With the rapid advancement of robotics technology, multi-legged bionic robots have demonstrated superior performance in applications such as disaster relief, reconnaissance, and logistics. The field of modern bionics has extended into robotics, where researchers replicate or reconstruct certain characteristics and functions of organisms in nature by studying animal structures, materials, functions, control strategies, and swarm behaviors. Among these, hexapod bionic robots stand out due to their strong terrain adaptability, high degrees of freedom, and flexibility, attracting widespread attention from research institutions worldwide. Gait control, as a key technology in the motion control of hexapod bionic robots, remains a challenging problem in the current field of robot locomotion. Existing gait control methods for hexapod bionic robots primarily include model-based approaches and control methods based on biological Central Pattern Generators (CPG). Model-based methods require precise modeling of robot parameters and the environment, leading to complex computations, whereas CPG-based methods mimic the rhythmic movements of higher animals to control limb motion. Physiological studies show that CPGs exist in the spinal cords of vertebrates or the thoracic-abdominal ganglia of invertebrates, and rhythmic movements such as chewing and running in various animals are achieved through CPGs. These movements are self-excited behaviors of lower neural centers, capable of autonomous oscillation without control from higher centers like the brain or sensory feedback, making CPG-based methods highly practical and valuable for research.
In multi-legged animals, each single-leg joint is controlled by an individual CPG neuron oscillator, or a single CPG oscillator serves as an oscillation center to control multiple joints of a limb. Artificial bionic CPGs combine multiple CPG oscillators into a network and adjust the coupling relationships between oscillators to achieve gait control for robots. Among oscillator-based CPGs, neuron oscillators are notable for their numerous parameters and clear biological significance. In this study, we propose a CPG gait control strategy for a hexapod bionic robot based on the Kimura neural oscillator. We first design the mechanical structure of the hexapod bionic robot inspired by a spider and perform kinematic analysis. Then, we establish an oscillator model based on the Kimura neural oscillator and tune its parameters. Next, we design a CPG network model according to the phase relationships of the six legs of the bionic robot. Finally, we conduct joint experiments using computer simulation tools and a prototype. The results show that the output signal amplitude and phase difference of the CPG network model generated from the Kimura neural oscillator are stable, meeting the gait control requirements of the hexapod bionic robot, thus providing a feasible solution for gait control in such systems.
The mechanical structure of a hexapod bionic robot plays a crucial role in providing protection and support, directly influencing gait motion control. We selected the spider, an arthropod, as the bionic object and designed the mechanical structure of the hexapod bionic robot by imitating the spider’s body structure and the proportions of single-leg joints. This approach led to a simple configuration capable of realizing various gaits for the bionic robot. The robot’s torso is octagonal, with six legs evenly distributed on both sides of the body. Each leg consists of three joints: the coxa, femur, and tibia, offering three degrees of freedom. The body length and width are 0.89 m and 0.89 m, respectively, while the lengths of the coxa, femur, and tibia are 0.02 m, 0.22 m, and 0.20 m, with a foot length of 0.015 m. The rotation angle ranges for the coxa, femur, and tibia are -90° to 90°, -60° to 60°, and 45° to 105°, respectively. The single-leg structure and 3D model of the bionic robot are illustrated in the figure above.
To analyze the motion of the bionic robot, we perform kinematic analysis using the Denavit-Hartenberg (D-H) coordinate system. For a single leg, let $\alpha$, $\beta$, and $\gamma$ represent the rotation angles of the hip, knee, and ankle joints, respectively. The coordinate system defines the Y-axis as the robot’s direction of travel, the X-axis perpendicular to Y, and the Z-axis perpendicular to the XOY plane. The segments $J_1$, $J_2$, and $J_3$ correspond to the coxa, femur, and tibia. Through D-H analysis, we derive the forward kinematics for a single leg of the bionic robot:
$$
\begin{bmatrix} p_{ix} \\ p_{iy} \\ p_{iz} \end{bmatrix} = \begin{bmatrix} \cos\alpha (J_1 + J_3\cos(\beta + \gamma) + J_2\cos\beta) \\ \sin\alpha (J_1 + J_3\cos(\beta + \gamma) + J_2\cos\beta) \\ J_3\sin(\beta + \gamma) + J_2\sin\beta \end{bmatrix}
$$
where $p_{ix}$, $p_{iy}$, and $p_{iz}$ are the coordinates of the foot tip relative to the body. The inverse kinematics, which compute the joint angles from the foot tip coordinates, are given by:
$$
\alpha = \arctan\left(\frac{p_{iy}}{p_{ix}}\right)
$$
$$
\beta = -\arctan\left(\frac{J_3\sin\gamma}{J_2 + J_3\cos\gamma}\right) + \arctan\left(\frac{z}{\sqrt{x^2 + y^2}}\right)
$$
$$
\gamma = \pm \arccos\left(\frac{x^2 + y^2 + z^2 – J_2^2 – J_3^2}{2J_2 J_3}\right)
$$
These equations enable precise control of the bionic robot’s leg movements for various gaits.
