Hysteresis in the gait transition of a quadruped investigated using simple body mechanical and oscillator network models

We investigated the dynamics of quadrupedal locomotion by constructing a simple quadruped model that consists of a body mechanical model and an oscillator network model. The quadruped model has front and rear bodies connected by a waist joint with a torsional spring and damper system and four limbs controlled by command signals from the oscillator network model. The simulation results reveal that the quadruped model produces various gait patterns through dynamic interactions among the body mechanical system, the oscillator network system, and the environment. They also show that it undergoes a gait transition induced by changes in the waist joint stiffness and the walking speed. In addition, the gait pattern transition exhibits a hysteresis similar to that observed in human and animal locomotion. We examined the hysteresis mechanism from a dynamic viewpoint.

Figure 1

Body mechanical model of quadruped model consisting of two bodies and four limbs.

Figure 2

Oscillator network model with four oscillators. Solid blue arrows indicate interactions among the oscillators based on the phase relationship ${\Delta }_{\mathit{ij}}$. The oscillator phases are modulated by tactile sensor information (dotted green arrows). The oscillator phases determine the limb joint kinematics (dashed black arrows).

Figure 3

Limb joint kinematics composed of swing and stance phases, which were determined from the length and orientation of the limb axis. The kinematics changes from the swing to stance phase when the limb tip contacts the ground. When the limb tip reaches the PEP, it moves into swing phase.

Figure 4

Footprint diagrams for walk, trot, and pace patterns in which the right and left limbs move out of phase [red: forelimbs (limbs 1 and 2); blue: hindlimbs (limbs 3 and 4)].

Figure 5

Gait transition induced by varying the waist joint stiffness using the parameter $f$. (a) The phase difference ${\Delta }_{13}$ plotted at foot contact of the right hindlimb. The quadruped model adopted the walk, trot, or pace pattern depending on $f$ and the initial state. A gait transition occurs between the walk and trot patterns at different joint stiffnesses depending on the direction of the stiffness change. The pace pattern did not change greatly with $f$. (b) The footprint diagram for the gait transition from the trot to walk pattern.

Figure 6

Gait transition induced by changing the walking speed using the duty factor $\beta$. This shows the phase relationship between the right forelimb and the right hindlimb ${\Delta }_{13}$, plotted at the foot contact of the right hindlimb, where the gait transition occurs between the walk and trot patterns at different walking speeds depending on the direction of the speed change.

Figure 7

Stability characteristics in the gait patterns. (a) Return maps of the phase difference ${\Delta }_{13}$ for various values of the parameter $f$ by plotting the relationship between phase difference ${\Delta }_{13n}$ for the $n$th step and phase difference ${\Delta }_{13n+1}$ for the next step during locomotion. Solid and open dots indicate stable and unstable gait patterns, respectively. (b) Stable and unstable gait patterns investigated by the return maps.

Figure 8

Effects of changing the physical parameters of the quadruped model on the hysteresis in the gait transition induced by changing the waist joint stiffness $f$. The mass (a), length (b), and width (c) of the front and rear bodies were varied.