Flexibility is known to improve the propulsive performance of flapping fins. Flapping fins generate forces oscillatory in nature and this paper reports an investigation on the effect of flexibility and other parameters such as heaving, pitching amplitudes and operating frequency in reducing the center of mass oscillations of bodies attached to flapping fins. A detailed theoretical investigation has been carried out to predict the optimal operating parameters along with the fin stiffness to reduce the COM oscillations for a given self-propelled speed (SPS). Some design guidelines have been proposed which reduce COM oscillations that aid in the development of aerial and underwater robotic vehicles.