are observed during hovering or when the animals rest on the substratum; brief bursts of high amplitude and high frequency fin movements are observed during rapid locomotion and maneuvering [11].
III. MECHANICAL FIN DESIGN
A. Design of Fin Rays
Fig. 3 shows an example of a ray-finned fish, a bowfin, with a long-based dorsal fin. The fin consists of fin-rays and a flexible membrane connecting them together. In median fins a set of muscles (usually six) for each fin-ray provide the latter with two degrees-of-freedom movement capability, while it has been suggested that certain fish can actively bend the rays of their median fins [4], [6].
Kier and Thompson [11] Suggest that fins of cuttlefish are supported by three-dimensional array of muscle; no fin ray is found in the fins of cuttlefish. Although the designs of actuators, both linear and rotary, are not exhaustive, they are however unable to model the complex musculature of the fins of the cuttlefish.

 

Fig. 4 Fin supported by rays. The base of each fin ray can be modelled by a universal joint (U) allowing two degrees of freedom movement for each fin ray [4], [6], [12].

Despite the complexity of the actual musculature, the fins of cuttlefish exhibit much the same undulations as that displayed by the fins of ray-finned fish with undulatory swimming mode. In order to simplify our modeling, the fin of cuttlefish is divided into many segments such that the fin looks similar to that of ray-finned fish. We may therefore conclude from the mechanical point of view that the fins of cuttlefish can also be supported by rigid structure such as fin rays as far as to perform undulatory movements. Fig. 4 shows fin diagram of any fish, including that of cuttlefish, performing undulations. Universal joint is (U) one possible joint that permits two degree of freedom movements at the base of each fin ray.
At the present stage, mechanical modeling of undulating fins, which include those of cuttlefish and ray-finned fish, is simplified to one degree of freedom from originally two degrees of freedom at the base of each fin ray. A servomotor serves as a muscle producing one degree of freedom at the base of each ray. A crank is attached at each servomotor to function as a fin ray. In order for the fin to exhibit undulations similar to that of any undulating fin, each servomotor is programmed so that the crank attached to the servomotor oscillates based on a sinusoidal function with a certain phase lead or lag defined by ? (Fig.5). Ten servomotors are used to model one module of mechanical fin.

Fig. 5 (a) Ten servomotors are spaced equally. (b) Isometric view of servomotors arrangement. Each servomotor serves as a muscle and a crank a fin ray.
B. Design of Flexible Membrane
Flexible materials such as plastic sheet, cloth, and thin rubber sheet are conventionally considered to model a membrane and connect all the cranks. One main disadvantage of such materials when used as flexible membrane is they assume any unpredictable shape which can disturb the water flow and movement of the fin. Thin rubber sheet is in fact an ideal material which must be used to model a flexible membrane. In this application, however, the rubber is required to elongate up to 200% of its initial length, and extra power is needed therefore in addition to the power required to push water.
Realizing the drawback of conventional materials, a mechanism was designed and built to model any sinusoidal waveform discretely by a series of straight lines – one straight line joins two points on a sinusoidal curve (Fig. 6).

(a)

Fig. 6 (a) Discrete model of a sinusoidal waveform developed by a series of straight lines joining two points every 45°. (b) Two cranks are connected by a straight line.

Fig. 7 (a) CAD picture of one fin segment. (b) Kinematic diagram of (a).

The mechanism will enable the flexible membrane to maintain a straight line between two cranks and at the same time extendible since the distance between two cranks changes as they oscillate. Fig. 7 shows the kinematic diagram of a linkage representing two cranks and a membrane. The linkage consists of five links and five lower pairs. Gruebler [12] states that degrees of freedom of a mechanism is

, (1)

where dof is the total degrees of freedom, n number of links including the frame, l number of lower pairs, and h umber of higher pairs. Kutzbach defines degrees of freedom as the number of actuators to be independently controlled in a mechanism. Equation (1) correctly anticipates that the linkage in Fig. 7 possesses two degrees of freedom since points A and E are each driven by a servomotor.
Proper material selection is inevitable in the design of the flexible membrane. Material density, strength, and water absorption capability are important criteria for selection. The material is required to have a density near that of water so that no or little work is done to lift and orientate the membrane [13]. A polymer, acrylic, was therefore chosen for the flexible membrane since its density is 1.2 times that of water.
IV. THEORY
An underwater robot was built based on undulatory swimming of cuttlefish and stingray. The robot is propelled by a pair of undulating fins; undulations of each fin are controlled by ten equally spaced servomotors to which are connected cranks of the same length, as shown in Fig. 1. Each fin consists of nine fin segments – one fin segment, shown as link BC in Fig. 8, is controlled two servomotors. Each fin segment is connected to the adjacent one by a revolute joint.

Fig. 8 Two-degree-of-freedom mechanisms constitute an undulating fin.

