Proprioceptive Mechanism for Bioinspired Fish Swimming

In this Letter, we propose a mechanism for driving bioinspired fish swimming locomotion based on proprioceptive sensing. Proprioception provides information about and representation of a body’s position, motion, and acceleration in addition to the usual five senses. We hypothesize that a feedback loop based on this “sixth” sense results in an instability, driving the locomotion. In order to test our assumptions, we use a biomimetic robot and compare the experimental results to a simple yet generic model with excellent agreement.

Figure 1

Proprioception scheme: the extension of the muscles providing the deformation is moderated by the integration of the information from the proprioceptive sensors.

Figure 2

(a) Side view of the fishlike robotic system: the white material is 3D printed using soft polymers. The tail is driven by two cables attached to a waterproof servomotor (blue piece). The swimmer is attached to a force sensor (vertical rod connected to the head). The thin white bar represents 3.5 cm. (b) Top view of the servomotor and the elastic tail.

Figure 3

(a) A typical temporal variation on the feedback parameter $\gamma$. (b) The dynamics of the angles $\alpha$ (blue) and $\varphi$ (orange) as the controlling parameter $\gamma$ is varied. The angular frequency $\omega$ is deduced from the period $2\pi /\omega$. (c) Temporal evolution of the lateral force $F_y$ (green) and of the thrust $T=-F_x$ (gray) with $F_x$ the longitudinal force. We also show the mean thrust $\overline{T}$ in steady state for $\gamma =1.65\mathrm{rad}{\mathrm{N}}^{-1}$ (dashed horizontal line). ${\alpha }_0$, ${\mathrm{\Phi }}_0$, and $F_0$ denote the peak-to-peak amplitudes of $\alpha (t)$, $\varphi (t)$, and $F_y(t)$, respectively. (See movie in the Supplemental Material [29]).

Figure 4

Oscillatory characteristics of the proprioceptive loop. (a) Angular amplitude of the tail oscillation ${\alpha }_0$ as a function of $\gamma$. (b) Angular frequency $\omega$ as a function of $\gamma$. The experimental measurements and their error bars are drawn in blue. In (a), the thin blue curve passing through the data is an interpolation with the function ${\alpha }_0={\mathrm{\Theta }}_{\exp }\sqrt{\gamma -{\gamma }_{c,\exp }}$ with ${\gamma }_{c,\exp }=0.72\mathrm{rad}{\mathrm{N}}^{-1}$ and ${\mathrm{\Theta }}_{\exp }=0.72{\mathrm{rad}}^{1/2}{\mathrm{N}}^{1/2}$. The numerical predictions are represented by thick black lines.

Figure 5

Swimming velocity $U$ as a function of the proprioceptive feedback parameter $\gamma$. The experimental points are shown as disks. The curve represents the fit $U=0.14\sqrt{\gamma -{\gamma }_c}$ in Systeme International units.