Abstract:
An underwater vehicle uses an undulatory fin propulsion system and a control apparatus to receive sensor data and control the operation of the vehicle in real time over a wide range of forward velocities. In an embodiment, a model of propulsive efficiency is used to achieve high values of propulsive efficiency, giving a lowered energy drain on the battery. Externally monitored information, such as that on flow velocity, is conveyed to the control apparatus residing in the vehicle&#39;s control unit, which in turn signals the propulsion system to adopt kinematics, such as fin frequency and amplitude, associated with optimal propulsion efficiency. In an embodiment, the model of propulsive efficiency is generated from a multilayer perception neural network model using data from aquatic species, such as undulatory fin propulsion in the knifefish ( Xenomystus nigri ), and a sensitivity analysis is used to lower the number of required inputs. Power savings could protract vehicle operational life and/or provide more power to other functions, such as communications.

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 12/824,184, filed on Jun. 27, 2010, now U.S. Pat. No. 8,548,656, issued on Oct. 1, 2013, which claims priority to U.S. Provisional Patent Application No. 61/222,059, filed on Jun. 30, 2009, both of which are incorporated herein by reference. This application is related to U.S. patent application Ser. No. 12/824,185, filed on Jun. 27, 2010, now U.S. Pat. No. 8,468,108, issued on Jun. 18, 2013, and to U.S. patent application Ser. No. 12/824,186, filed on Jun. 27, 2010, both of which are incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to underwater vehicles, and more specifically to methods and systems that lead to increased propulsive efficiency of autonomous underwater vehicles (AUVs) over a range of velocities. 
     BACKGROUND 
     Much research and development on autonomous underwater vehicles (AUVs) has focused on larger vehicles driven by propellers (see for example “Trends in Biorobotic Autonomous Undersea Vehicles,”  IEEE Journal of Oceanic Engineering, Vol.  30 , No.  1, pp. 109-139, January 2005 by P. R. Bandyopadhyay, incorporated herein by reference). 
     Research and development has also focused on vehicles with undulatory body motions (see for example “An Efficient Swimming Machine,”  Scientific American, Vol.  272 , Issue  3, pp. 64-70, March 1995 by M. S. Triantafyllou et al., “The Geometric Mechanics of Undulatory Robotic Locomotion,”  The International Journal of Robotics Research, Vol.  17 , No.  7, pp. 683-701, July 1998 by J. Ostrowski et al., “Hydrodynamics of Fishlike Swimming,”  Annual Review of Fluid Mechanics, Vol.  32, pp. 33-53, 2000 by M. S. Triantafyllou et al., “Nonlinear Control Methods for Planar Carangiform Robot Fish Locomotion,”  Proceedings of the  2001  IEEE International Conference on Robotics and Automation , pp. 427-434, 2001 by K. A. Morgansen et al., and “Design and Dynamic Analysis of Fish Robot: PoTuna,”  Proceedings of the  2004  IEEE International Conference on Robotics and Automation , pp. 4887-4892, April 2004 by E. Kim et al., each of which is incorporated herein by reference). 
     More recently, developments in the design and propulsion of biomimetic autonomous underwater vehicles (AUVs) have focused on boxfish as models (see for example “Biomimetric Micro Underwater Vehicle with Oscillating Fin Propulsion: System Design and Force Measurement,”  Proceedings of the  2005  IEEE International Conference on Robotics and Automation , pp. 3312-3317, April 2005 by X. Deng et al., incorporated herein by reference). In this paper a biomimetic system concept design, fabrication details and experimental force measurements on prototype boxfish-inspired vehicles are presented. 
     While swimming mechanics in boxfish are well understood (e.g. routine swimming performance, maneuverability and stability, carapace hydrodynamics, drag and lift, and vortical flow self-correcting forces), little attention has been given to the functional design and operation of boxfish-inspired vehicles (see for example “Does A Rigid Body Limit Maneuverability?,”  The Journal of Experimental Biology, Vol.  203, pp. 3391-3396, 2000 by J. A. Walker, “Boxfishes and Unusually Well-Controlled Autonomous Underwater Vehicles,”  Physiological and Biological Zoology, Vol.  73 , No.  6, pp. 663-671, 2000 by M. S. Gordon et al., “Boxfishes (Teleoste: Ostraciidae) As A Model System For Fishes Swimming With Many Fins: Kinematics,”  The Journal of Experimental Biology, Vol.  204, pp. 1459-1471, 2001 by J. R. Hove et al., “Fish Functional Design and Swimming Performance,”  Journal of Fish Biology, Vol.  65, pp. 1193-1222, 2004 by R. W. Blake, and “Evidence of Self-Correcting Spiral Flows in Swimming Boxfishes,”  Bioinspiration  &amp;  Biomimetrics, Vol.  3, 2008 by I. K. Bartol et al., each of which is incorporated herein by reference). 
     In particular, while boxfish-inspired vehicles have many potential advantages in operating in complex environments (e.g. high maneuverability and stability), limited battery life and payload capacities are likely functional disadvantages. In particular, boxfish employ undulatory median and paired fins during routine swimming which are characterized by high hydromechanical Froude efficiencies (≈0.9) at low forward speeds. However, current boxfish-inspired vehicles are propelled by a low aspect ratio, ‘plate-like’caudal fin (ostraciiform tail) which can be shown to operate at a relatively low maximum Froude efficiency (≈0.5) and is mainly employed as a rudder for steering and in rapid swimming bouts (e.g. escape responses). 
     BRIEF SUMMARY OF THE INVENTION 
     An autonomous underwater vehicle (AUV) uses a model of propulsive efficiency to achieve high values of propulsive efficiency over a range of forward velocities, giving a lowered energy drain on the battery. Externally monitored information, such as that on flow velocity, is conveyed to an apparatus residing in the vehicle&#39;s control unit, which in turn signals the locomotive unit to adopt kinematics, such as fin frequency and amplitude, associated with optimal propulsion efficiency. In an embodiment, the model of propulsive efficiency is generated from a multilayer perception neural network model using data from aquatic species, such as undulatory fin propulsion in the knifefish ( Xenomystus nigri ), and a sensitivity analysis is used to lower the number of required inputs. Power savings could protract vehicle operational life and/or provide more power to other functions, such as communications. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates the result of a sensitivity analysis of propulsive efficiency to different parameters. 
         FIG. 2  illustrates a model of propulsive efficiency as a function of two parameters. 
         FIG. 3  illustrates an apparatus for computing efficiency as a function of two parameters. 
         FIG. 4A  illustrates an apparatus for computing a parameter as a function of efficiency and a second parameter. 
         FIG. 4B  illustrates an apparatus for computing a parameter as a function of efficiency and a second parameter. 
     
