Kinematics Modeling and Experiments of Pectoral Oscillation Propulsion Robotic Fish

Journal of Bionic Engineering 6 (2009) 174–179

Kinematics Modeling and Experiments of Pectoral Oscillation Propulsion Robotic Fish Shao-bo Yang, Jing Qiu, Xiao-yun Han Institute of Mechatronical Engineering and Automatization, National University of Defense Technology, Changsha 410073, P. R. China

Abstract A robotic fish driven by oscillating fins, “Cownose Ray-I”, is developed, which is in dorsoventrally flattened shape without a tail. The robotic fish is composed of a body and two lateral fins. A three-factor kinematic model is established and used in the design of a mechanism. By controlling the three kinematic parameters, the robotic fish can accelerate and maneuver. Forward velocity is dependent on the largest amplitude and the number of waves in the fins, while the relative contribution of fin beat frequency to the forward velocity of the robotic fish is different from the usual result. On the other hand, experimental results on maneuvering show that phase difference has a stronger effect on swerving than the largest amplitude to some extent. In addition, as propulsion waves pass from the trailing edge to the leading edge, the robotic fish attains a backward velocity of 0.15 m·sí1. Keywords: robotic fish, pectoral oscillation propulsion, largest amplitude, number of waves, fin beat frequency, phase difference Copyright © 2009, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(08)60114-6

1 Introduction Swimmers which mainly use pectoral fins for swimming, use the fins in two ways: undulation and oscillation. Undulation of the pectoral fins, termed “rajiform” locomotion[1], is characterized by having more than one complete wavelength on the fins at a time[2]. Oscillation of the pectoral fins, termed “mobuliform” locomotion[2], is similar to flapping of birds’ wings; the fins move up and down with less than half a wavelength along the fins. In 1979, Blake classified oscillation of the pectoral fins into two main types[3]: a rowing action (drag-based mode) and a flapping action (lift-based mode). According to Vogel[4], each oscillation type has its own advantages. Drag-based mode is more efficient at low speed, while lift-based mode is more efficient at high speed. These two types differ in the direction of pectoral oscillation. When a fish swims using the drag-based mode, the direction of pectoral oscillation is almost parallel to the direction of swimming of the fish. While, when the fish swims using the lift-based mode, the oscillation direction is perpendicular to the swimming direction. Corresponding author: Shao-bo Yang E-mail: [email protected]

Fish using the lift-based mode, such as the Manta ray, are extremely maneuverable and very efficient during cruising. Similar to the Manta ray, the Cownose ray (Rhinoptera bonasus) (Fig. 1) belongs to the Myliobatiformes, cartilaginous fish characterized by a dorsoventrally compressed rhombic profile and an elongated tail. Its pectoral fins are greatly enlarged and fused to the cranium, forming large, highly modified, wing-like structures. In order to gracefully propel itself, the Cownose ray moves its fins up and down in a plane that is roughly perpendicular to the main axis of the body (Fig. 2).

Fig. 1 Cownose ray (Heine[5]).

Yang et al.: Kinematics Modeling and Experiments of Pectoral Oscillation Propulsion Robotic Fish

500 mm. After the robotic fish is adjusted to neutral buoyancy, the total weight is about 1 kg.

z O

x y

175

t=0

t = T/2

t = T/6

t = 2T/3

t = T/3

t = 5T/6

Fig. 2 Steady swimming of Cownose ray.

Previous studies of the Cownose ray were focused on the kinematics of the specimen. Heine[5] documented the kinematics of the Cownose ray, and concluded that the only variable strongly correlated with swimming speed was the maximum speed of the upstroke of the wing tip. Rosenberger[6] explored the dichotomy between undulatory and oscillatory locomotion by comparing the kinematics of pectoral fin locomotion in eight species of botoids that are different in their swimming behavior, phylogenetic position and lifestyle. Kinematic data showed that R. bonasus increases swimming velocity by increasing wave speed and fin-tip velocity. Natural selection has ensured that the mechanical systems evolved in the Cownose ray are of high efficiency. The ability to maneuver and the advantage of noiseless propulsion could be of additional significance. Recently, biologists have shown much interest in this area and have shed new light especially on the working median or paired fins, such as pectoral propulsion[7–13]. In this study, we introduce a robotic fish, “Cownose Ray-I”. We then discuss in detail the relative contributions of kinematic variables such as the number of waves, the fin beat frequency and the largest amplitude of fin ray to the robot’s forward velocity and maneuverability.

