Abstract
Generating locomotive gaits is a very important work for bioinspired robots and has received wide attentions among scientists and engineers. The central pattern generator (CPG) neural system located in spinal cord is a very important imitating object to produce rhythm patterns and control locomotion of animals. This paper proposes a theoretical method to construct a novel style CPG controller for the quadruped locomotive gait based on the delay-coupled VDP oscillators. The controller system consists of four VDP oscillators scheduled with unidirectional ring structure. By using the Hopf bifurcation analysis, we obtain parameter conditions for rhythm generation and present dynamical classification of periodic rhythm with different spatiotemporal patterns, which corresponds to the gait of the quadruped locomotion. Assisting with switching network structure, the delayed VDP-CPG controller presents six types of classical locomotive gaits, i.e., pronk, lateral-sequence (L-S) walk, diagonal-sequence (D-S) walk, bound, pace, and trot, in a wide range of parameter areas. The numerical simulations are illustrated to agree with theoretical analysis. The presented approach herein provides a frame of dynamical analysis to build the CPG controller producing rhythm controlling signals of the quadruped locomotion gaits.
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References
Grillner, S., El Manira, A.: Current principles of motor control, with special reference to vertebrate locomotion. Physiol. Rev. 100(1), 271–320 (2020)
Wang, J., Chen, W., Xiao, X., Xu, Y., Li, C., Jia, X., Meng, M.Q.-H.: A survey of the development of biomimetic intelligence and robotics. Bio. Intel. Robot. 1, 100001 (2021)
Lobato-Rios, V., Ramalingasetty, S.T., Özdil, P.G., Arreguit, J., Ijspeert, A.J., Ramdya, P.: NeuroMechFly, a neuromechanical model of adult Drosophila melanogaster. Nat. Methods 19(5), 620–627 (2022)
Yu, J., Tan, M., Chen, J., Zhang, J.: A survey on CPG-inspired control models and system implementation. IEEE Trans. Neural Netw. Learn. Syst. 25(3), 441–456 (2013)
Kinugasa, T., Sugimoto, Y.: Dynamically and biologically inspired legged locomotion: a review. J. Robot. Mechatron. 29(3), 456–470 (2017)
Ryczko, D., Simon, A., Ijspeert, A.J.: Walking with salamanders: from molecules to biorobotics. Trends Neurosci. 43(11), 916–930 (2020)
Kimura, H., Fukuoka, Y., Cohen, A.H.: Biologically inspired adaptive walking of a quadruped robot. Philos. Trans. R. Soc. A 365, 153–170 (2007)
Kimura, H., Fukuoka, Y., Cohen, A.H.: Adaptive dynamic walking of a quadruped robot on natural ground based on biological concepts. Int. J. Robot. Res. 26(5), 475–490 (2007)
Maufroy, C., Kimura, H., Takase, K.: Towards a general neural controller for quadrupedal locomotion. Neural Netw. 21, 667–681 (2008)
Owaki, D., Horikiri, S., Nishii, J., Ishiguro, A.: Tegotae-based control produces adaptive inter- and intra-limb coordination in bipedal walking. Front. Neurorobot. 15, 629595 (2021)
Zhu, W., Guo, X., Owaki, D., Kutsuzawa, K., Hayashibe, M.: A survey of sim-to-real transfer techniques applied to reinforcement learning for bioinspired robots. IEEE Trans. Neural Netw. Learn. Syst. 34, 9552429 (2021)
Koseki, S., Kutsuzawa, K., Owaki, D., Hayashibe, M.: Multimodal bipedal locomotion generation with passive dynamics via deep reinforcement learning. Front. Neurorobot. 16, 1054239 (2023)
