Paper Information

Simulation of autonomous rhythm and gait generation in quadrupedal locomotion with hindlegs

Abstract

Intending to reduce the complexity in quadrupedal locomotion planning and control through emergence even more, we propose a legged locomotion controller inspired by the sensorimotor functions observed in the cat spinal cord. In simulations using a hind-legged biped robot, we demonstrate the emergence of a rhythm and gait closely resembling the belt-driven locomotion observed in spinal cats. We also show the self-propulsive locomotion of this hind-legged biped, based on the hypothesis that the brainstem output directs the intensity of muscle contraction (locomotion power) to the spinal cord. Furthermore, we meticulously examine the dynamic interaction with the environment, considering the fixture's presence, and document the entire process in detail. We show that the mechanisms underlying the emergence of legged locomotion are the interactions between leg phase switching, self-excited oscillations of the body or trunk, and leg load spatiotemporal pattern. Its temporal pattern determines the rhythm, and its spatial pattern determines the gait.

Locomotion of a spinal cat with hindlimbs on a treadmill

Spinal Cat Movie (.mp4 with narration) from the video "The basal ganglia and brainstem locomotor control, at 1min. 31sec., E. Garcia-Rill eds., 1989"

(at 6sec.) a spinal cat supported manually at the chest and the tail
(6sec.-- ) walking in the gait of out-of-phase coordination at slow belt speed
(15sec.-- ) onset of slow motion
(at 22sec.) slight pitching motion of the trunk induced by the increase of belt speed
(24-27sec.) running of out-of-phase (the narration used 'walk to trot' at 26sec.)
(28sec.-- ) running of in-phase (slight phase difference) with pitching motion of the trunk and kicking motion of legs at medium belt speed
(46sec.-- ) running of in-phase (exactly) with large pitching motion of the trunk at high belt speed

Reference

H. Forssberg, S. Grillner and J. Halbertsma: ``The locomotion of the low spinal cat. I. Coordination within a hindlimb,'' Acta Physiol. Scand., vol.108, pp.269--281, 1980. [DOI:10.1111/j.1748-1716.1980.tb06533.x]
H. Forssberg, S. Grillner, J. Halbertsma and S. Rossignol: ``The locomotion of the low spinal cat. II. Interlimb coordination,'' Acta Physiol. Scand., vol.108, pp.283--295, 1980. [doi:10.1111/j.1748-1716.1980.tb06534.x]

Locomotion of midbrain (and thalamic) cats on a treadmill

Midbrain Cats Movie (.mp4 with narration). The reproduction (the trot gait only) of the mid-brain cat experiments (Shik et al. 1966) by Jordan and Steeves (1976) including the spontaneous hindlimbs stepping of a thalamic cat, from the video "The basal ganglia and brainstem locomotor control, E. Garcia-Rill eds., 1989"

Simulation of a hind-legged biped in belt-driven locomotion on a treadmill without belt speed adaptation

(Figure 12) Increasing belt speed from 0.14(m/s) to 0.3(m/s): from walking to running in out-of-phase real time (.mp4)

Oscillation in pitching of the trunk was induced after the increase of belt speed (check in slow motion).
The frequency of pitching was double of the frequency of the step cycle. This is because pitching was entrained with the stepping of each leg and both legs were in out-of-phase coordination.

(Figure 14) Increasing belt speed from 0.14(m/s) to 0.42(m/s): from out-of-phase walking to in-phase running real time (.mp4)

The source of the rhythm generation was autonomously switched from rolling to pitching of the trunk.
The frequency of pitching was equal to the one of the step cycle. This is because pitching was entrained with the stepping of each leg and both legs were in in-phase coordination.
The TD timing of both legs became in-phase on the 2nd step cycle after the change of belt speed (check in slow motion).