2 Material and methods 2.1 Structure of “Cownose Ray-I” The robotic fish[14] (see Figs. 3a and 3b) based on the principle of “likeness in shape” is dorsoventrally flattened without a tail, including a main body and two pectoral fins. The body is made of aluminum, while the triangular pectoral fins are made of elastic silica gel. The length (distance from snout to posterior disc margin) of the robotic fish is 300 mm, and the span (distance between the two fin-tips when the pectoral fins are flat) is

(a) Design of robotic fish

(b) Model of robotic fish

Fig. 3 “Cownose ray-I” robotic fish.

The work units and the joints between the motors and the fin rays are modular. The robot is powered by eight directly jointed eudipleural Futaba 3003 servomotors. When each motor on the pectoral fins is in single control according to the given rule, a propulsion wave will travel along the chord length. The location of the servo-motors can be adjusted along the length of the body, so the center of gravity of the body and the arrangement of the pectoral fins are also adjustable. 2.2 Kinematic modeling A kinematic model is necessary for an in-depth study of propulsion, and it is also a useful bridge between fish and machine. Assumptions for the kinematic model are as follows: (1) The thickness of the pectoral fin is ignored; (2) The fin rays are perpendicular to the direction of the main axis of the body; (3) The model has no tail. In the robotic fish, the propulsive waves along the pectoral fins are generated by controlling the oscillation of the fin rays. Therefore, the kinematics of the pectoral fins may be described in terms of the oscillation up and down of the fin rays. Moreover, it can be supposed that the plane formed by the oscillation of the fin rays is

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perpendicular to the main axis of the body during swimming. The local reference frame in the fin rays is set up (see Fig. 3a): the origin is at the juncture of the fin rays and the servo-motors, and the positive direction of the x-axis points to the front of the model. 2.3 Experimental setup In the water tank (length × width × height: 4.0 m × 1.8 m × 1.5 m), the depth of the water is 0.6 m and the water is kept still before the robotic fish swims. A DCR-TRV60E Sony video camera was used to record the swimming sequences of the robotic fish, and each frame was processed to analyze its movement. Fig. 4 is taken from a typical swimming sequences.

3 Results 3.1 Kinematic modeling The oscillation kinematic model of the pectoral fins can be incorporated into the formulation of the fin rays. It is assumed that the fin rays oscillate in a sine wave, and the angle between the fin rays and the y-axis is ș(t). Therefore, the kinematic formula of a single fin ray is given by

Ti  t Ti 0  T max sin(2ʌft  (i  1)M ) ,

(1)

where și(t) is the amplitude of the ith fin ray, și0 is the original angle of the ith fin ray, șmax is the largest amplitude of the fin ray, f is the beat frequency of the fin rays, and ij is the phase difference between the adjacent fin rays. The rotating speed of the ith fin ray is

Z

2ʌf T max cos(2ʌft  (i  1)M ) .

(2)

Then, the largest tangential speed of the fin rays in the plane parallel to the oxz plane is Vmax

Fig. 4 Robotic fish swimming in the tank.

Because of the phase difference between neighboring fin rays, the sine wave along the pectoral fins travels from the leading edge to the trailing edge. As the propulsive wave propagates backward, the water around the pectoral fins reacts to propel the robotic fish forward. The velocity is dependent on the hydrodynamic force, which is related to the shape of the pectoral fins and the swimming mode. However, the largest amplitude of the fin rays may affect the shape of the pectoral fins. The number of waves, the phase difference between the adjacent fin rays, and the fin beat frequency may change the speed of the propulsive wave. Therefore, the forward velocity is a function of these three kinematic variables. To explore how the three variables control the swimming of the robotic fish, we studied forward swimming (by symmetrical oscillation of the two pectoral fins in the number of waves, fin beat frequency and the largest amplitude) and maneuverability (swerving and backward swimming by asymmetric oscillation of the two pectoral fins).

