Effect of flapping orientation on caudal fin propelled bio-inspired underwater robots

Abstract

Aquatic animals and mammals in nature, in particular, the Body and/or Caudal Fin (BCF) swimmers swim either by flapping their fins in the sideways direction or the dorso-ventral direction. Not much literature is available on the effects of the performance of these robots based on the choice of its flapping orientation. In this research, it is found that dorso-ventral flapping could lead to better self-stabilizing effects and lesser energy consumption compared to sideways flapping. It is also found that the choice of dorso-ventral flapping offers the possibility of controlling the body’s oscillation amplitude while flapping. This is an appealing advantage for underwater surveying robots carrying cameras and sensors as controlled body oscillations could yield better results from its payloads. The main body of results is obtained with simulations for underwater vehicle dynamics with the coefficients of the REMUS underwater vehicle, while stability analysis for a generalised case is also presented.

Appendices

Appendix A

See Tables 1 and 2.

Table 2 Descriptions of all the symbols used in the equations

Appendix B

Stability for a generalised case

The study conducted in the article for stability in sideways flapping and dorso-ventral flapping is for a specific set of body shape parameters. For the sake of completeness, a generalised case is also presented here for obtaining the relation between stability and body hydrodynamic coefficients of underwater robots. This analytical derivation is adapted Edward V. Lewis 1989 (Lauder 2015).

Consider a swimmer following a straight-line motion confined to a single plane (e.g., considering forces acting only in the x and y plane—see Fig. 1). The forces X (surge), Y (sway) and N (yaw) in different directions equations can be written as:

$$ Y = F_{Y} \left( {u, v, \dot{u}, \dot{v}, r, \dot{r}} \right) $$

(14a)

$$ N = F_{N} \left( {u, v, \dot{u}, \dot{v}, r, \dot{r}} \right) $$

(14b)

where u is the velocity in the x-direction, v is the velocity in the y-direction, $\dot{u}$ is the acceleration in the x-direction, $\dot{v}$ is the acceleration in the y direction, r is the angular velocity about yaw axis, $\dot{r}$ is the angular acceleration about the yaw axis.

For obtaining the expression defining the dynamic stability of the system, the above equations are reduced to useful mathematical expressions using Taylor series expansions about a given initial equilibrium point u₁, v₁, $\dot{u}_1,$$\dot{v}_1,$r₁, $\dot{r}_1$ as shown below:

$$ X = F_{X} \left( {u_{1} , v_{1} , \dot{u}_{1} , \dot{v}_{1} , r_{1} , \dot{r}_{1} } \right) + \left( {u - u_{1} } \right)\frac{\delta X}{\delta u} + \left( {v - v_{1} } \right)\frac{\delta X}{\delta v} + \ldots + \left( {\dot{r} - \dot{r}_{1} } \right)\frac{\delta X}{{\delta \dot{r}}} $$

(15a)

$$ Y = F_{Y} \left( {u_{1} , v_{1} , \dot{u}_{1} , \dot{v}_{1} , r_{1} , \dot{r}_{1} } \right) + \left( {u - u_{1} } \right)\frac{\delta Y}{\delta u} + \left( {v - v_{1} } \right)\frac{\delta Y}{\delta v} + \ldots + \left( {\dot{r} - \dot{r}_{1} } \right)\frac{\delta Y}{{\delta \dot{r}}} $$

(15b)

$$ N = F_{n} \left( {u_{1} , v_{1} , \dot{u}_{1} , \dot{v}_{1} , r_{1} , \dot{r}_{1} } \right) + \left( {u - u_{1} } \right)\frac{\delta N}{\delta u} + \left( {v - v_{1} } \right)\frac{\delta N}{\delta v} + \ldots + \left( {\dot{r} - \dot{r}_{1} } \right)\frac{\delta N}{{\delta \dot{r}}} $$

(15c)

