# Dynamics and simulation of biorobotic UUV/Bioroboto (UUV) dinamika IR modeliavimas.

1. Introduction

Natural swimmers have evolved to beautifully utilize physical principles from unsteady hydrodynamics to achieve high maneuverability and efficiency. Man-made unmanned underwater vehicles (UUVs), on the other hand, have been conceived and operated for decades to either remain safely within the realm of steady hydrodynamics--where the design of vehicle body, actuation mechanisms, and control system is simpler to understand and implement --or to avoid the issue altogether by using a number of thrusters to push an arbitrarily-shaped body through the water. This has often resulted in efficient cruising, but not in efficient maneuvering [1].

A number of numerical investigations about the dynamics of the biorobotic UUV. Mukund Narasimhan and Sahjendra N. Singh [2, 3] provided the open-loop control and feedback linearizing yaw plane control of a multifin biorobotic rigid underwater vehicle. WANG Zhao-li et al. [4] provided the Hydrodynamic analysis of the pectoral-fins in viscous flows, and LAO Yi-jia et al. [5] and LIU Zhen et al. [6] gives some experimental results of oscillating flexible caudal fin. Haibin Xie et al. [7] and Shao-bo Yang et al. [8] studied dynamic and kinematics of the robotic fish.

This paper presents an innovative propulsion system approach for a new vehicle, resulting in efficient maneuvering through the exploitation of unsteady hydrodynamics. The vehicle swims by coordinating the motion of four biology-inspired high-lift flapping hydrofoils that are attached to its rigid hull. Because of this connection to biology, this vehicle has been named biorobotic UUV. Section 2 introduces the dimensional motion model of biorobotic UUV. Section 3 presents the model for fluid dynamics of flapping hydrofoil. Section 4 provides the simulation results of the motion of the biorobotic UUV which is propelled and controlled by four flapping hydrofoils. Finally, section 5 draws conclusions.

2. Dimensional motion model of biorobotic UUV

A earth-fixed frame [S.sub.E] (o, [x.sub.0], [y.sub.0], [z.sub.0]), and a Biorobotic UUV body-fixed frame [S.sub.B] (B, x, y, z), as shown in Fig. 1, are defined, and their transform matrixes are [C.sup.B.sub.E], and [C.sup.E.sub.B]. The Biorobotic UUV's velocity vector is considered to be V = [[[v.sup.T], [[omega].sup.T]].sup.T], v = [[[v.sub.x], [v.sub.y], [v.sub.z]].sup.T], [omega] = [[[[omega].sub.x], [[omega].sub.y], [[omega].sub.z]].sup.T], and the earth-fixed position/orientation vector is R = [[[r.sup.T], [[THETA].sup.T]].sup.T], r = [[[x.sub.0], [y.sub.0], [z.sub.0]].sup.T], [THETA] = [[[theta], [psi], [phi]].sup.T]. The Dimensional Motion Model of Biorobotic UUV is built, as shown in Eqs. (1) and (2) [2, 9-10]. And a detailed explanation of this can be found in the literature [9] and [10].

[FIGURE 1 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The right hand term in Eq. (1) is the vector of external forces and moments, such as weight, buoyant, fluid dynamics which produced by the motion of UUV, and disturbed force which produced by the environment. In this paper, we suppose that the Biorobotic UUV motions in the enough deep, enough big and enough quiescent water, so the disturbed force which produced by the environment is ignorable. And the fluid dynamics which produced by the motion of UUV can be described by the mass forces and drag forces of the main body and the fluid dynamics of the flapping hydrofoil [11]. The mass forces and drag forces of the main body can be computed by the model which is detailed described in the literature [1] and [10]. And the model of the fluid dynamics of the flapping hydrofoil is built in the next section.

3. Fluid dynamics of flapping hydrofoil

According to Green theorem, the perturbation velocity potential [PHI](t) in inviscid, incompressible, and irrotational flow with non-uniform velocity V(x, y, z,t) of an arbitrary field point P(x, y, z,t) can be expressed as an integral on the boundary surface of flow field S, which is composed of flapping hydrofoil surface [S.sub.B], wake surface [S.sub.W] and outer control surface [S.sub.[infinity]], [11-13]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where R(P,Q) is the distance between field point P (x, y, z, t) and boundary point Q ([x.sub.0], [y.sub.0], z), [partial derivative]/[partial derivative][n.sub.Q] is normal derivative to S at point Q.

The perturbation potential [PHI](t) should satisfy the following boundary conditions:

[??][PHI](t) [right arrow] 0, [S.sub.[infinity]] [infinity]; (4)

[partial derivative][PHI](t)/[partial derivative][n.sub.Q] = -[V.sub.0] (x, y, z, t) x [n.sub.Q] on [S.sub.B]; (5)

[P.sup.+] - [P.sup.-] = 0; (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where, [n.sub.Q] is unit normal vector on the flapping hydrofoil surface, [Q.sub.t] is the point on the wake surface and superscripts + and - are used respectively to mark the values of upper and lower sides of wake.

