Construction and Modelling of a Carangiform Robotic Fish

We describe an experimental testbed which we have constructed to study carangiform fish-like swimming. We mimic a carangiform fish with a three-link robot, the tailfin link of which acts as a flat-plate hydrofoil. We experiment with sinusoidal signals to the joint actuators. The experimental results agree well with a simulation assuming quasi-steady flow around the tailfin. 1. Motivation Many fish and marine mammals are very impressive swimmers: so impressive, in fact, that some have argued that fishlike swimmers are superior, in one way or another, to conventional man-made water vehicles. For decades researchers have tried to better understand how fish swim so effectively, [1, 2, 3] and a few have contemplated the mechanical imitation of fish, with the idea of building vehicles which are faster, more efficient, more stealthy, and/or more maneuverable than propeller-driven craft. [4] Perhaps the best-known robotic fish was built by Triantafyllou et al. [5] Their highly sophisticated robot had many actuated degrees of freedom and represented a fairly faithful reproduction of tuna swimming. They found that by undulating its tail and body the tuna was able to reduce the drag it experienced while being moved through the water. This evidence tends to bolster the notion that fish swimming is particularly efficient. It may still be an open question whether robot fish can really outperform efficient propeller designs, which themselves have received much research and labor. The prospect of increased efficiency for robot submersibles remains tantalizing, however, since these submersibles typically run on electric batteries of limited duration, and anything which enabled them to operate for a longer period would be of significant value. Increased stealth is another potential motive for building robot fish. Fish typically do not cause the noisy cavitation sometimes experienced by propellers. Indeed from a biological standpoint there is good reason to suppose that fish have evolved to swim quietly and stealthily. Ahlborn et al. [6] built an artificial fishtail to mimic fishlike swimming; they observed that the alternating creation and destruction of vortices in the wake behind the fish was not only an efficient way to swim, but also helped guard against detection by predators. Furthermore, many fish are highly maneuverable. Some fish can perform a 180 degree turn within a fraction of their own body length. This is not generally possible for boats or ships, which typically have large turning radii. We suspect that improved agility may be the biggest advantage that robot fish enjoy over their propeller-driven cousins. Finally, we have an abstract theoretical interest in understanding aquatic locomotion and hopefully unifying it in a single mathematical framework with other, terrestrial forms of locomotion in which an animal or robot uses quasiperiodic changes of internal shape variables to generate gross body motion. Past work at Caltech has brought the tools of differential geometry to bear on the control of other forms of undulatory locomotion, [7, 8] and we hope that these tools can be extended to fishlike swimming. But before we can proceed with this program, we must develop a model of fish locomotion with some experimental validation, which is the topic of this paper. 2. Description of the Model There are a wide variety of fish morphologies and at least a few different types of fish locomotion. We focus on attempting to mimic the swimming of the carangiform fishes, fast-swimming fishes which resemble tuna and mackerel. Carangiform fishes typically have large, high-aspect-ratio tails, and they swim using only motions of the rear and tail, while the forward part of the body remains relatively immobile. For our model we consider an idealized carangiform fish that consists of only three links: a rigid body in front, a large wing-like tail at the rear, and a slender stem or peduncle which connects the two. The three rigid links are connected by rotational joints with joint angles θ1 and θ2. See Figure 1. We continue to idealize the model by supposing that we can neglect threedimensional effects and regard the problem as essentially planar. In particular, we assume that the large tail can be considered as a rectangular flat plate (although the tails of real carangiform fish are often lunate in shape.) We will presume the tail experiences a hydrodynamic lift force derived from quasi-steady two-dimensional wing theory. The peduncle we will regard as hydrodynamically negligible. The forward rigid body will experience a drag force quadratic in and opposed to its velocity. There is a distance lb between the body’s center of mass and the peduncle. The peduncle has length lp. The tailfin has chord lt and area A. Let ~ le be a unit vector pointing in the direction of the leading edge of the tailfin hydrofoil. In a coordinate frame aligned with the principal axes of the fish’s body, then: ~ le = − (cos(θ2), sin(θ2), 0) (1) The body of the fish has instantaneous translational velocity ẋ and ẏ along its longitudinal and lateral axes; it also has instantaneous rotational velocity φ̇.