## 6 Conclusions  
In this paper, the biological neuron axon could continuously elongate and turn, and the growth of the back part did not change the shape of the previously used grown part. Combined with the law of gravitation, random motion, the Rall cable model and the Matsuoka oscillator model, a local CPG self-growing network with multiple physical characteristics was established. The fusion of a self-growing network and CPG has been realized, the control ability of self-growing network was presented, and the motion control of a quadruped robot was completed. Our conclusions are as follows. (1) An axon self-growing algorithm based on universal gravitation and random motion was established, the Rall Cable Model for studying the potential distribution of an axon membrane was analyzed and simplified, and multiple physical properties of the synapse (including resistance, capacitance, axon length and diameter, etc.) were combined with the Rall Cable Model to establish a simplified model of single neuron signal transmission, as shown in Equation (19). The multi-physical properties obtained by the neuron self-growth model (such as the synapse length calculated by Equation (9) and the resistance R1 calculated by Equation (22)) were added to Equation (17) to calculate parameters \(\tau\) and b of Equation (19). On this basis, the entire synapse system of the neuron's self-growth is taken as a whole, as well as the key parameters in the control model: the delay time constant Tr and the synapse, as shown in Equations (25) and (26). The connection weight corresponds to the axon  
Figure 18: Trot gait diagram of experimental prototype. length and axon diameter in the growth model, which allows the combination of the growth and development process of the neuron and the control function. (2) By analyzing the Matsuoka oscillator model (shown in Equation (27)) and comparing the difference between the mathematical model of its neuron monomer and the model built in this article, a local CPG network topology is proposed as shown in Figure 10. This topology is regarded as the goal of network self-growth. A neural network growth model based on gravitational field control is established, meaning that the network can grow the target topology from any initial position of the neuron, as shown in Figure 11, to obtain the rhythm output signal corresponding to the angular displacement of the hip joint of the quadruped robot, as shown in Figure 12.  
(3) The quadruped robot motion control simulation prototype and experimental prototype are used as shown in Figure 13: the output signal of the local CPG network (as shown in Figure 16) is used as the hip joint control signal of the simulation prototype and the experimental prototype to control the quadruped robot simulation. The prototype and the experimental prototype realized walking with a trot gait, as shown in Figures 17 and 18, which verified the feasibility of the local CPG network self-growth model with multi-physical characteristics proposed in this paper for the gait control of a quadruped robot. Since the gait control of the quadruped robot by the CPG network in this article is still an open-loop control system, it was not possible to change the gait of the robot through environmental feedback information or to adapt to the complex road conditions and change the rhythm and mode of the network output accordingly. Therefore, in order to give the self-growth-based network model higher practical value when applied to the motion control of a footed robot, the research work that needs to be done in the next stage is as follows:  
(1) Continue to improve the self-growth model of neurons, considering more physical characteristics that can be linked to the control model parameters, so that the built model can be constantly close to the complexity of the biological neural network in terms of its microstructure and control function. For example, in this paper, the diameter of the neuron axon is constant.