Swimming on limit cycles with nonholonomic constraints*

Reduced order mathematical models are particularly useful when designing real time control algorithms, which inherently need to be computationally efficient. The complexities of the fluid-body interaction between a swimming robot and its environment make finding a reduced order model crucial for the implementation of real time control. The Chaplygin sleigh, a well known nonholonomic system, serves as an inspiration for such a low dimensional model.

We show through experiments that the steady state dynamics of a swimming fish shaped body propelled by a periodic torque are confined to a limit cycle that is topologically similar to the limit cycle in the dynamics of the dissipative Chaplygin Sleigh. Using an unsteady version of the panel method potential flow solver and the harmonic balance approach we obtain a surrogate model of a Chaplygin sleigh with equivalent dynamics. We demonstrate the utility of such a surrogate model by designing a control torque that steers the fish shaped body in a desired direction while simultaneously tracking a prescribed speed.