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Unlike traditional discrete robots, compliant robots are made of a continuous flexible body whose material distribution is such that a minimum set of input forces can exploit resultant modes of vibration for locomotion (Valdivia y Alvarado and Youcef- Toumi, 2006, 2008). In the case of fish swimming, the design and synthesis process is summarized as follows:

a) The desired swimming mode is chosen from a range of classical swimmers, (e.g. anguilliform, carangiform, or thunniform). Reviews of fish swimming characteristics can be found in (Lighthill, 1975) and (Videler, 1993). Based on the desired mode shape, the design-intent body motions are identified (see fig. 3-1). For the carangiform mode, studied herein, the spine motions are given by

h(x, t; f ) = 12y(x; f ) cos(2πf t − kx) (3.2.1)

body lateral deflection at a distance x from the nose (which, for a carangiform swimmer, is different for each frequency), k = 2π/0.9L is the wavenumber, and L is the body length. These target body motions, are shown in figure 3-1. The design-intent flapping frequency is also selected in this step. For the robots studied herein, the design-frequency is fd = 2.7 Hz. The robotic fish are able to

swim at other flapping frequencies, though typically with reduced performance. b) The body geometry, including fin shape and placement, is dictated by the selected swimming mode. The top panel of figure 3-2 is a schematic of the carangiform-type swimmers studied herein.

c) The material and actuation distributions are found by solving the governing equation for body dynamics, given the desired kinematics (3.2.1). The body dynamics are governed by a modified Bernoulli-Euler beam equation (Valdivia y Alvarado, 2007)

(m + ma)∂ 2h ∂t2 = ∂2 ∂x2  M (x, t) − EI∂∂x2h2 − µI ∂3h ∂t∂x2  (3.2.2)

where m(x) and ma(x) are the mass and added mass per unit length of an

infinitesimal section of the body at position x, I(x) is the section moment of inertia, and E(x) and µ(x) are the material elasticity and viscosity, respectively. The servomotor is commanded by a square-wave input signal and applies a concentrated moment at position x = a. This actuation can be approximated with a sine wave and delta function:

M (x, t) ≈ M0δ(x − a) sin(2πfdt) (3.2.3)

Using equations (3.2.2) and (3.2.3), the material properties, E and µ, as well as the actuator moment, M0, and position, a, are determined, which result in

L = 14.8 cm

a = 7.62 cm

2.54 cm

b = 4.32 cm

Figure 3-2: The carangiform swimming robot used in the PIV experiments consists of a compliant body with an embedded actuator. Power and control signal are carried by umbilical cord (Valdivia y Alvarado, 2007). (top) schematic, (left) isometric view, (right) robot A.

d) The prototype’s body is cast using silicone and urethane gel compounds matching the desired material properties.

This approach yields simple and robust devices. Further discussion regarding this design process is detailed in (Valdivia y Alvarado, 2007).

3.3

Materials and methods

Two nearly-identical prototypes, robots A and B, were used for the flow visualization experiments. They were designed to mimic the swimming motions and performance of carangiform swimmers. The body form based on these desired motions is shown in figure 3-2.

The two prototypes have a body length from snout to tail tip of L = 14.8 cm, are composed of elastomer materials of average elasticity E = 97835 Pa and viscosity µ = 92.3 Pa·s, and are powered by single servomotor. The servomotor applies a moment M0 = 0.1 Nm to a plate located at a distance a = 7.6 cm from the prototype’s snout.

The prototypes have a body mass of 68 grams and are close to neutral buoyancy. The two robots were identical in design and differed only due to construction. Robot B was slightly tail-heavy, whereas robot A swam at nearly level trim. The flow features of robot A were characterized using high-speed particle image velocimetry (PIV). Unfortunately, robot A was retired at the conclusion of the PIV experiments due to mechanical failure (after over one hundred hours of swimming), so robot B was used to qualitatively illustrate the wake using dye visualization.

Quantitative measurements were made using high-speed particle image velocimetry (PIV) (Raffel et al, 2002). The robotic fish was allowed to swim freely in a tank seeded with 93 µm particles. A horizontal laser sheet was positioned such that it was at the fish mid-plane. A high-speed camera imaged from below at 100 fps, yielding a time-step between frames of 0.01 s. Image resolution was 1260×1024 pixels, and the field of view was 16.6 cm x 13.5 cm, giving a 75.9 px/cm zoom.

