Analysis of the gait generation principle by a simulated quadruped model with a CPG incorporating vestibular modulation

Abstract

This study aims to understand the principles of gait generation in a quadrupedal model. It is difficult to determine the essence of gait generation simply by observation of the movement of complicated animals composed of brains, nerves, muscles, etc. Therefore, we build a planar quadruped model with simplified nervous system and mechanisms, in order to observe its gaits under simulation. The model is equipped with a mathematical central pattern generator (CPG), consisting of four coupled neural oscillators, basically producing a trot pattern. The model also contains sensory feedback to the CPG, measuring the body tilt (vestibular modulation). This spontaneously gives rise to an unprogrammed lateral walk at low speeds, a transverse gallop while running, in addition to trotting at a medium speed. This is because the body oscillation exhibits a double peak per leg frequency at low speeds, no peak (little oscillation) at medium speeds, and a single peak while running. The body oscillation autonomously adjusts the phase differences between the neural oscillators via the feedback. We assume that the oscillations of the four legs produced by the CPG and the body oscillation varying according to the current speed are synchronized along with the varied phase differences to keep balance during locomotion through postural adaptation via the vestibular modulation, resulting in each gait. We succeeded in determining a single simple principle that accounts for gait transition from walking to trotting to galloping, even without brain control, complicated leg mechanisms, or a flexible trunk.

Appendix

The neural oscillator model used for each leg of our quadruped models was governed by the following set of differential equations. These equations are based on Matsuoka’s neural oscillator model Matsuoka (1985, 1987):

$$\begin{aligned} T_r \dot{u}_{\mathrm{e}i} + u_{\mathrm{e}i}&= - \sum _{j=1}^4 (p_{ij}[u_{\mathrm{e}j}]^{+} + q_{ij}[u_{\mathrm{f}j}]^+) + s \nonumber \\&- b v_{\mathrm{e}i} + \hbox {feed}1_{\mathrm{e}i} + \hbox {feed}2_{\mathrm{e}i} , \end{aligned}$$

(1)

$$\begin{aligned} T_a \dot{v}_{\mathrm{e}i} + v_{\mathrm{e}i}&= y_{\mathrm{e}i}, \end{aligned}$$

(2)

$$\begin{aligned} y_{\mathrm{e}i}&= [u_{\mathrm{e}i}]^+ = \max \,\,(u_{\mathrm{e}i}, 0), \end{aligned}$$

(3)

$$\begin{aligned} T_r \dot{u}_{\mathrm{f}i} + u_{\mathrm{f}i}&= - \sum _{j=1}^4 (p_{ij}[u_{\mathrm{f}j}]^{+} + q_{ij}[u_{\mathrm{e}j}]^+) + s \nonumber \\&- b v_{\mathrm{f}i} + \hbox {feed}1_{\mathrm{f}i} + \hbox {feed}2_{\mathrm{f}i}, \end{aligned}$$

(4)

$$\begin{aligned} T_a \dot{v}_{\mathrm{f}i} + v_{\mathrm{f}i}&= y_{\mathrm{f}i}, \end{aligned}$$

(5)

$$\begin{aligned} y_{\mathrm{f}i}&= [u_{\mathrm{f}i}]^+ = \max \,\,(u_{\mathrm{f}i}, 0), \end{aligned}$$

(6)

$$\begin{aligned} P_{i,j}&= \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0 &{} \beta &{} \alpha &{} 0 \\ \beta &{} 0 &{} 0 &{} \alpha \\ \alpha &{} 0 &{} 0 &{} \beta \\ 0 &{} \alpha &{} \beta &{} 0 \end{array} \right] , Q_{i,j} = \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} \gamma &{} 0 &{} 0 &{} 0 \\ 0 &{} \gamma &{} 0 &{} 0 \\ 0 &{} 0 &{} \gamma &{} 0\\ 0&{}0 &{}0 &{}\gamma \end{array} \right] \nonumber \\ \end{aligned}$$

(7)

and

$$\begin{aligned} y_i = y_{\mathrm{f}i} - y_{\mathrm{e}i}, \end{aligned}$$

(8)

where the suffix $i$ denotes the number of leg (i.e., 1: left fore; 2: left hind; 3: right fore; 4: right hind); the suffixes e and f mark the extensor and flexor, respectively. The terms $u_{\mathrm{e}i}$ and $u_{\mathrm{f}i}$ indicate an inner state modeled on a membrane potential of the extensor and flexor neurons of the $i$th leg, respectively, and $v_{\mathrm{e}i}$ and $v_{\mathrm{f}i}$ indicate the variable representing the degree of fatigue or adaptation in the extensor and flexor neurons of the $i$th leg, respectively. Meanwhile, $s$ is a tonic descending signal strength; $\hbox {feed}1_{\{\mathrm{e,f}\}i}$ and $\hbox {feed}2_{\{\mathrm{e,f}\}i}$ represent sensory feedback inputs (e.g., $\hbox {feed}1_{\mathrm{e}i}$ is a feedback to the extensor neuron of the $i$th leg); $y_{\mathrm{e}i}$ and $y_{\mathrm{f}i}$ are the outputs of the extensor and flexor neurons of the $i$th leg, which are discontinuous terms via Eqs. (3) and (6); and $y_i$ is the output of the oscillator of the $i$th leg, as shown by the patterns in Fig. 3 in the main text. The constants $T_r$ and $T_a$ are time constants of the inner state and the adaptation variable to determine the intrinsic frequency of the oscillator, and $b$ is a constant related to the recurrent inhibition of the inner state. Finally, $\sum _{j=1}^4 (p_{ij}[u_{\{\mathrm{e,f}\}j}]^+ + q_{ij}[u_{\{\mathrm{f,e}\}j}]^+)$ represents the inhibitory synaptic inputs from the neurons of the legs, such as in Fig. 5 in the main text, where their strengths $p_{ij}$ and $q_{ij}$ are shown in Eq. (7). The essential gait pattern is produced by this term.