For a body moving with a uniform velocity, initially $\dot{u}_1,$$\dot{v}_1,$ r₁, $\dot{r}_1,$ vanish because there is no acceleration while at equilibrium when moving with constant speed. Moreover, since most AUV’s are symmetric about the x–z plane, the cross-flow terms $\partial{y}/ \partial{u}=\partial{y}/ \partial \dot{u}=0$

Therefore, a change in forward velocity or acceleration does not produce any transverse force on the swimmer, and hence this term is neglected. Since we are assuming that the swimmer moves with uniform velocity in x-direction, there can be no transverse force acting on the swimmer at the initial state i.e., $F_{y}$$\left( {u_{1} ,v_{1} ,\dot{u}_{1} ,\dot{v}_{1} ,r_{1} , \dot{r}_{1} } \right)$=0. Thus, equation (15a, 15b, 15c) can be reduced to,

$$ Y = \left( {\frac{\partial Y}{\partial v}} \right)v + \left( {\frac{\partial Y}{{\partial \dot{v}}}} \right)\dot{v} + \left( {\frac{\partial Y}{\partial r}} \right) r + \left( {\frac{\partial Y}{{\partial \dot{r}}}} \right)\dot{r} $$

(16a)

Similarly, the yawing moment can be written as:

$$ N = \left( {\frac{\partial N}{\partial v}} \right)v + \left( {\frac{\partial N}{{\partial \dot{v}}}} \right)\dot{v} + \left( {\frac{\partial N}{\partial r}} \right) r + \left( {\frac{\partial N}{{\partial \dot{r}}}} \right)\dot{r} $$

(16b)

Since we are considering the effect of transverse disturbance, the relevant terms are only the sway force and yaw moment. Hence, the effect on surge force is not discussed here. The equations obtained here are for the forces with respect to the axis fixed on the vehicle; hence their effect on the acceleration is given below:

$$ X = M \left( { \dot{u} - v r } \right) $$

(17a)

$$ Y = M \left( { \dot{v} + u r} \right) $$

(17b)

$$ N = I_{Z} \dot{r} $$

(17c)

Substituting (16a) and (16b) in (17a) and (17b) and simplifying, we get the following equations,

$$ - \left( {\frac{\partial Y}{\partial v}} \right)v + \left( {M - \frac{\partial Y}{{\partial \dot{v}}}} \right)\dot{v} - \left( {\frac{\partial Y}{\partial r} - M u_{1} } \right)r {-} \left( {\frac{\partial Y}{{\partial \dot{r}}}} \right)\dot{r} = 0 $$

(18a)

$$ - \left( {\frac{\partial N}{\partial v}} \right)v - \left( {\frac{\partial N}{{\partial \dot{v}}}} \right)\dot{v} - \left( {\frac{\partial N}{\partial r}} \right)r + \left( {I_{Z} - \left( {\frac{\partial N}{{\partial \dot{r}}}} \right)} \right)\dot{r} = 0 $$

(18b)

In Eq. (18a), all the terms have units of force and in Eq. (18b) all the terms have units of the moment. Non-dimensionalizing by dividing the first equation with $\left( {\rho /2} \right)L^{2} V^{2}$ and second equation with $\left( {\rho /2} \right)L^{3} V^{2}$ and using primed symbols for non-dimensionalized terms this yields:

$$ - Y_{{v{\prime }}} v{\prime } + \left( {M{\prime } - Y_{{\dot{v}}} } \right)\dot{v}{\prime } - \left( {Y_{{r{\prime }}} - M{\prime }} \right)r{\prime } - Y_{{\dot{r}{\prime }}} \dot{r}{\prime } = 0 $$

(19a)

$$ - N_{{v{\prime }}} v{\prime } - N_{{\dot{v}}} \dot{v}{\prime } - N_{{r{\prime }}} r{\prime } + \left( {I_{z}^{{\prime }} - N_{{\dot{r}{\prime }}} } \right)\dot{r}{\prime } = 0 $$

