Transition and formation of the torque pattern of undulatory locomotion in resistive force dominated media

In undulatory locomotion, torques along the body are required to overcome external forces from the environment and bend the body. These torques are usually generated by muscles in animals and closely related to muscle activations. In previous studies, researchers observed a single traveling wave pattern of the torque or muscle activation, but the formation of the torque pattern is still not well understood. To elucidate the formation of the torque pattern required by external resistive forces and the transition as kinematic parameters vary, we use simplistic resistive force theory models of self-propelled, steady undulatory locomotors and examine the spatio-temporal variation of the internal torque. We find that the internal torque has a traveling wave pattern with a decreasing speed normalized by the curvature speed as the wave number (the number of wavelengths on the locomotor's body) increases from 0.5 to 1.8. As the wave number increases to 2 and greater values, the torque transitions into a two-wave-like pattern and complex patterns. Using phasor diagram analysis, we reveal that the formation and transitions of the pattern are consequences of the integration and cancellation of force phasors.