The generation of walking patterns is central to bio-inspired robotics and has been attained using methods encompassing diverse numerical as well as analog implementations. Here, we demonstrate the possibility of synthesizing viable gaits using a paradigmatic low-dimensional non-linear entity, namely, the Rössler system, as a dynamical unit. Through a minimalistic network wherein each instance is univocally associated with one leg, it is possible to readily reproduce the canonical gaits as well as generate new ones via changing the coupling scheme and the associated delays. Varying levels of irregularity can be introduced by rendering individual systems or the entire network chaotic. Moreover, through tailored mapping of the state variables to physical angles, adequate leg trajectories can be accessed directly from the coupled systems. The functionality of the resulting generator was confirmed in laboratory experiments by means of an instrumented six-legged ant-like robot. Owing to their simple form, the 18 coupled equations could be rapidly integrated on a bare-metal microcontroller, leading to the demonstration of real-time robot control navigating an arena using a brain–machine interface.
REFERENCES
1.
2.
A.
Biewener
and T.
Daniel
, “A moving topic: Control and dynamics of animal locomotion
,” Biol. Lett.
6
, 387
–388
(2010
).3.
M.
Frasca
, P.
Arena
, and L.
Fortuna
, Bio-inspired Emergent Control of Locomotion Systems
(World Scientific
, 2004
), Vol. 48
.4.
S.
Grillner
and P.
Wallen
, “Central pattern generators for locomotion, with special reference to vertebrates
,” Annu. Rev. Neurosci.
8
, 233
–261
(1985
). 5.
T.
Orlovsky
, G. N.
Orlovskiĭ
, T.
Deliagina
, and S.
Grillner
, Neuronal Control of Locomotion: From Mollusc to Man
(Oxford University Press
, 1999
).6.
E.
Marder
and D.
Bucher
, “Central pattern generators and the control of rhythmic movements
,” Curr. Biol.
11
, R986
–R996
(2001
). 7.
O. E.
Rössler
, “Adequate locomotion strategies for an abstract organism in an abstract environment–A relational approach to brain function,” in Physics and Mathematics of the Nervous System, edited by M. Conrad, W. Güttinger, and M. Dal Cin (Springer, Berlin, Heidelberg, 1974), pp. 342–369.8.
O. E.
Rössler
, “Deductive biology–Some cautious steps
,” Bull. Math. Biol.
40
, 45
–58
(1978
).9.
A.
Billard
and A. J.
Ijspeert
, “Biologically inspired neural controllers for motor control in a quadruped robot,” in Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks. IJCNN 2000. Neural Computing: New Challenges and Perspectives for the New Millennium (IEEE, 2000), Vol. 6, pp. 637–641.10.
M. A.
Lewis
, R.
Etienne-Cummings
, M. J.
Hartmann
, Z. R.
Xu
, and A. H.
Cohen
, “An in silico central pattern generator: Silicon oscillator, coupling, entrainment, and physical computation
,” Biol. Cybern.
88
, 137
–151
(2003
). 11.
K.
Matsuoka
, “Mechanisms of frequency and pattern control in the neural rhythm generators
,” Biol. Cybern.
56
, 345
–353
(1987
). 12.
R.
Campos
, V.
Matos
, and C.
Santos
, “Hexapod locomotion: A nonlinear dynamical systems approach,” in IECON 2010-36th Annual Conference on IEEE Industrial Electronics Society (IEEE, 2010), pp. 1546–1551.13.
W.
Li
, W.
Chen
, X.
Wu
, and J.
Wang
, “Parameter tuning of CPGs for hexapod gaits based on genetic algorithm,” in 2015 IEEE 10th Conference on Industrial Electronics and Applications (ICIEA) (IEEE, 2015), pp. 45–50.14.
P.
Arena
, L.
Fortuna
, and M.
Frasca
, “Attitude control in walking hexapod robots: An analogic spatio-temporal approach
,” Int. J. Circuit Theory Appl.
