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Table 1 Models of symmetric swimmer encountered in the literature

From: Sub-Riemannian geometry, Hamiltonian dynamics, micro-swimmers, copepod nauplii and copepod robot

∙ Scallop \(\dot x_{0} = \frac {\dot \theta \,\sin \theta }{2-\cos ^{2}\theta }\)
∙ Symmetric 2-links Copepod \(\dot x_{0} = \frac {\sum _{i=1}^{2} \dot \theta _{i} \sin \theta _{i}}{\sum _{i=1}^{2} \left (1+\sin ^{2}\theta _{i}\right)}\)
∙ Symmetric Takagi n-links equal lengths L=1 \(\dot x_{0} = \frac {(1+2r)\,\sum _{i=1}^{n} \dot \theta _{i} \sin \theta _{i}}{3/2\,r\log (2/r)\sum _{i=1}^{n} \left (1+\sin ^{2}\theta _{i}\right)}\)
∙ Symmetric Purcell (Avron-Raz), \(\mathrm {d} x_{0} = \frac {\mathrm {d} \xi _{1}+\mathrm {d} \xi _{2}}{4+c-\xi _{1}^{2}-\xi _{2}^{2}}\)
2-links equal lengths L=1   \(\xi _{i} = \cos \theta _{i},\; c=l_{0}/2\)
   Limit case: c=r=0 Takagi = Purcell.