Aguilar, J. et al. A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems. Rep. Prog. Phys. 79, 110001 (2016).
Aguilar, J. et al. Collective clog control: optimizing traffic flow in confined biological and robophysical excavation. Science 361, 672–677 (2018).
Brandenbourger, M., Locsin, X., Lerner, E. & Coulais, C. Non-reciprocal robotic metamaterials. Nat. Commun. 10, 4608 (2019).
Savoie, W. et al. A robot made of robots: emergent transport and control of a smarticle ensemble. Sci. Robot. 4, eaax4316 (2019).
Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. Living crystals of light-activated colloidal surfers. Science 339, 936–40 (2013).
Aubret, A., Martinet, Q. & Palacci, J. Metamachines of pluripotent colloids. Nat. Commun. 12, 6398 (2021).
Miskin, M. Z. et al. Electronically integrated, mass-manufactured, microscopic robots. Nature 584, 557–561 (2020).
Tan, T. H. et al. Odd dynamics of living chiral crystals. Nature 607, 287–293 (2022).
Sanchez, T., Chen, D. T., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012).
Needleman, D. & Dogic, Z. Active matter at the interface between materials science and cell biology. Nat. Rev. Mater. 2, 17048 (2017).
Bricard, A., Caussin, J. B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).
Bililign, E. et al. Motile dislocations knead odd crystals into whorls. Nat. Phys. 18, 212–218 (2022).
Shankar, S., Souslov, A., Bowick, M. J., Marchetti, M. C. & Vitelli, V. Topological active matter. Nat. Rev. Phys. 4, 380–398 (2022).
Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013).
Ball, P. Animate materials. MRS Bull. 46, 553–559 (2021).
Chen, Y., Li, X., Scheibner, C., Vitelli, V. & Huang, G. Realization of active metamaterials with odd micropolar elasticity. Nat. Commun. 12, 5935 (2021).
Scheibner, C. et al. Odd elasticity. Nat. Phys. 16, 475–480 (2020).
Marder, E. & Bucher, D. Central pattern generators and the control of rhythmic movements. Curr. Biol. 11, R986–R996 (2001).
Peyret, G. et al. Sustained oscillations of epithelial cell sheets. Biophys. J. 117, 464–478 (2019).
Gilpin, W., Bull, M. & Prakash, M. The multiscale physics of cilia and flagella. Nat. Rev. Phys. 2, 74–88 (2020).
Lavi, I., Piel, M., Lennon-Duménil, A.-M., Voituriez, R. & Gov, N. Deterministic patterns in cell motility. Nat. Phys. 12, 1146–1152 (2016).
Shankar, S. & Mahadevan, L. Active hydraulics and odd elasticity of muscle fibres. Nat. Phys. 20, 1501–1508 (2024).
Keber, F. C. et al. Topology and dynamics of active nematic vesicles. Science 345, 1135–1139 (2014).
Giomi, L. & DeSimone, A. Spontaneous division and motility in active nematic droplets. Phys. Rev. Lett. 112, 147802 (2014).
Baconnier, P. et al. Selective and collective actuation in active solids. Nat. Phys. 18, 1234–1239 (2022).
Miskin, M. et al. Graphene-based bimorphs for micron-sized, autonomous origami machines. Proc. Natl Acad. Sci. USA 115, 466–470 (2018).
McDonald, A. & Clerk, A. A. Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics. Nat. Commun. 11, 5382 (2020).
Mathew, J. P., Pino, J. D. & Verhagen, E. Synthetic gauge fields for phonon transport in a nano-optomechanical system. Nat. Nanotechnol. 15, 198–202 (2020).
Wanjura, C. C. et al. Quadrature nonreciprocity in bosonic networks without breaking time-reversal symmetry. Nat. Phys. 19, 1429–1436 (2023).
Liu, T., Ou, J.-Y., MacDonald, K. & Zheludev, N. Photonic metamaterial analogue of a continuous time crystal. Nat. Phys. 19, 986–991 (2023).
Nirody, J., Duran, L., Johnston, D. & Cohen, D. Tardigrades exhibit robust interlimb coordination across walking speeds and terrains. Proc. Natl Acad. Sci. USA 118, e2107289118 (2021).
Kramar, M. & Alim, K. Encoding memory in tube diameter hierarchy of living flow network. Proc. Natl Acad. Sci. USA 118, e2007815118 (2021).
Full, R., Earls, K., Wong, M. & Caldwell, R. Locomotion like a wheel? Nature 365, 495–495 (1993).
Biewener, A. & Patek, S. Animal Locomotion (Oxford Univ. Press, 2018).
Ijspeert, A., Crespi, A., Ryczko, D. & Cabelguen, J.-M. From swimming to walking with a salamander robot driven by a spinal cord model. Science 315, 1416–1420 (2007).