The analysis of the fin will be simplified to one fin segment, which is controlled by two servomotors A and D whose angular displacements with respect to x-axis are specified by ??? and ??? as shown in Fig. 8. Bear in mind that the linkage in Fig. 1 has two degrees of freedom and BC is a link whose length varies according to the orientations of crank AB and DC.
Consider the enlarged fin segment in Fig. 8 moving forward parallel to x-axis at constant speed V, and the two servomotors performing harmonic oscillations according to the following functions:

Anderson et al. [14] suggest that under these conditions, the fin segment is subject to time-varying force K(t). The servomotors have to provide torque inputs of Q1(t) and Q2(t). If m is the period of oscillation of the fin segment, let F be the time-averaged value of K(t), and P the average input power per cycle:

The time-average useful force along x- axis, Thrust T, is given by:

Collectively, for n number of segments, T assumes greater value which is theoretically nT; in this case any hydrodynamic interaction is ignored as propulsive waves propagate along the fins. The propulsive efficiency, ?, is defined to be the ratio of useful power over input power P, as

Despite common undulatory swimming technique, each of the species has a unique way in making propulsive wave by using their body and/or fin(s) – hereinafter called swimming gaits – depending on their body stiffness, shape and length of fin(s), and many more factors.
The most well known example of a species with undulatory swimming gait is eel; other examples include shark and needlefish, which are all anguilliform swimmers. They undulate from one-third to almost all of their bodies, depending on the speed, often with one or more complete waves present at a time. Lighthill [15] proposes (8) to represent body undulations of a swimming eel shown in Fig. 9:

where A is the tail beat amplitude, ? is a parameter defining how fast the amplitude grows from head to tail, k defines how many number of waves present (equal to ???wave length), and V is the body wave propagation speed.

Fig. 9 (a) Coordinate systems for elongated body modeling. (b) Amplitude envelope defining the boundary of undulations. (c) Eel swimming from position 1 to position 3 with undulation amplitude growing from head to tail.

Fig. 10 Morphology of stingray and estimated amplitude envelope of its pectoral fins.

Here, we are interested in applying Lighthill’s proposal of the swimming gait of eel – undulation amplitude grows from head to tail – to the present mechanical system. Interested reader may refer to [15] for a detailed discussion on elongated body theory.
A different yet interesting swimming gait is found in rajiform swimmer such as stingray. Thrust is produced by large undulations along pectoral fins, which span from the anterior to the posterior of the fish. The amplitude envelope of the undulations increases from the anterior part to the fin apex and decreases toward the posterior as shown in Fig. 10. Such amplitude envelope can be explained from its triangular-shaped pectoral fins; they are narrower toward anterior and posterior of the fish. The swimming of stingray is often likened to the flight of birds due to the design of the fins that allows them to be flapped up and down.
Cuttlefish, although not included by Breder’s classification, is also observed using its fin for low speed swimming. Besides using water jet propulsion, some wide-finned cuttlefish however swim like a rajiform swimmer i.e. by passing large amplitude undulations along their fins and also flapping their fins to for high-speed locomotion. The amplitude envelope of the fin undulations of those cuttlefish is similar to that of rajiform swimmer as shown in Fig. 3. It is however observed that in smaller and narrow-finned cuttlefish, the amplitude envelope is fairly constant along the fins, as shown in Fig. 11.

 

Fig. 12 One module of undulating fin consisting of ten equally spaced servomotors attached to a lightweight structure, ten cranks representing ten fin rays, and nine fin segments.
In order to investigate which of the suggested amplitude envelope provides the maximum efficiency for undulating fin propulsion, one module of undulating fin (Fig. 12) was designed and built. The fin can accommodate 10-cm peak-to peak amplitude, various wave speeds, amplitude envelopes, and wavelengths. The servomotors are programmable and can accept various time-dependent and time-independent inputs that allow the fin to act as a propulsion system and a control surface [16], respectively. Being modular, some of the fin segments can be detached from the module, and hence changing the effective fin length. Biorobot counterparts of various fish swimming with undulating fin(s) were considered, such as bowfin, south american electric fish, knifefish, and cuttlefish, to name a few.
A. Gymnotus Carapo Robot
Due to the nature of the mechanical system being modular, some experiments was successfully conducted to mimic an underwater species, called gymnotus carapo or south american electric fish which swims with a long-based anal fin by constructing a biomimetic underwater robot shown in Fig. 13. Two experiments were done: first experiment used 12 cm fin width, and the second one 20 cm.
Although no quantitative data was obtained from the experiment, it was however found that the robot moves faster with wider fin. Wave amplitude and speed were also found to be proportional to the thrust produced by the undulating fin. At this point in time, no data has yet determined the effect of wavelength on the propulsive thrust. Qualitative observations during the experiments suggest that shorter wavelength is mandatory for low speed swimming as it contributes significantly to the stability of the robot or in other words small (undetected by eye) body oscillations. Therefore depending on the fin length of the robot, it is necessary to have as many waves present along the fin as possible in order to achieve fairly stable robot for cruising at low speed.
Fig. 13 Gymnotus Carapo robot.
It must be noted however that the theory proposed earlier in this report is unable to predict which envelope shape gives highest efficiency because of neglected hydrodynamic interaction when the fin segments oscillate; the theory only provides qualitative insights. From the experiments done on the first prototype, biomimetic of gymnotus carapo, it can be established that thrust T is:

where w is the fin width, A wave amplitude, V wave speed, and ? defines wavelength of undulations.
B. Untethered Cuttlefish Robot
The encouraging results from the first experiment drove the research further to building a second prototype mimicking a cuttlefish, a marine animal that rhythmically undulates its fins for locomotion and maneuvring.
Previous researches in the area of undulating fin include the development of a 1-m long by 1/3-m deep undulating fin device actuated by parallel bellows which allow fin bending [17], and the development of a 53-cm long by 5-cm deep rigid ribbon fin structure used by a black ghost knifefish [18]. Our emphasis is to establish a research platform in a form of a robot consisting of a pair of undulating fins with variable amplitude envelope, peak-to-peak amplitude, wavelength, fin length, and speed.
Instead of modeling the fin muscle of the cuttlefish, its fins were designed with a support structure similar to that of ray-finned fish. One fin length was divided into nine segments driven by ten digital servomotors representing the muscles at the base of fin rays. In order to achieve 10-cm peak-to peak amplitude of undulation when actuators were positioned every 90° along the wavelength, the length of each segment was determined to be 9 cm. The total fin length is therefore 81 cm, with a fin width of 20 cm. One pair of fins was fabricated and assembled together with other structural elements to form a biomimetic cuttlefish robot.

The cuttlefish robot was built completely untethered by having on board power supply and controller. Twenty servomotors (Futaba S3801) were controlled by a customized control board comprising six microcontrollers (Basic Stamp BS2SX), which were arranged and synchronized in multilevel communications for efficient programming. The microcontrollers proved to be sufficient in performing all the required calculations, which accommodate almost all experimental variables such as amplitude envelope, peak-to-peak amplitude, speed, and wavelength, in 15 ms.
VI. CONCLUSION AND FUTURE WORK
Although all the fin rays seem to oscillate independently, they are however driven by the same sinusoidal function but at different phase lead or lag. A mechanism can be developed to enable the fin rays to oscillate sinusoidally at a certain phase lead or lag, and therefore reducing the number of actuators, cost, and weight of prototype. In addition to actuators, it is interesting to take note studies done by [19] which suggests the possibility of tissue fibres in the fin of cuttlefish storing elastic energy during fin bending, thus allowing the fin to function as a harmonic oscillator and thereby increasing the efficiency of the fins during locomotion. More thorough study on cuttlefish locomotion must be done in order to render proper and efficient control action on the current robot.
Due to the nature of the fin being made of acrylic, a rigid material, and independent control of each fin ray, the application of this fin is not limited only to oscillating fins. Besides sinusoidal function with time-dependent input, the robot microcontroller also accept other time-independent functions which allow the fin of the robot to generate a particular curve, depending on input function, which can be used as control surface while flying in water [16]. All fish, including cuttlefish, squid and ray, use their fins as control surface to maneuver when they have gained speed.
The future of biomimetic application relies on the advanced development of actuators. Emerging biologically inspired materials such as Electro-active Polymer (EAP) and shape-memory polymer (SMP) suggest avenues worth pursuing which then may replace current motor-based actuators, such as servomotors. EAP has a potential in modeling the fish muscles, including undulating fins [20], [21].
Despite the absence of quantitative data from the initial experiments, the results obtained from many observations in the experiments encourage and inspire us to proceed further in investigating factors contributing to propulsive thrust and efficiency of the undulating fins.
Future research works will develop other variants of biomimetic robots utilizing the modular mechanical fin and investigate the meneuverability and control aspect of a biomimetic robot with only one undulating fin. Future focus will be on data acquisition from experiments, which will be used to analyze the deployability – feasibility for deployment – of the cuttlefish robot.

ACKNOWLEDGMENTS

The authors wish to thank Dr. F.M.J. Nickols for his valuable feedback in the early stage of the development of the first prototype. He also generously provided all waterproofed casings for the servomotors used in the cuttlefish robot. We thank technicians of Computer Numerical Control Laboratory for assistance in manufacturing components of the cuttlefish robot. We also want to acknowledge the support from Professor Gerald Seet, the Director of Robotics Research Center (RRC), and the technicians at the RRC on water tank, electronic equipment, and other facilities.
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Undulating Fins

Undulating Side Fins