(Figure 17) The reversibility of gait transitions in the step-by-step increase of belt speed from 0.14(m/s) via 0.3(m/s) to 0.42(m/s) and in the decrease of the opposite order: slow motion (.mp4)

Simulation of a hind-legged biped in belt-driven locomotion on a treadmill with belt speed adaptation

(Figure 18) Increasing belt speed from 0.2(m/s) to 0.5(m/s): from out-of-phase walking to in-phase running slow motion (.mp4), real time

(Figure B1) Increasing belt speed from 0.2(m/s) to 0.3(m/s): from walking to running in out-of-phase real time (.mp4), slow motion

Simulation of a hind-legged biped in slef-propulsive locomotion on the floor

(Figure 19) Increasing locomotion power (upper center command) from 1.4 to 2.8: from out-of-phase walking to in-phase running slow motion (.mp4), real time

(Figure B2) Increasing locomotion power from 1.4 to 2.5: from walking to running in out-of-phase real time (.mp4), slow motion

Related Materials

Section 2.1

conventional studies

(Kimura et al. 1990) Gait transition from the trot to pace of 'Collie2' (mp4 with narration, PDF)
(Kimura et al. 1999) Gait transition from the pronk to bound of 'Patrush' using the neural oscillators (mp4)
(Fukuoka et al. 2003) Gait transition from the walk to trot of 'Tekken1' (mp4)

emergence of rhythm and gait based on the self-excited body oscillation

(Ono et al. 2001) Self-excited walking of a biped in the sagittal plane on a flat floor (mp4)
(Maufroy et al. 2010) Simulation of self-excitedly generated walking (mp4) and adaptation against the lateral perturbation (mp4)
(Maufroy et al. 2012) Experiment of self-excitedly generated walking (mp4) and adaptation against the lateral perturbation (mp4)

(Miura and Shimoyama 1984) Dynamic walking of a stilt-typed biped robot (mp4 with narration)

Section 2.2

The hindlimbs stepping of the preparation with the brainstem and spinal cord in vitro by the chemical MLR stimulation and without afferent input (mp4 with narration)
- from the video "The basal ganglia and brainstem locomotor control, at 11min. 53sec., E. Garcia-Rill eds., 1989"
- Despite of this result, it is not clear how the sensorimotor functions modulates the spontaneous rhythm without the higher central signal in a spinal cat.

Section 2.4

(Ono et al. 2001) Self-excited walking of a biped in the sagittal plane on a flat floor (mp4)
(Maufroy et al. 2010) Simulation of self-excitedly generated walking (mp4) and adaptation against the lateral perturbation (mp4)
(Maufroy et al. 2012) Experiment of self-excitedly generated walking (mp4) and adaptation against the lateral perturbation (mp4)

Section 9.3

"Our objective in the subsequent step is to elucidate the autonomous generation of the rhythm and gait of a quadruped ..."

Autonomous rhythm and gait transition in self-propulsive locomotion of a quadruped on the floor. The torsion joint of the trunk is fixed.
The transitions were elicited when the locomotion power increased. In the generation and transition of the rhythm and gait, the flexor half-center activity (\alpha_F^self) and the phase variable of the rhythm generator (\phi^self) of the leg itself, and also \alpha_F^* (* : contralateral, ipsilateral, or diagonal legs) of other legs were merely used for LOT (lift-off timing) determination conditions. The upper center command (Cup) meaning the locomotion power was used for the feedforward functions in the pattern formation. The application of the proposed method to a quadruped has just been started. Even though locomotion speed in the trot gait is not enough, the validity of the framework in this study could be shown.

Generation of the walk gait (slow motion .mp4, Cup=1, 0.19 m/s). The SLFLO (\alpha_F^self > 0) for forelegs and the SLFLO (\alpha_F^self > p) with the additional condition (\phi_st^self > q) for hindlegs was the only one LOT determination condition employed, and no explicit interleg coordination was incorporated.
From the walk via trot in walking to trot in running (slow motion .mp4). The first trigger (explicit interleg coordination) for the walk-to-trot transition arose when the Cup was increased to 1.4. The second trigger for the running transition arose when the Cup was increased to 1.6. Afterward, the Cup was gradually increased to 2.2 (0.30 m/s). The diagonal interleg coordination was incorporated for the trot gait and those triggers. Moreover, the ipsilateral one was incorporated for the trigger of the walk-to-trot transition.

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