Thandiackal, R. et al. Emergence of robust self-organized undulatory swimming based on local hydrodynamic force sensing. Sci. Robot. 6, eabf6354 (2021).
Choi, S. et al. Learning quadrupedal locomotion on deformable terrain. Sci. Robot. 8, eade2256 (2023).
Cui, H. et al. Design and printing of proprioceptive three-dimensional architected robotic metamaterials. Science 376, 1287–1293 (2022).
He, Q. et al. A modular strategy for distributed, embodied control of electronics-free soft robots. Sci. Adv. 9, eade9247 (2023).
Saintyves, B., Spenko, M. & Jaeger, H. A self-organizing robotic aggregate using solid and liquid-like collective states. Sci. Robot. 9, eadh4130 (2024).
IEEE Spectrum. A compilation of robots falling down at the DARPA Robotics Challenge. YouTube https://www.youtube.com/watch?v=g0TaYhjpOfo (2015).
Burden, S., Libby, T., Jayaram, K., Sponberg, S. & Donelan, J. Why animals can outrun robots. Sci. Robot. 9, eadi9754 (2024).
Verhey, K. & Hammond, J. Traffic control: regulation of kinesin motors. Nat. Rev. Mol. Cell Biol. 10, 765–777 (2009).
Chen, Y., Ju, L., Rushdi, M., Ge, C. & Zhu, C. Receptor-mediated cell mechanosensing. Mol. Biol. Cell 28, 3134–3155 (2017).
Zehr, E. P. & Stein, R. B. What functions do reflexes serve during human locomotion?. Prog. Neurobiol. 58, 185–205 (1999).
Purcell, E. M. Life at low Reynolds number. Am. J. Phys. 45, 3–11 (1977).
Fruchart, M., Scheibner, C. & Vitelli, V. Odd viscosity and odd elasticity. Annu. Rev. Condens. Matter Phys. 14, 471–510 (2023).
Bililign, E. et al. Motile dislocations knead odd crystals into whorls. Nat. Phys. 18, 212–218 (2021).
Poncet, A. & Bartolo, D. When soft crystals defy Newton’s third law: nonreciprocal mechanics and dislocation motility. Phys. Rev. Lett. 128, 048002 (2022).
Braun, O. & Kivshar, Y. Nonlinear dynamics of the Frenkel–Kontorova model. Phys. Rep. 306, 1–108 (1998).
Ijspeert, A. Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21, 642–653 (2008).
Ryu, H. & Kuo, A. An optimality principle for locomotor central pattern generators. Sci. Rep. 11, 13140 (2021).
Fruchart, M., Hanai, R., Littlewood, P. B. & Vitelli, V. Non-reciprocal phase transitions. Nature 592, 363–369 (2021).
Mandal, R. et al. Learning dynamical behaviors in physical systems. Preprint at https://arxiv.org/abs/2406.07856 (2024).
Martinez Alvarez, V. M., Barrios Vargas, J. E. & Foa Torres, L. E. F. Non-Hermitian robust edge states in one dimension: anomalous localization and eigenspace condensation at exceptional points. Phys. Rev. B 97, 121401 (2018).
Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).
Ghatak, A., Brandenbourger, M., van Wezel, J. & Coulais, C. Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial. Proc. Natl Acad. Sci. USA 117, 29561–29568 (2020).
Helbig, T. et al. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020).
Xiao, L. et al. Non-Hermitian bulk–boundary correspondence in quantum dynamics. Nat. Phys. 16, 761–766 (2020).
Coulais, C., Fleury, R. & van Wezel, J. Topology and broken hermiticity. Nat. Phys. 17, 9–13 (2020).
Bergholtz, E., Budich, J. & Kunst, F. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).
Scheibner, C., Irvine, W. T. M. & Vitelli, V. Non-Hermitian band topology and skin modes in active elastic media. Phys. Rev. Lett. 125, 118001 (2020).
Zhou, D. & Zhang, J. Non-Hermitian topological metamaterials with odd elasticity. Phys. Rev. Res. 2, 023173 (2020).
Landau, L. et al. Theory of Elasticity. Course of Theoretical Physics (Elsevier Science, 1986).
Duan, Q. et al. PyPop7: a pure-Python library for population-based black-box optimization. J. Mach. Learn. Res. 25, 1–28 (2024).
Loshchilov, I., Glasmachers, T. & Beyer, H.-G. Large scale black-box optimization by limited-memory matrix adaptation. IEEE Trans. Evol. Comput. 23, 353–358 (2019).
Veenstra, J. et al. Adaptive locomotion of active solids. Zenodo https://doi.org/10.5281/zenodo.13832206 (2025).
Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017).
Shaat, M. & Park, H. Chiral nonreciprocal elasticity and mechanical activity. J. Mech. Phys. Solids 171, 105163 (2023).