30
, 349
–362
(2002
). 15.
P.
Arena
, L.
Fortuna
, M.
Frasca
, and L.
Patané
, “A CNN-based chip for robot locomotion control
,” IEEE Trans. Circuits Syst. I: Regul. Pap.
52
, 1862
–1871
(2005
). 16.
P.
Arena
, L.
Fortuna
, M.
Frasca
, and G.
Vagliasindi
, “A wave-based CNN generator for the control and actuation of a lamprey-like robot
,” Int. J. Bifurcation Chaos
16
, 39
–46
(2006
). 17.
P.
Arena
, C.
Bonomo
, L.
Fortuna
, M.
Frasca
, and S.
Graziani
, “Design and control of an ipmc wormlike robot
,” IEEE Trans. Syst. Man Cybern. Part B Cybern.
36
, 1044
–1052
(2006
). 18.
A. J.
Ijspeert
, A.
Crespi
, and J.-M.
Cabelguen
, “Simulation and robotics studies of salamander locomotion
,” Neuroinformatics
3
, 171
–195
(2005
). 19.
K.
Nakada
, T.
Asai
, and Y.
Amemiya
, “An analog CMOS central pattern generator for interlimb coordination in quadruped locomotion
,” IEEE Trans. Neural Networks
14
, 1356
–1365
(2003
). 20.
J.
Duysens
and A.
Forner-Cordero
, “A controller perspective on biological gait control: Reflexes and central pattern generators
,” Annu. Rev. Control
48
, 392
–400
(2019
). 21.
L.
Matthey
, L.
Righetti
, and A. J.
Ijspeert
, “Experimental study of limit cycle and chaotic controllers for the locomotion of centipede robots,” in 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2008), pp. 1860–1865.22.
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
–398
(1976
). 23.
O. E.
Rössler
, “Chaotic behavior in simple reaction systems
,” Z. Naturforsch. A
31
, 259
–264
(1976
). 24.
J. C.
Sprott
, Elegant Chaos: Algebraically Simple Chaotic Flows
(World Scientific
, 2010
).25.
L.
Minati
, M.
Frasca
, P.
Oświecimka
, L.
Faes
, and S.
Drożdż
, “Atypical transistor-based chaotic oscillators: Design, realization, and diversity
,” Chaos
27
, 073113
(2017
). 26.
S.
Boccaletti
, A. N.
Pisarchik
, C. I.
Del Genio
, and A.
Amann
, Synchronization: From Coupled Systems to Complex Networks
(Cambridge University Press
, 2018
).27.
A.
Buscarino
, L.
Fortuna
, M.
Frasca
, and G.
Muscato
, “Chaos does help motion control
,” Int. J. Bifurcation Chaos
17
, 3577
–3581
(2007
). 28.
A.
Buscarino
, C.
Camerano
, L.
Fortuna
, and M.
Frasca
, “Chaotic mimic robots
,” Philos. Trans. R. Soc. A
368
, 2179
–2187
(2010
). 29.
L.
Minati
, “Across neurons and silicon: Some experiments regarding the pervasiveness of nonlinear phenomena
,” Acta Phys. Pol. B
49
, 2029
(2018
). 30.
L.
Minati
, M.
Frasca
, N.
Yoshimura
, and Y.
Koike
, “Versatile locomotion control of a hexapod robot using a hierarchical network of nonlinear oscillator circuits
,” IEEE Access
6
, 8042
–8065
(2018
). 31.
O.
Rossler
, “An artificial cognitive-plus-motivational system,” in Progress in Theoretical Biology, edited by R. Rosen (Academic Press, 1981), pp. 147–160.32.
D. M.
Wilson
, “Insect walking
,” Annu. Rev. Entomol.
11
, 103
–122
(1966
). 33.
F.
Delcomyn
, “Insect walking and robotics
,” Annu. Rev. Entomol.