Thus the integral Eq. (3) can be written as [11]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [DELTA][PHI] is the potential jump across the wake surface, which can be expressed by:

[DELTA][PHI](t) = [PHI][(t).sup.+]-[PHI]S[(t).sup.-]. (9)

For the unsteady problem, the velocity potential [PHI](t) changes with the time. And combining the Kutta pressure condition:

[([DELTA]p).sub.TE](t) = [p.sup.+.sub.TE] (t) - [p.sup.-.sub.TE] (t) = 0, (10)

the integral Eq. (8) can be uniquely solved by means of numerical iterative method.

The perturbation velocities V (x, y, z, t) on the flapping hydrofoil surface are evaluated by differentiating the velocity potential on the flapping hydrofoil surface. So the total velocity is:

[V.sub.t] (x,y,z,t) = [V.sub.0] (x,y,z,t) + V (x,y,z,t). (11)

By Bernoulli's theorem the pressure on the flapping hydrofoil surface can be expressed as [12]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The hydrodynamic characteristics are obtained by integrating the pressure on the flapping hydrofoil surface.

Eq. (8) is a integral for determining the potential [PHI] on the flapping hydrofoil surface SB and the normal derivative [partial derivative]/[partial derivative][n.sub.Q] on flow field S at point Q. The flapping hydrofoil surface [S.sub.B] and the wake surface [S.sub.W] can be individually divided into certain element with the number of [N.sub.B] or [N.sub.W]. And each element has a certain serial number [N.sub.j], (j = 1, 2, ..., [N.sub.B] + [N.sub.W]). The field point [P.sub.i]([x.sub.i], [y.sub.i], [z.sub.i]) is placed on the shape center of every element, and the term [PHI](P) in Eq. (8) merges into other terms. The potential at the nodes of elements can be tackled by employing the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

4. Simulation

For demonstration of the model of biorobotic UUV dimensional motion and the model for fluid dynamics of flapping hydrofoil, a numerical example of the motion of the biorobotic UUV is presented. The biorobotic UUV has a compressed body propelled and controlled by four flapping hydrofoils. The four flapping hydrofoils' collocation on the UUV is shown in Fig. 2. And the biorobotic UUV which was simulated in this paper has specifications of 10.0 m length, 3200 kg weight, 1.5 m wide.

Each flapping hydrofoil is assumed to undergo a flapping motion described as [14]:

[[theta].sub.i] = [A.sub.i] sin (2[pi] [f.sub.i]t + [[phi].sub.i]) + [[theta].sup.0.sub.i], (14)

where [[theta].sub.i] is the flapping angle of the ith hydrofoil, [A.sub.i] is the crest value of [[theta].sub.i], [[phi].sub.i] is the original phase angle, and [[theta].sup.0.sub.i] is the central angle of the flapping hydrofoil.

[FIGURE 2 OMITTED]

According the fluid dynamics model of flapping hydrofoil, the fluid dynamics of flapping hydrofoil was calculated. The Fig. 3 is the contours of pressure.

The biorobotic UUV can complete the motion of change depth through changing the central flapping angles of the two frontal flapping hydrofoils. And the biorobotic UUV also can complete the motion of change depth through changing the central flapping angles of all frontal flapping hydrofoils. The two motions of change depth are simulated.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The Fig. 4 is the depth curve and pitching angle curve when the biorobotic UUV changing depth form 10 to 20 m through changing the central flapping angles of the two frontal flapping hydrofoils. And the Fig. 5 is the depth curve and pitching angle curve when the biorobotic UUV changing depth form 10 to 20 m through changing the central flapping angles of all frontal flapping hydrofoils (Figs. 6 and 7). The simulation results indicate that the biorobotic UUV can steadily complete the motion of change depth through using the two frontal flapping hydrofoils or using all four flapping hydrofoils. The biorobotic UUV can change depth form 10 to 20 m in 300 s through using the two frontal flapping hydrofoils and in 170 s through using all four flapping hydrofoils, so the biorobotic UUV has excellent maneuverability at low velocity, and especially the biorobotic UUV can expedite change depth through all four lapping hydrofoils.

5. Conclusions

The model of biorobotic UUV dimensional motion and the model for fluid dynamics of flapping hydrofoil were presented in this paper for analyzing the dynamics of biorobotic UUV. The simulation results show that:

1) the biorobotic UUV can steadily complete the motion of change depth through using the two frontal flapping hydrofoils or using all four flapping hydrofoils,

2) the biorobotic UUV has excellent maneuverability at low velocity.

Received April 25, 2012

Accepted September 05, 2013

References

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DU Xiao-xu, Northwestern Polytechnical University, Youyi Xilu 127,710072 Xi'an, China, E-mail: nwpudxx@163.com

SONG Bao-wei, Northwestern Polytechnical University, Youyi Xilu 127,710072 Xi'an, China, E-mail: songbaowei@nwpu.edu.cn

PAN Guang, Northwestern Polytechnical University, Youyi Xilu 127,710072 Xi'an, China, E-mail: panguang601@163.com

http://dx.doi.org/10.5755/j01.mech.19.5.5531>
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