(19b)

where,

$$ M{\prime } = \frac{M}{{\rho \frac{{L^{3} }}{2}}};v{\prime } = \frac{v}{V};\dot{v}{\prime } = \frac{{\dot{v}L}}{{V^{2} }};I_{z}^{{\prime }} = \frac{{I_{Z} }}{{\rho \frac{{L^{2} }}{2}}};r{\prime } = \frac{rL}{V};\dot{r}{\prime } = \frac{{\dot{r}L^{2} }}{{V^{2} }} $$

$$ Y_{{v{\prime }}} = \frac{{Y_{v} }}{{\left( {\frac{\rho }{2}} \right)L^{2} V}};Y_{{r{\prime }}} = \frac{{Y_{r} }}{{\left( {\frac{\rho }{2}} \right)L^{3} V}};N_{{v{\prime }}} = \frac{{N_{v} }}{{\left( {\frac{\rho }{2}} \right)L^{3} V}};N_{{r{\prime }}} = \frac{{N_{r} }}{{\left( {\frac{\rho }{2}} \right)L^{4} V}} $$

$$ Y_{{\dot{v}{\prime }}} = \frac{{Y_{{\dot{v}}} }}{{\left( {\frac{\rho }{2}} \right)L^{3} }};Y_{{\dot{r}{\prime }}} = \frac{{Y_{{\dot{r}}} }}{{\left( {\frac{\rho }{2}} \right)L^{4} }};N_{{\dot{v}{\prime }}} = \frac{{N_{{\dot{v}}} }}{{\left( {\frac{\rho }{2}} \right)L^{4} }};N_{{\dot{r}{\prime }}} = \frac{{N_{{\dot{r}}} }}{{\left( {\frac{\rho }{2}} \right)L^{5} }} $$

(Y_v represents ∂Y/∂v, N_v represents ∂N/∂v and so on).

Using only these linear terms, solutions to yaw and sway terms can be obtained to analyse the swimmer’s stability. Equations B.6(a) and B.6(b) are two simultaneous differential equations of the first order in two unknowns—the transverse velocity component v’ and the yaw angular velocity component r’. The standard solutions of the first-order equation are as follows:

$$ v{\prime } = V_{1} {\text{e}}^{{\sigma_{1} t}} + V_{2} {\text{e}}^{{\sigma_{2} t}} $$

(20a)

$$ r{\prime } = R_{1} {\text{e}}^{{\sigma_{1} t}} + R_{2} {\text{e}}^{{\sigma_{2} t}} $$

(20b)

where new parameters V₁, V₂, R₁ and R₂ are constants of integration; $\sigma_{1}$ and $\sigma_{2}$ are known as stability indices.

In the above equation, if both $\sigma_{1}$ and $\sigma_{2}$ are negative, both v′ and r′ will approach zero with the passage of time. In this case, the swimmer would reach a straight-line motion without yawing or drifting in the transverse direction. However, if both of them are positive, v′ and r′ will increase continuously over time leaving the swimmer spiralling and never returns back to its original course. The stability of the vehicle also depends on the magnitude of these indices. If they have high magnitude and are both negative then the vehicle is more stable and will reach the straight-line motion more rapidly.

These indices can be found out by substituting equations B.7(a) and B.7(b) in equation B.6, and simplification gives a quadratic equation in σ as shown below:

$$ A \sigma^{2} + B \sigma + C = 0 $$

(21)

where

$$ A = \left( {I_{z}^{{\prime }} - N_{{\dot{r}{\prime }}} } \right)M{\prime }\dot{v}{\prime };B = - \left( {I_{z}^{{\prime }} - N_{{\dot{r}{\prime }}} } \right)Y_{{v{\prime }}} - M{\prime }N_{{r{\prime }}} ; $$

$$ C = Y_{{v{\prime }}} N_{{r{\prime }}} - \left( {Yr{\prime } - M{\prime }} \right)N_{{v{\prime }}} $$

By finding out these values of A, B and C we can obtain the stability indices as shown below:

$$ \sigma_{1,2} = \frac{{ - B \pm \sqrt {B^{2} - 4AC} }}{ 2A} $$

(22)