Decentralized Control of Earthworm-like RobotBased on Tegotae Function

Takeshi Kano*, Akio Ishiguro*,***Research Institute of Electrical Communication, Tohoku University, Japan

{tkano,ishiguro}@riec.tohoku.ac.jp**Japan Science and Technology Agency, CREST, Japan

1 IntroductionEarthworms can move on unstructured terrain by propa-

gating bodily waves of contraction and expansion from headto tail with contracted parts anchored to the ground (Fig.1) [1]. This locomotion has attracted the attention of en-gineers because earthworm-like robots can be used in ar-eas where legged or wheeled robots are unable to functionproperly, such as disaster areas and lumen in human body.Accordingly, several extant studies focused on developingearthworm-like robots in recent years [2–5].

With respect to the above-described applications, it isdesirable that earthworm-like robots have adaptability andfault tolerance. A key to satisfy these requirements is relatedto an autonomous decentralized control in which a non-trivial macroscopic function emerges through the interactionof individual components. However, the above requirementswere not ensured sufficiently in the previous studies, and thisindicates that there is scope for further improvement in termsof control schemes.

In this study, a decentralized control scheme forearthworm-like robots was proposed by focusing on Tego-tae, which is a Japanese concept that describes the extentto which a perceived reaction matches an expectation [6–8].Specifically, a Tegotae function that quantifies Tegotae wasdefined, and a control scheme in which Tegotae is increasedat each body part was designed. The proposed controlscheme was validated via simulations.

2 ModelThe mass points are concatenated one-dimensionally

via parallel combinations of dampers and real-time tunablesprings (RTSs) (Fig. 2). An RTS is a spring in which theresting length can be changed arbitrarily [9]. The restingand real length of the RTS that connects the ith and i + 1thmass points are denoted by l̄i+ 1

2and li+ 1

2, respectively.

The friction between the mass points and the ground isdescribed as viscous friction, and the friction coefficient ofthe ith mass point, ηi, is given by ηi = b/l̃n

i , where l̃i =(li+ 1

2+ li− 1

2)/2, and b and n denote positive constants. Thus,

the friction coefficient increases as the RTS contracts, andthis mimics the property of a real earthworm (Fig. 1(a)).

An oscillator is implemented in each RTS, and the rest-ing length of the i + 1

2 th RTS, l̄i+ 12, is determined based on

the corresponding oscillator phase, θi+ 12, as follows:

li+ 12

= L+asinθi+ 12, (1)

Tim

e

Contraction

Expansion

Direction of

motion

(b)

Bristles

Longitudinal muscle

Circular muscle

(a)

Figure 1: A schematic of (a) body structure and (b) locomotion.

i i− 1i+ 1

Head

1

Tail

N

θi+

1

2

: RTS : Damper : Oscillator

Figure 2: A schematic of the proposed model.

where a and L denote positive constants.The time evolution of the oscillator phase is given by the

following expression:

θ̇i+ 12

= ω +σ∂Ti+ 1

2

∂θi+ 12

, (2)

where ω denotes the intrinsic angular frequency, and σ de-notes a positive constant specifying the feedback strength.The function Ti+1/2 denotes the Tegotae function, and it isdefined as follows:

Ti+ 12

= −sinθi+ 12·Si, (3)

where Si denotes the reaction force from the environmentalong the direction of motion. It corresponds to the shearstress acting on the body of a real earthworm (as shown inthe bottom part of Fig. 3).

The meaning of the Tegotae function Ti+1/2 can be ex-plained as follows. When the ith mass point anchors to theground and obtains a propulsive reaction force, i.e., Si is pos-itive, it is desirable for the RTS that connects the ith and(i + 1)th mass points to contract. Hence, l̄i+1/2 becomessmall or −sinθi+1/2 becomes large to pull the posterior partof the body forward. Therefore, the Tegotae function canbe defined as the product of (−sinθi+1/2) and Si. The sec-ond term on the right-hand side of Eq. (2) operates suchthat the oscillator phase is modified to increase the Tegotae.Specifically, the Tegotae-based sensory feedback converges

P32

The 8th International Symposium

on Adaptive Motion of Animals and Machines(AMAM2017)—92—

(a) (b)

Contraction

Expansion

0

π

2

3

2π

π

Direction of

motion

Side view Side view

0

π

2

3

2π

π

Si > 0 Si < 0

Direction of

motion

Figure 3: Effect of local sensory feedback: (a) Si > 0 and (b)Si < 0.

970000

940000

910000

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850000

820000

790000

760000

730000

Time step Direction of motion

(b)

1

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1.0

0.0

0 200000 400000 600000 800000 1000000 -1.0 30

Time step

Seg

men

t n

um

ber

(a)

sinφi+1/2

Figure 4: (a) Locomotion of the simulated earthworm. Con-tracted and expanded segments are shown in green andred, respectively. (b) The spatiotemporal plot of phase.The color indicates the value of sinθi+ 1

2.

the phase to 3π/2 and π/2 when Si is positive and negative,respectively (Fig. 3).

3 SimulationSimulations were conducted to validate the proposed

control scheme. The parameter values were determined bytrial-and-error. The initial value of θi+ 1

2(1 ≤ i ≤ N − 1) is

set as 0. Figure 4 shows snapshots and the spatiotemporalplot of the phase under a uniform environment. The bodyeffectively moved forward by propagating the wave of con-traction from the head to the tail as in the case of a real earth-worm immediately after the simulation commenced (Fig 4.(a)). The oscillator phase also propagated from the headto the tail (Fig. 4(b)). Figure 5 shows the result when asimulated earthworm traversed a terrain with nonuniformfriction. The simulated earthworm moved effectively byexploiting high friction areas (Fig. 5(a)). The oscillatorphases were modified to adapt to changes in the friction ofthe ground (Fig. 5(b)). Figure 6 shows the result when afew of the RTSs were replaced with passive springs to in-vestigate the fault tolerance of the proposed control scheme.A wave of contraction was generated from the head to thetail, and it passed through the area in which the RTSs werereplaced with passive springs. Thus, the adaptability andfault tolerance of the proposed scheme were confirmed.

AcknowledgmentsThis work was supported by Japan Science and Tech-

nology Agency, CREST. The authors thank Prof. YoshifumiSaijo and Hironori Chiba of the Graduate School of Biomed-ical Engineering, Tohoku University.

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(a) (b)

1

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1.0

0.0

0 200000 400000 600000 800000 1000000 -1.0 30

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Seg

men

t n

um

ber

sinφi+1/2

Figure 5: (a) Locomotion of the simulated earthworm on a terrainwith nonuniform friction. Dense colors indicate highfriction areas. (b) The spatiotemporal plot of phase.

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(b)

1

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1.0

0.0

sinφi+1/2

0 200000 400000 600000 800000 1000000 -1.0 30

Time step

Seg

men

t n

um

ber

(a)

Figure 6: (a) Locomotion of the simulated earthworm when sev-eral RTSs are replaced by passive springs. Segmentsreplaced by passive springs are shown in black. (b)The spatiotemporal plot of phase. The black regionsrepresent the segments replaced by passive springs.

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