Modeling biological neuron oscillators is essential for simulating the physiological process where neurons react to stimuli. The biological CPG motion control system consists of a high-level control signal source, CPG, and effectors. The high-level source, such as the cerebral cortex or basal ganglia, handles motion planning, while the CPG, as a lower neural center, generates periodic rhythmic signals through self-excitation. Biological neuron cells, the basic units in motion control, are electrically excitable cells that typically exhibit excitatory and inhibitory states, communicating through mutual excitation and inhibition. A neuron cell comprises a cell body, dendrites for receiving signals, and an axon for transmitting signals. Due to the plasticity of synaptic connections between neuron cells, altering these connections allows control of different rhythmic movements in animals.
For the hexapod bionic robot, we model the CPG network based on CPG principles, studying the coupling relationships between oscillators to achieve gait control. We select the Kimura neural oscillator, an improvement over the Matsuoka neuron oscillator. The Matsuoka oscillator models neural cell activity to study muscle control laws, incorporating an adaptation term to simulate fatigue characteristics in leaky integrator differential equations. Its mathematical model is:
$$
T_r \frac{du}{dt} + u + bv = s
$$
$$
T_a \frac{dv}{dt} + v = y
$$
$$
y = g(u – \theta)
$$
$$
g(x) = \max(0, x)
$$
where $u$ is the neuron membrane potential (active term), $v$ is the fatigue term, $s$ is external input, $T_r$ is rise time, $T_a$ is delay time, $b$ is self-inhibition coefficient, $\theta$ is output threshold, and $y$ is output signal. Simulation shows that the Matsuoka oscillator outputs only positive values and non-periodic signals, making it unsuitable for directly controlling joint motors in a bionic robot.
The Kimura oscillator addresses this limitation by designing a pair of mutually inhibiting extensor and flexor neuron cells. It adds sensory feedback and linearly combines outputs from two Matsuoka oscillators, better simulating biological properties. The Kimura oscillator model is:
$$
T_a \frac{dv^e_i}{dt} = -v^e_i + y^e_i
$$
$$
T_r \frac{du^e_i}{dt} = -u^e_i – a_1 y^e_i – a_2 v^e_i – \sum_{j=1}^n \omega_{ij} y^e_j – \text{Feed}^e_i + s^e
$$
$$
T_a \frac{dv^f_i}{dt} = -v^f_i + y^f_i
$$
$$
T_r \frac{du^f_i}{dt} = -u^f_i – a_1 y^f_i – a_2 v^f_i – \sum_{j=1}^n \omega_{ij} y^f_j – \text{Feed}^f_i + s^f
$$
$$
y^e_i = \max(u^e_i, 0)
$$
$$
y^f_i = \max(u^f_i, 0)
$$
$$
y_i = y^f_i – y^e_i
$$
Here, $e$ and $f$ denote extensor and flexor neuron cells, $u^e_i$, $u^f_i$ are active terms, $v^e_i$, $v^f_i$ are fatigue terms, $s^e$ and $s^f$ are external inputs determining amplitude, $T_r$ and $T_a$ are time constants, $a_1$ is mutual inhibition coefficient, $a_2$ is self-inhibition coefficient, $\omega_{ij}$ is connection weight from oscillator $j$ to $i$, $\text{Feed}^e_i$ and $\text{Feed}^f_i$ are feedback terms, and $y_i$ is the output of oscillator $i$. Simulation confirms that the Kimura oscillator produces periodic signals suitable for joint control in bionic robots.