2ʌfbT max ,

(3)

where b is the maximum length of the fin rays. As the rotating phase of the fin rays along the same pectoral fin follows in order, the propulsive wave with a wave number less than 0.5 moves along the pectoral fins during swimming. However, the length of the wave is restricted by the dynamic response of the fin rays. For example, for the sine input, the restriction to the rotating speed of fin rays is[15]:

Z d

nr Tm , D  nr2 K m2 / Ra

(4)

where nr is the ratio of rotating speed of the servo-motor, Tm is the torque output of the motor, D is the total frictional damping, Km is the restoring torque of the motor, and Ra is the resistance of the armature. If the rotating speed accords with Eq. (4), the fin rays should follow the sine input. Once the rotating speed exceeds the restriction, the largest amplitude of the fin rays will be greatly attenuated. 3.2 Kinematic experiments 3.2.1 Surging When the robotic fish is swimming in the water tank, the maximum amplitude of the fin rays is changed from 15 degree to 60 degree, and the number of waves is

Yang et al.: Kinematics Modeling and Experiments of Pectoral Oscillation Propulsion Robotic Fish

changed from 0.1 to 0.5, and the fin beat frequency from 0.5 Hz to 2 Hz. To explore the relative contribution of any of the three variables to the forward velocity, we discuss how the three variables influence the forward velocity while other parameters are kept constant (see Fig. 5): (1) Relative contribution of the number of waves to forward velocity N is the number of waves present along the fin in a period, so that

N

L/O,

(5)

tween 0.8 Hz to 1 Hz the velocity of the robotic fish is nearly constant. As the fin beat frequency exceeds 1 Hz the forward velocity decreases dramatically. (3) Relative contribution of the maximum amplitude to the forward velocity The forward velocity of the robotic fish increases with the maximum amplitude of the fin rays, when which is smaller than 45Û. As the maximum amplitude further increases from 45Û to 60Û, the forward velocity is almost constant. 3.2.2 Maneuverability The asymmetric oscillation of the two pectoral fins controls swerving. First, the difference of the largest amplitude of the two pectoral fins is explored. With fin beat frequency of 1 Hz and the wave number of 0.4, when the maximum amplitude of the left pectoral fin is 45Û and that of the right is 9Û, the robotic fish turns right with the rotating speed of 22.5Û per second and a gyration radius of one body length. Second, the effect of phase difference between the adjacent fin rays is explored. With fin beat frequency of 1Hz and the largest amplitude of 45Û, the robotic fish finishes the gyration of 360Û with the rotating speed of 45Û per second without the gyration radius. In this case, the propulsion waves along the left and the right pectoral fins propagate in the reverse direction, which illustrates the excellent maneuverability of the robotic fish.

The largest amplitude șmax (degree)

Number of waves N / Flapping frequency f (Hz)

where L is the largest chord length of the pectoral fins and Ȝ is the wavelength of propulsive wave along the pectoral fins. In the study, the number of waves can be controlled by the phase difference between adjacent fin rays with ij = 360Û × N/3. For example, when the number of waves along the pectoral fin is 0.4, the phase difference is 48Û. As the wave number of the Cownose ray is less than 0.5 when cruising, the maximum wave number in our experiment does not exceed 0.5. When N ” 0.4 the forward velocity increases with N. However, as N increases from 0.4 to 0.5 the forward velocity decreases.. (2) Relative contribution of fin beat frequency to the forward velocity First, the forward velocity increases with the fin beat frequency. Then, as the fin beat frequency is be-

177

Fig. 5 Effect of the three kinematic parameters on the forward velocity.

