters were set to generate walk ($\beta=0.75$, $T=3.8\text{s}$) and trot ($\beta=0.5$, $T=2.9\text{s}$) gaits. Figure 11 (conceptual) shows the consistent periodic footfall pattern for the walk gait after an initial transient. The COM displacement plots (Figure 12) reveal stable forward progression for both gaits. The average forward speed for the walk gait was $0.115 \text{ m/s}$, and for the trot gait, it was $0.131 \text{ m/s}$. The lateral displacement (Y-axis) was minimal for walk ($<\pm5\text{mm}$) but more pronounced for trot ($<\pm16\text{mm}$), indicating the yaw disturbance. The vertical displacement (Z-axis) was very small, confirming stable contact without bouncing.
4.2. Rolling Gait Performance
After morphing to the rolling configuration, the body/tail traveling wave actuation was applied. The COM trajectory (Figure 13) shows linear displacement along the rolling direction (Y-axis). The achieved rolling speed was approximately $0.34 \text{ m/s}$, which is about 2.6 times faster than the fastest crawling gait, demonstrating the advantage of this mode for rapid traversal over suitable terrain.
4.3. Effectiveness of Active Tail Optimization
To quantify the optimization, the trot gait was simulated with and without the active tail swing. Without the tail, the robot exhibited a peak yaw angle deviation of approximately $9.2^\circ$ over several strides. With an optimally phased tail swing (amplitude ~$5.7^\circ$), the peak yaw deviation was reduced to about $4.8^\circ$ (Figure 14). Consequently, the lateral drift of the COM was significantly reduced from a maximum of $16.39\text{mm}$ to $4.2\text{mm}$ (Figure 15). This clearly validates the proposed tail swing strategy as an effective method for gait optimization in this bionic robot.
5. Conclusion
This work presented the comprehensive design and development of a novel crocodile-inspired bionic robot featuring metamorphic legs for multi-modal locomotion. The key contributions are: 1) A bio-inspired mechanical design incorporating a metamorphic leg mechanism that enables switching between a sprawling crawling configuration and a rigid rolling configuration, greatly enhancing the bionic robot’s environmental adaptability. 2) Implementation of a Hopf oscillator-based CPG network for generating stable and rhythmic walk and trot gaits, providing a robust foundation for locomotion control. 3) A detailed stability analysis and the proposal of an innovative active tail-swinging strategy to compensate for internal yaw disturbances during trotting, effectively optimizing the gait for straighter and more efficient travel. 4) Extensive dynamic simulations confirming the feasibility and performance of all designed locomotion modes.
The results demonstrate that the bionic robot successfully achieves stable walking, optimized trotting, and fast rolling, mimicking the versatile locomotion of its biological counterpart. Future work will focus on the physical implementation of the metamorphic mechanism, integrating sensor feedback for adaptive CPG control, and exploring more complex multi-modal gait transitions for navigating highly complex terrains. This research underscores the potential of combining bio-inspired design, metamorphic structures, and neuromechanical control principles for creating advanced, adaptable bionic robots.