49
, 51
–70
(2004
). 34.
S.
Mitra
, M. A.
Riley
, and M. T.
Turvey
, “Chaos in human rhythmic movement
,” J. Mot. Behav.
29
, 195
–198
(1997
). 35.
Q.
Li
, J.
Guo
, and X.-S.
Yang
, “Bifurcation and chaos in the simple passive dynamic walking model with upper body
,” Chaos
24
, 033114
(2014
). 36.
J. J.
Vidal
, “Toward direct brain-computer communication
,” Ann. Rev. Biophys. Bioeng.
2
, 157
–180
(1973
). 37.
L. F.
Nicolas-Alonso
and J.
Gomez-Gil
, “Brain computer interfaces: A review
,” Sensors
12
, 1211
–1279
(2012
). 38.
R. A.
Ramadan
and A. V.
Vasilakos
, “Brain computer interface: Control signals review
,” Neurocomputing
223
, 26
–44
(2017
). 39.
R.
Rossler
and O. E.
Rossler
, “Is the mind-body interface microscopic?
,” Theor. Med.
14
, 153
–165
(1993
). 40.
L.
Minati
, N.
Yoshimura
, and Y.
Koike
, “Hybrid control of a vision-guided robot arm by EOG, EMG, EEG biosignals and head movement acquired via a consumer-grade wearable device
,” IEEE Access
4
, 9528
–9541
(2016
). 41.
L.
Minati
, N.
Yoshimura
, M.
Frasca
, S.
Drożdż
, and Y.
Koike
, “Warped phase coherence: An empirical synchronization measure combining phase and amplitude information
,” Chaos
29
, 021102
(2019
). 42.
M.
Golubitsky
, I.
Stewart
, P.-L.
Buono
, and J.
Collins
, “Symmetry in locomotor central pattern generators and animal gaits
,” Nature
401
, 693
–695
(1999
). 43.
M.
Golubitsky
, I.
Stewart
, P.-L.
Buono
, and J.
Collins
, “A modular network for legged locomotion
,” Physica D
115
, 56
–72
(1998
). 44.
A.
Pisarchik
, R.
Jaimes-Reátegui
, J.
Villalobos-Salazar
, J.
Garcia-Lopez
, and S.
Boccaletti
, “Synchronization of chaotic systems with coexisting attractors
,” Phys. Rev. Lett.
96
, 244102
(2006
). 45.
G.
Tirabassi
, R.
Sevilla-Escoboza
, J. M.
Buldú
, and C.
Masoller
, “Inferring the connectivity of coupled oscillators from time-series statistical similarity analysis
,” Sci. Rep.
5
, 1
–14
(2015
). 46.
A.
Mejia Tobar
, R.
Hyoudou
, K.
Kita
, T.
Nakamura
, H.
Kambara
, Y.
Ogata
, T.
Hanakawa
, Y.
Koike
, and N.
Yoshimura
, “Decoding of ankle flexion and extension from cortical current sources estimated from non-invasive brain activity recording methods
,” Front. Neurosci.
11
, 733
(2018
). 47.
H.
Nomura
, N.
Wakami
, and S.
Kondo
, “Non-linear technologies in a dishwasher,” in Proceedings of 1995 IEEE International Conference on Fuzzy Systems (IEEE, 1995), Vol. 5, pp. 57–58.48.
S. M.
Hosseinalipour
, A.
Tohidi
, M.
Shokrpour
, and N. M.
Nouri
, “Introduction of a chaotic dough mixer part A: Mathematical modeling and numerical simulation
,” J. Mech. Sci. Technol.
27
, 1329
–1339
(2013
). 49.
K.
Pearson
and R.
Franklin
, “Characteristics of leg movements and patterns of coordination in locusts walking on rough terrain
,” Int. J. Rob. Res.
3
, 101
–112
(1984
). © 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