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In addition, at a fin beat frequency of 1 Hz (maximum amplitude of 45Û and wave number of 0.4), as the propulsive waves pass from the trailing to the leading edge, the robotic fish attains a forward velocity of 0.15 m·sí1, which is faster than the forward velocity 0.13 m·sí1 in the same case. The maneuverability of the robotic fish is summarized in Table 1. Table 1 Maneuverability of robotic fish Phase difference (degree)

Largest amplitude (degree˅

Left fin Right fin

Left fin Right fin

Swimming action

Rotating speed (degree·sí1) /speed (m·sí1)

48

48

45

9

Turn right and forward surging

23

48

48

9

45

Turn left and forward surging

23

48

í48

45

45

Turn right without gyration radius

45

í48

48

45

45

Turn left without gyration radius

45

í48

í48

45

45

Backward surging

í0.15

motors will run over the restriction given in Eq. (4), and the dynamic response of the fin rays will lead to the decrease in the largest amplitude as well as the forward velocity. Finally, as the largest amplitude increases from 45Û to 60Û, the forward velocity is nearly constant because the interaction between adjacent fin rays restricts the amplitude of movement of the fin rays. Besides, the experiment of maneuverability shows that the phase difference of the propulsive wave along the two pectoral fins will cause the robotic fish to swerve, as well as the difference of the largest amplitude of the two pectoral fins. In addition, the robotic fish attains a forward velocity of 0.15 m·sí1, larger than the forward velocity of 0.13 m·sí1 in the same case, which may result partly from the over restriction of the weight of the wires controlling the swimming of the robotic fish, and partly from the different shapes of the leading edge and the trailing edge devoting to the different model’s speeds.

4 Discussion

5 Conclusions

Both the wave number and the largest amplitude have strong effect on the forward velocity. In Eq. (3), șmax v Vmax, the largest amplitude will increase with the tangential speed of the fin rays, fin-tip speed. It seems as if the forward velocity increases with fin-tip speed. Therefore, the fin-tip speed may cause larger forward velocity of the fish with pectoral oscillation propulsion, which agrees with the observations of the forward velocity of the Cownose ray by Heine[5] in 1992. Nevertheless, the fin beat frequency does not have the same effect on the fish using pectoral oscillatory propulsion. The results of the study are partially different from the conventional results, which are presented as follows. First, as the wave number increases to 0.5, the forward velocity becomes less. Because the interaction between the adjacent fin rays on the pectoral fins increases with the number of waves, the pectoral fins will be tightened and the largest amplitude of the fin rays will decrease. Second, when the fin beat frequency is between 0.8 Hz and 1 Hz the forward speed will be greater than at any other frequency. At lower frequency the robotic fish heaves up and down much more, therefore the lift force needed to heave will increase and the propulsive force and forward velocity will be reduced. By contrast when fin beat frequency increases to 2 Hz the frequency of the

In the study, a robotic fish propelled by oscillation of pectoral fins, “Cownose Ray-I”, is developed. When the rigid fin rays are driven by eight eudipleural servo-motors producing a sine wave, a propulsion wave with the wave number of less than 0.5 is propagated along the fins. In the water, the robotic fish is capable of freely surging forward and backward, as well as rapidly swerving without the gyration radius. The relative contribution of each of the three kinematic variables to the forward velocity is explored while the others are kept constant. The speed of robotic fish is obviously dependent on the largest amplitude and the wave number as well as phase differences between the adjacent fin rays, while the fin beat frequency has a different effect on the fish using the pectoral oscillatory propulsion. The results of the study are partly different from conventional results, which may result from the restriction of the structure of fin rays and the function of the servo-motors. The swerving of the robotic fish is achieved by the asymmetric phase of left and right pectoral fins, which implies that the effect of the phase difference of the propulsive wave between two pectoral fins on swerving is stronger than that of the asymmetric largest amplitude of the two pectoral fins. It is obvious that the slower forward velocity of the

Yang et al.: Kinematics Modeling and Experiments of Pectoral Oscillation Propulsion Robotic Fish

robotic fish in the study and the defects existing in the model design suggest that the model could be improved. The frequency analysis and undulating study for a loaded flexible beam model[16–19] should be explored. Also, the hydrodynamic forces on the model swimming[20] should be investigated.

Acknowledgement The authors would like to thank Dr. Dai-bing Zhang, Dr. Hai-bing Xie and Mr. Long-xin Lin for their assistance in the experiments. The supports of National Natural Science Foundation of China (No.50405006) and the supports of the innovation foundation of graduate students of National University of Defense Technology (No.B060302) are also gratefully acknowledged.

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