Sun, T., Xiong, X., Dai, Z., Owaki, D., Manoonpong, P.: A comparative study of adaptive interlimb coordination mechanisms for self-organized robot locomotion. Front. Robot. AI 8, 638684 (2021)
Golubitsky, M., Stewart, I., Buono, P.-L., Collins, J.: A modular network for legged locomotion. Phys. D 115(1–2), 56–72 (1998)
Golubitsky, M., Stewart, I., Buono, P.-L., Collins, J.: Symmetry in locomotor central pattern generators and animal gaits. Nature 401(6754), 693–695 (1999)
Buono, P.-L., Golubitsky, M.: Models of central pattern generators for quadruped locomotion I. Primary gaits. J. Math. Biol. 42(4), 291–326 (2001)
Buono, P.-L.: Models of central pattern generators for quadruped locomotion II. Secondary gaits. J. Math. Biol. 42(4), 327–346 (2001)
Stewart, I.: Spontaneous symmetry-breaking in a network model for quadruped locomotion. Int. J. Bifurcat. Chaos 27(14), 1730049 (2017)
In, V., Kho, A., Longhini, P., Neff, J.D., Palacios, A., Buono, P.-L.: Meet ANIBOT: the first biologically-inspired animal robot. Int. J. Bifurcat. Chaos 32(01), 2230001 (2022)
Barrio, R., Lozano, Á., Rodríguez, M., Serrano, S.: Numerical detection of patterns in CPGs: gait patterns in insect movement. Commun. Nonlinear Sci. 82, 105047 (2020)
Dutta, S., Parihar, A., Khanna, A., Gomez, J., Chakraborty, W., Jerry, M., Grisafe, B., Raychowdhury, A., Datta, S.: Programmable coupled oscillators for synchronized locomotion. Nat. Commun. 10(1), 1–10 (2019)
Wang, Y., Xue, X., Chen, B.: Matsuoka’s CPG with desired rhythmic signals for adaptive walking of humanoid robots. IEEE Trans. Cybern. 50(2), 613–626 (2018)
Liu, G.L., Habib, M.K., Watanabe, K., Izumi, K.: Central pattern generators based on Matsuoka oscillators for the locomotion of biped robots. Artif. Life Robot. 12(1), 264–269 (2008)
Fukui, T., Matsukawa, S., Habu, Y., Fukuoka, Y.: Gait transition from pacing by a quadrupedal simulated model and robot with phase modulation by vestibular feedback. Robotics 11(1), 3 (2021)
Espinal, A., Rostro-Gonzalez, H., Carpio, M., Guerra-Hernandez, E.I., Ornelas-Rodriguez, M., Puga-Soberanes, H., Sotelo-Figueroa, M.A., Melin, P.: Quadrupedal robot locomotion: a biologically inspired approach and its hardware implementation. Comput. Intel. Neurosci. (2016). https://doi.org/10.1155/2016/5615618
Zhong, G., Chen, L., Jiao, Z., Li, J., Deng, H.: Locomotion control and gait planning of a novel hexapod robot using biomimetic neurons. IEEE Trans. Control Syst. Technol. 26(2), 624–636 (2017)
Wang, B., Zhang, K., Yang, X., Cui, X.: The gait planning of hexapod robot based on CPG with feedback. Int. J. Adv. Robot. Syst. 17(3), 1729881420930503 (2020)
Liu, J., Tong, Y., Liu, J.: Review of snake robots in constrained environments. Robot. Auton. Syst. 141, 103785 (2021)
Wu, X., Ma, S.: Neurally controlled steering for collision-free behavior of a snake robot. IEEE Trans. Control Syst. Technol. 21(6), 2443–2449 (2013)
Qiao, G., Zhang, Y., Wen, X., Wei, Z., Cui, J.: Triple-layered central pattern generator-based controller for 3D locomotion control of snake-like robots. Int. J. Adv. Robot. Syst. 14(6), 1729881417738101 (2017)
