Modeling and analysis of a new locomotion control neural networks

Abstract

Experimental data have shown that inherent bursting of the neuron plays an important role in the generation of rhythmic movements in spinal networks. Based on the mechanism that the spinal neurons of a lamprey generate this inherent bursting, this paper builds a simplified inherent bursting neuron model. A new locomotion control neural network is built that takes advantage of this neuron model and its performance is analyzed mathematically and by numerical simulation. From these analyses, it is found that the new control networks have no restriction on their topological structure for generating the oscillatory outputs. If a network is used to control the motion of bionic robots or build the model of the vertebrate spinal circuitry, its topological structure can be constructed using the unit burst generator model proposed by Grillner. The networks can also be easily switched between oscillatory and non-oscillatory output. Additionally, inactivity and saturation properties of the new networks can also be developed, which will be helpful to increase the motor flexibility and environmental adaptability of bionic robots.

Appendix A

For the mechanical model of the lamprey, the parameters are shown in Table 3.

Table 3 The parameters of the mechanical model of the lamprey

The muscles are simulated using a combination of a spring and a damper (Ekeberg 1993). Its model can be written as follows:

$$\begin{aligned} T=\alpha (M_f -M_e )+\beta (M_f +M_e +\xi )\Delta \phi +\delta \Delta \dot{\phi } \end{aligned}$$

(1)

where $M_f$ are the motoneuron activities of the flexor muscles; $M_e$ are the motoneuron activities of the extensor muscles; $\Delta \phi$ is the difference between the actual angle of the joint and its resting angle; and $\alpha$, $\beta$, $\xi$, and $\delta$ indicate the gain, the stiffness gain, the tonic stiffness, and the damping coefficient of the muscles, respectively. The parameters of all of the muscles are the same here: $\alpha =1$ N m; $\beta =0.3$ N m; $\xi =10$, and $\delta =5$ N m s.

The fluid forces $F_i ,\,(i=1,\ldots ,41)$ of the links of the body can be calculated as

$$\begin{aligned} F_i =\frac{1}{2}\rho l_i h_i C_i v_i ^{2},(i=1,\ldots , 41) \end{aligned}$$

(2)

where $\rho$ is the density of the water; $l_i$ is the length of the link i of the body; $h_i$ is the height of the link i of the body; $C_i$ is the drag coefficient of the link i of the body (here $C_i =1$ for all links of the body); $v_i$ is the normal velocity of the midpoint of the link i.