The sensory feedback inputs in Eqs. (1) and (4) are defined as

$$\begin{aligned} \begin{aligned} \hbox {feed}1_{\mathrm{e}i}&= k_1(\theta _i - \theta _0),\quad \hbox {and}\\ \hbox {feed}1_{\mathrm{f}i}&= -k_1(\theta _i - \theta _0) \end{aligned} \end{aligned}$$

(9)

where $k_1 C \theta _i$ and $\theta _0$ are the weight, the present hip joint angle of the $i$th leg, and the origin of each hip joint angle, respectively, as shown in Fig. 16. Next:

$$\begin{aligned} \begin{aligned} \hbox {feed}2_{\mathrm{e}i}&= \sigma (leg)\ k_2 \phi ,\quad \hbox {and}\\ \hbox {feed}2_{\mathrm{f}i}&= -\sigma (leg)\ k_2 \phi \end{aligned} \end{aligned}$$

(10)

and:

$$\begin{aligned} \sigma (leg) = \left\{ \begin{array}{l@{\quad }l} 1, &{} \text{ if }\quad \hbox {leg} \,\hbox {is a fore leg} (i=1\,or\,3); \\ -1, &{} \text{ if }\quad \hbox {leg} \,\hbox {is a hind leg} (i=2\,or\,4) \end{array} \right. \end{aligned}$$

where $\phi$ denotes a body inclination angle around the pitch axis as shown in Fig. 16, and $k_2$ is the strength of vestibular modulation.

Fig. 16

A diagram of the quadruped model

The equations of the proportional-derivative (PD) controllers used in the leg controller section in Fig. 4 in the main text will now be introduced. The PD control of each hip joint is shown as:

$$\begin{aligned} \tau _{i} = K^\tau _{pi\cdot \{\mathrm{sw}, \mathrm{st}\}} (\theta _{di\cdot \{\mathrm{sw}, \mathrm{st}\}} - \theta _i) - K^\tau _{vi\cdot \{\mathrm{sw}, \mathrm{st}\}} \dot{\theta }_i, \end{aligned}$$

(11)

where $\tau _{i}, \,\theta _i$ and $\dot{\theta }_i$ are the output torque, the present angle, and the present angular velocity of the hip joint of the $i$th leg in Fig. 16, respectively. The terms $\theta _{di\cdot \{\mathrm{sw}, \mathrm{st}\}}$ are the target hip joint angles in the swing/stance phase (Fig. 4 in the main text), and $K^\tau _{pi\cdot \{\mathrm{sw}, \mathrm{st}\}}$ and $K^\tau _{vi\cdot \{\mathrm{sw}, \mathrm{st}\}}$ represent proportional and derivative gains (P and D gains) in the swing/stance phase. The PD control of each linear knee joint is shown as:

$$\begin{aligned} F_{i} = K^F_{pi\cdot \{\mathrm{sw}, \mathrm{st}\}} (l_{di\cdot \{\mathrm{sw}, \mathrm{st}\}} - l_i) - K^F_{vi\cdot \{\mathrm{sw}, \mathrm{st}\}} \dot{l}_i, \end{aligned}$$

(12)

where $F_{i}, \,l_i$ and $\dot{l}_i$ are the output force, the present length, and the present velocity of the linear knee joint of the $i$th leg in Fig. 16, respectively; $l_{di\cdot \{\mathrm{sw}, \mathrm{st}\}}$ are the target lengths of the linear knee joints in the swing/stance phase (Fig. 4 in the main text), and $K^F_{pi\cdot \{\mathrm{sw}, \mathrm{st}\}}$ and $K^F_{vi\cdot \{\mathrm{sw}, \mathrm{st}\}}$ represent P and D gains in the swing/stance phase.

The values of the parameters in the equations were empirically determined such that the quadruped model could walk and run safely at each chosen speed. For a selected speed, all the values are constant. However, the values of $s, T_r, \theta _{d \cdot \mathrm{st}}, K^\tau _{pi\cdot \mathrm{st}}, K^F_{pi\cdot \mathrm{st}}$ and $K^F_{pi\cdot \mathrm{sw}}$ are set according to the selected speed. All other values are constant irrespective of changes in the speed. For example, we show the values of the parameters in Table 1 when the quadruped model is walking with a walk gait at 0.3 m/s.

Table 1 Values of the parameters used in a simulation where the quadruped model is walking with a walk gait at 0.3 m/s