Manzoor, S., Khan, U., Ullah, I.: Serpentine and rectilinear motion generation in snake robot using central pattern generator with gait transition. Iran J. Sci. Technol. Trans. Electr. Eng. 44(3), 1093–1103 (2020)
Zhu, Y., Zhou, S., Gao, D., Liu, Q.: Synchronization of non-linear oscillators for neurobiologically inspired control on a bionic parallel waist of legged robot. Front. Neurorobotics 13, 59 (2019)
Liu, X., Iwasaki, T.: Design of coupled harmonic oscillators for synchronization and coordination. IEEE Trans. Autom. Control 62(8), 3877–3889 (2017)
Zhao, J., Iwasaki, T.: Orbital stability analysis for perturbed nonlinear systems and natural entrainment via adaptive Andronov-Hopf oscillator. IEEE Trans. Autom. Control 65(1), 87–101 (2019)
Kohannim, S., Iwasaki, T.: Design of coupled Andronov-Hopf oscillators with desired strange attractors. Nonlinear Dyn. 100(2), 1659–1672 (2020)
Futakata, Y., Iwasaki, T.: Entrainment to natural oscillations via uncoupled central pattern generators. IEEE Trans. Autom. Control 56(5), 1075–1089 (2010)
Wu, A., Iwasaki, T.: Pattern formation via eigenstructure assignment: general theory and multi-agent application. IEEE Trans. Autom. Control 63(7), 1959–1972 (2017)
Rowat, P.F., Selverston, A.I.: Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network. J. Neurophysiol. 70(3), 1030–1053 (1993)
Tani, J., Qiu, J., Yamaguchi, E.: Emergence of the rhythmic movement of a dragonfly wing model. JSME Int. J. Ser. C 41(4), 689–694 (1998)
Dutra, M.S., de Pina Filho, A.C., Romano, V.F.: Modeling of a bipedal locomotor using coupled nonlinear oscillators of Van der Pol. Biol. Cybern. 88(4), 286–292 (2003)
Liu, C., Chen, Q., Zhang, J.: Coupled Van Der Pol oscillators utilised as central pattern generators for quadruped locomotion, pp. 3677–3682
Jasni, F., Shafie, A.A.: Van Der Pol central pattern generator (VDP-CPG) model for quadruped robot, pp. 167–175
Yu, H., Gao, H., Ding, L., Li, M., Deng, Z., Liu, G.: Gait generation with smooth transition using CPG-based locomotion control for hexapod walking robot. IEEE T. Ind. Electron. 63(9), 5488–5500 (2016)
Yu, H., Gao, H., Deng, Z.: Enhancing adaptability with local reactive behaviors for hexapod walking robot via sensory feedback integrated central pattern generator. Robot. Auton. Syst. 124, 103401 (2020)
Zaier, R., Eldirdiry, O.: Legged robot design and Van der Pol oscillator based control approach. Int. J. Model. Identif. 38(3–4), 282–290 (2021)
Barron-Zambrano, J.H., Torres-Huitzil, C.: Two-phase GA parameter tunning method of CPGs for quadruped gaits, pp. 1767–1774
Li, W., Chen, W., Wu, X., Wang, J.: Parameter tuning of CPGs for hexapod gaits based on genetic algorithm, pp. 45–50
Ausborn, J., Snyder, A.C., Shevtsova, N.A., Rybak, I.A., Rubin, J.E.: State-dependent rhythmogenesis and frequency control in a half-center locomotor CPG. J. Neurophysiol. 119(1), 96–117 (2018)
Aminzare, Z., Srivastava, V., Holmes, P.: Gait transitions in a phase oscillator model of an insect central pattern generator. SIAM J. Appl. Dyn. Syst. 17(1), 626–671 (2018)
Song, Z., Xu, J., Zhen, B.: Multitype activity coexistence in an inertial two-neuron system with multiple delays. Int. J. Bifurcat. Chaos 25(13), 1530040 (2015)
Song, Z., Zhen, B., Hu, D.: Multiple bifurcations and coexistence in an inertial two-neuron system with multiple delays. Cogn. Neurodynamics 14(3), 359–374 (2020)