A time-series of PIV images were captured for each of three trials at selected flapping frequencies between 1 and 4 Hz. Flapping frequency, f , tail flapping amplitude, H, and spine location, h(x, t), were determined from these raw images. The time-series of particle images were then processed using the LaVision DaVis 7.1 software package. The output was a velocity field of 79×64 vectors, with

w H t0 t t1 t2 U fish tail

y x

Figure 3-3: Composite wake used to compute wake geometry. The locations of tail maximum excursions and vortex centroids are recorded for three flapping cycles. Wake width, w, streamwise spacing (i.e. stride length), `, flapping amplitude, H, and swimming speed, U , are computed from the composite wake.

approximately 70 vectors along the length of the fish body. The data were post- processed in Matlab to determine vorticity, circulation, and wake geometry.

The procedure used to determine vortex circulation and wake geometry is similar to that used by Streitlien and Triantafyllou (1998) in the study of flapping foils. Namely, we form a composite wake from three or more tail flap cycles by freezing each vortex in its shed position, and we make measurements on the composite wake (see figure 3-3). This composite wake allows us to use 2D classical vortex dynamics theory to predict the forces on the fish (3.4.4). While this model ignores three- dimensional effects, we show in Section 3.4.4 that it does successfully predict the swimming performance of the fish. Streitlien and Triantafyllou (1998) define a vortex as a simply-connected region of same-signed vorticity which is above some threshold. In this experiment we used a threshold of 4 s−1, which is approximately 10% of the maximum vorticity level for many trials. The circulation, Γ, and centroid of the vorticity constituting each discrete vortex, (xc, yc), is computed by evaluating the

zeroth and first moments of the vorticity, ω, respectively

Γ =XωδA , xc = Γ1 P xωδA , yc = Γ1

X

where the summation is performed over the field points constituting the vortex, and δA = (16 px)2 = 0.044 cm2 is the box size. Equation (8.1.8) is evaluated in five time-

steps about vortex shedding, and the mean values are used to form the composite wake (this time-average smoothes out any small fluctuations in the PIV data). The lateral width, w, and streamwise spacing (i.e. stride length), `, are computed from the composite wake, and the circulation, Γ, is the mean of the magnitudes of all vortex circulations. Streitlien and Triantafyllou (1998) reported acceptable agreement between the measured thrust of a flapping foil and that computed using this procedure with equation (3.4.3).

Swimming speed is defined as

U = f ` (3.3.2)

where f is the flapping frequency, which is identical to the vortex shedding frequency. Swimming speed computed using equation (3.3.2) was, for all trials, within 3% of the value calculated by inspecting the movement of a feature of the body in several frames.

Qualitative flow visualization was performed using dye. A fluorescent dye mixture was painted onto the caudal fin and allowed to shed freely into the flow as the robot swam. The mixture consisted of fluorescein dye, polyvinyl acetate (adhesive), dimethicone (viscous thickener), butylene glycol (hygroscopic substance and solubilizer), and other solubilizers. The dye was illuminated using incandescent flood lamps fitted with blue cinema gels and imaged using a video camera at 30 fps. Images were post processed by performing a band-pass filter on the light intensity levels, and by inverting the color spectrum (so the green dye appears magenta in the images herein).

0 0.25 0.5 0.75 1 −0.05 0 0.05 −0.05 0 0.05 −0.05 0 0.05 f/fd = 0.37 f/fd = 1.04 f/fd = 1.58 x/L h(x,t)/L time 0 T/2 T

Figure 3-4: Spline positions (measured from the raw PIV images) illustrate the kinematics of one flapping cycle in the low-frequency (f /fd = 0.37), nominal

(f /fd = 1.04), and high-frequency flapping (f /fd = 1.58) regimes. In the nominal

case, the kinematics resemble carangiform swimming, whereas in the low- and high- frequency flapping cases, the kinematics are altered. The time-step between body tracings is 0.04 s in the f /fd = 0.37 case and 0.01 s in the other two cases. The

aspect ratio of the axes is 2:1.