Fig 1.
Interlimb phase relationships for locomotion speed.
A: Ipsilateral relative phases (fore leg–hind leg) for dogs and sheep versus Froude number (where the locomotion speed increases as the Froude number increases) [3]. Dogs change their phase relationship suddenly at a Froude number of approximately 0.5, while sheep change their phase relationship smoothly based on locomotion speed. B: Ipsilateral relative phases ((fore leg–hind leg)/2) of stick insects for gait cycle (where the locomotion speed decreases as the gait cycle increases) [4]. Data points and error bars show the average values and the errors of the mean values of the measured results, respectively. Stick insects change their phase relationships smoothly based on locomotion speed in a manner similar to sheep.
Fig 2.
A: Robot; B: Model.
Fig 3.
Locomotion control system using phase oscillators.
Each oscillator controls the movement of a single leg. Contralateral oscillators are set to have alternate phases. Each oscillator is affected by the touch sensor signal.
Fig 4.
Leg movement based on oscillator phase.
A: Oscillator phase. B: Desired leg movement. AEP and PEP represent the anterior extreme position and the posterior extreme position, respectively.
Fig 5.
Relative phases (ψ1, ψ2) of the robot simulation plotted at the foot contact of Leg 2 and the basins of attraction.
Relative phases are plotted for six initial conditions with (A) β = 0.5 and (B) β = 0.65. Six different markers represent the results for the six initial conditions. Irrespective of β, the robot established two different gaits (i.e., direct and retrograde wave gaits) that were dependent on the initial conditions. The basins of attraction for the two different gaits are plotted for (C) β = 0.5 and (D) β = 0.65. The red circles and green x points in (C) and (D) converge to the direct wave gaits and the retrograde wave gaits, respectively. The direct wave gaits have larger size of basins than the retrograde wave gait.
Fig 6.
Relative phases and maximum eigenvalue of gaits obtained for duty factor β in computer simulations.
A: Relative phase ψ1. B: Relative phase ψ2. C: Maximum eigenvalue. Two stable gaits were found for each duty factor (which were direct and retrograde wave gaits). The relative phases of each of the gaits changed smoothly with changing locomotion speed (duty factor β).
Fig 7.
Footprint diagrams of the gaits obtained at duty factors of β = 0.5 and 0.65 in computer simulations.
A: Direct wave gait. B: Retrograde wave gait.
Fig 8.
Relative phases (ψ1, ψ2) of the robot experiments plotted at foot contacts of Leg 2.
Relative phases are plotted for six initial conditions with (A) β = 0.5 and (B) β = 0.575. The six different markers represent the results for the six initial conditions. Irrespective of the value of β, the robot established two different gaits that were dependent on the initial conditions.
Fig 9.
Relative phases (A) ψ1 and (B) ψ2 of the gaits obtained for duty factor β in the robot experiments and the computer simulations.
Two stable gaits were obtained in the robot experiments: direct and retrograde wave gaits. The computer simulations used high and low feedback gains. When the feedback gain was reduced, the simulation results became much closer to the robot experimental results.
Fig 10.
Simple physical model with rigid body and six massless spring legs.
The body is represented by a flat plate here to show the geometric relationships between the model and the variables more clearly.
Fig 11.
Relative phases A (ψ1) and B (ψ2) of the direct and retrograde wave gaits from the simple model.
The relative phases are derived with both high stiffness (d* K* = 50)(solid line) and low stiffness (d* K* = 5)(dashed line) for s*/a* = 0.3. When the stiffness decreases, the relative phases ψ1 and ψ2 move away from 2βπ and 2(1 − β)π in a similar manner to the robot model in Fig 9. In addition, the relative phases of each gait change smoothly with changes in locomotion speed (duty factor β), as per the simulation.
Fig 12.
Maximum absolute eigenvalues of the direct and retrograde wave gaits of the simple model (Analysis) and the robot simulation (Simulation) for duty factor β.
A: s*/a* = 0, B: 0.15, and C: 0.3.
Table 1.
Physical parameters of the robot.
Fig 13.
Touchdown and liftoff events for the direct wave gait.
Black and grey legs represent the stance and swing legs, respectively. Events Ti and T(i + 3) (i = 1, 2, 3) have axial symmetry.
Fig 14.
Evolution of the oscillator phases as a result of sensory feedback at each event.
The sensory feedback provided at each event changes the relative phases.