Yao, S., Ding, L., Song, Z., Xu, J.: Two bifurcation routes to multiple chaotic coexistence in an inertial two-neural system with time delay. Nonlinear Dyn. 95(2), 1549–1563 (2019)
Dhamala, M., Jirsa, V.K., Ding, M.: Enhancement of neural synchrony by time delay. Phys. Rev. Lett. 92(7), 074104 (2004)
Owaki, D., Kano, T., Nagasawa, K., Tero, A., Ishiguro, A.: Simple robot suggests physical interlimb communication is essential for quadruped walking. J. R. Soc. Interface 10(78), 20120669 (2013)
Owaki, D., Ishiguro, A.: A quadruped robot exhibiting spontaneous gait transitions from walking to trotting to galloping. Sci. Rep. 7(1), 1–10 (2017)
Patrick, S.K., Noah, J.A., Yang, J.F.: Interlimb coordination in human crawling reveals similarities in development and neural control with quadrupeds. J. Neurophysiol. 101(2), 603–613 (2009)
Zhang, L., Stepan, G.: Bifurcations in basic models of delayed force control. Nonlinear Dyn. 99, 99–108 (2020)
Molnar, T.G., Dombovari, Z., Insperger, T., Stepan, G.: On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction. Proc. R. Soc. A 473, 20170502 (2017)
Kadar, F., Stepan, G.: Nonlinear dynamics and safety aspects of pressure relief valves. Nonlinear Dyn. (2023). https://doi.org/10.1007/s11071-023-08484-w
Song, Y., Xu, J., Zhang, T.: Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling. Chaos 21(2), 023111 (2011)
Zhang, S., Xu, J., Chung, K.-W.: Stability analysis of alternating wave solution in a Stuart-Landau system with time delay. Commun. Nonlinear Sci. 99, 105808 (2021)
Ohgane, K., Ei, S.-I., Mahara, H.: Neuron phase shift adaptive to time delay in locomotor control. Appl. Math. Model. 33(2), 797–811 (2009)
Verdaasdonk, B., Koopman, H.F., Van der Helm, F.C.: Resonance tuning in a neuro-musculo-skeletal model of the forearm. Biol. Cybern. 96(2), 165–180 (2007)
Lu, Q., Wang, X., Tian, J.: A new biological central pattern generator model and its relationship with the motor units. Cogn. Neurodynamics 16(1), 135–147 (2022)
Zhu, Y., Wu, Y., Liu, Q., Guo, T., Qin, R., Hui, J.: A backward control based on σ-Hopf oscillator with decoupled parameters for smooth locomotion of bio-inspired legged robot. Robot. Auton. Syst. 106, 165–178 (2018)
Liu, L., Zhang, C.: Dynamic properties of VDP-CPG model in rhythmic movement with delay. Math. Biosci. Eng. 17(4), 3190–3202 (2020)
Song, Z., Xu, J.: Self-/mutual-symmetric rhythms and their coexistence in a delayed half-center oscillator of the CPG neural system. Nonlinear Dyn. 108(3), 2595–2609 (2022)
Song, Z., Huang, X., Xu, J.: Spatiotemporal pattern of periodic rhythms in delayed Van der Pol oscillators for the CPG-based locomotion of snake-like robot. Nonlinear Dyn. 110, 3377–3393 (2022)
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This research is supported by the National Natural Science Foundation of China under Grant Nos. 12172212 and 11932015 and the Fundamental Research Funds for the Central Universities (No. 22120220588).
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Song, Z., Zhu, J. & Xu, J. Gaits generation of quadruped locomotion for the CPG controller by the delay-coupled VDP oscillators. Nonlinear Dyn 111, 18461–18479 (2023). https://doi.org/10.1007/s11071-023-08783-2
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DOI: https://doi.org/10.1007/s11071-023-08783-2