We perform parameter tuning for the Kimura oscillator using a single-parameter method to determine values applicable to gait control for the hexapod bionic robot. We analyze the effects of external stimulus $s$, delay time $T_a$, rise time $T_r$, self-inhibition coefficient $a_2$, and mutual inhibition coefficient $a_1$ on oscillator output. The results are summarized in the table below, which shows how each parameter influences the amplitude, frequency, and stability of the output signals, crucial for adapting the bionic robot to different terrains.
| Parameter | Effect on Output | Typical Range for Bionic Robot |
|---|---|---|
| External stimulus $s$ | Determines output amplitude; higher $s$ increases amplitude. | 5 to 50 for stable oscillation |
| Delay time $T_a$ | Affects signal decay; larger $T_a$ slows response. | 0.5 to 2.0 seconds |
| Rise time $T_r$ | Influences signal rise; smaller $T_r$ speeds up response. | 0.5 to 1.5 seconds |
| Self-inhibition $a_2$ | Controls fatigue; higher $a_2$ reduces oscillation damping. | 10 to 100 |
| Mutual inhibition $a_1$ | Governs interaction between neurons; critical for rhythm generation. | 0 to 2.5 |
Based on this analysis, we finalize the Kimura oscillator parameters as: $s=5$, $T_r=1$, $T_a=1$, $a_2=50$, $a_1=2$. These values ensure stable periodic outputs for controlling the bionic robot’s joints.
Gait control for the hexapod bionic robot involves coordination among six legs, each with three joints. To simplify CPG network complexity, we establish relationships between joints rather than using one oscillator per joint. Based on the Kimura oscillator, the gait of the bionic robot is determined by the connection weight matrix $\omega_{ij}$. We design a fully connected network topology for tripod gait, where oscillators CPG1, CPG3, and CPG5 (corresponding to legs 1, 3, 5) mutually excite, and CPG2, CPG4, and CPG6 (legs 2, 4, 6) mutually excite, while oscillators of adjacent legs from different groups mutually inhibit. The connection weight matrix for tripod gait in the bionic robot is:
$$
\omega_{ij} = \begin{bmatrix}
0 & -1 & 1 & -1 & 1 & -1 \\
-1 & 0 & -1 & 1 & -1 & 1 \\
1 & -1 & 0 & -1 & 1 & -1 \\
-1 & 1 & -1 & 0 & -1 & 1 \\
1 & -1 & 1 & -1 & 0 & -1 \\
-1 & 1 & -1 & 1 & -1 & 0
\end{bmatrix}
$$
This symmetric matrix uses $1$ for excitation, $-1$ for inhibition, and $0$ for self-connections. In tripod gait, the six legs are divided into two groups: group X (legs 1, 3, 5) and group Y (legs 2, 4, 6), which alternate between swing and support phases, controlled by two sets of signals.
We simulate the phase relationships for tripod gait using computer tools. The simulation outputs show that group X legs have identical waveforms, group Y legs have identical waveforms, and the phase difference between groups is half a cycle, consistent with tripod gait requirements for the bionic robot. The stability of these signals is verified through amplitude and frequency analysis, ensuring reliable performance in real-world applications for the hexapod bionic robot.
To validate the feasibility of the CPG network model based on Kimura oscillators, we map the gait control signals to joint angles and test them on an experimental platform for the hexapod bionic robot. The platform, version 2.0, is inspired by spiders and built to verify the proposed CPG model. We input tripod gait parameters into the CPG network and send control commands via an upper computer. The gait experiment involves initial standing, followed by tripod gait progression, and termination upon command. Observations confirm that the bionic robot successfully executes tripod gait with X and Y leg groups alternating swings, demonstrating stable motion without falls or irregularities.
The results indicate that the CPG network model generates consistent and stable output signals, enabling smooth locomotion for the bionic robot. This validates the effectiveness of the Kimura oscillator-based approach for gait control in hexapod bionic robots, highlighting its potential for adaptive and robust performance in diverse environments.
In conclusion, we have developed a CPG network model based on the Kimura biological neuron oscillator for gait control in a hexapod bionic robot. This model mimics the excitation and inhibition interactions between neuron cells, applying them to robot locomotion. Experimental results show that the model effectively controls tripod gait with stable output signals, proving its feasibility and robustness for bionic robot applications. Future work will focus on extending this model to more complex gaits, such as wave and ripple gaits, and incorporating sensory feedback for adaptive terrain negotiation in bionic robots. Additionally, we plan to explore real-time parameter adjustment for dynamic environments, further enhancing the versatility of hexapod bionic robots in practical scenarios.
The integration of biological principles into robotics through CPG models opens new avenues for developing agile and resilient bionic robots. By continuing to refine oscillator parameters and network topologies, we aim to achieve higher levels of autonomy and efficiency, making bionic robots invaluable tools in fields ranging from exploration to emergency response. The success of this study underscores the importance of interdisciplinary approaches, combining neuroscience, engineering, and computer science to advance the capabilities of bionic robots.