    
    
     DETAILED DESCRIPTION 
     A neural network model is used to develop a model of propulsive efficiency over a range of flow velocities. In an underwater vehicle, control unit circuitry is designed using this model to optimize propulsive efficiency relative to flow velocity. External information from a sensory system on flow velocity sends information to a control unit including a CPU that in turn sends control signals to a locomotive unit allowing actuators to operate at maximum efficiency relative to perceived flow conditions. Improving propulsive efficiency results in saving battery power that could protract vehicle life allow for a larger payload or provide additional power for other functions, such as communications. In an embodiment, a vehicle utilizing the model of propulsive efficiency has a substantially rigid structure, has an undulatory fin based propulsion system and is an autonomous underwater vehicle (AUV). In alternative embodiments, other vehicle body structures could be used, other types of propulsive systems could be used, and the vehicle could be under remote control rather than being autonomous. 
     In an experimental design, three circuit modules have been design based on multilayer perception neural network models of the hydrodynamic efficiencies of swimming in the knifefish ( Xenomystus nigri ) (see “Evaluation Using Multilayer Perception Neural Networks: A Case Study of Undulatory Median Fin Swimming in the Knifefish  Xenomystus Nigri,” Journal of Fish Biology, Vol.  71, pp. 1203-1207, 2007 by Li et al., incorporated herein by reference). 
     Specifically, a neural network model was implemented into circuitry by (1) construction of networks with the optimal topology based on experimental data; (2) sensitivity analysis of determinant factors for the Froude efficiency η p  (i.e. forward velocity U, fin lateral velocity W=∂h/∂t where h is the distance displaced by the fin&#39;s trailing edge, and lateral velocity of pushing on the water w=(∂h∂t −1 )+U(∂h∂x −1 )); (3) validation and testing of network predictions and (4) implementation of the neural network into three circuit modules. 
     Neural Network Model 
     Neural networks are characterized by distinct topologies of nonlinear differentiable activation functions in neurons consisting of input, output and hidden layers such that each neuron of a layer is connected to all others in the next layer (see for example “A Logical Calculus of the Ideas Immanent in Nervous Activity,”  Bulletin of Mathematical Biology, Vol.  52 , No.  1/2, pp. 99-115, 1990, reprinted from 1943, by W. W. McCulloch et al.). 
     Following the work presented in Li et al. (2007), the optimal neural network configuration was adopted from a neighborhood of 1-3 layers of 1-10 neurons. The momentum learning method was used, which is an advanced adaptive patterning learning technique progressed from the classic gradient descent method (see for example. “Electric Utility Coal Quality Analysis Using Artificial Neural Network Techniques,”  Neurocomputing, Vol.  23, pp. 195-206, 1998 by H. Salehfar et al.). 
     Gradient descent incorporates an error back prorogation algorithm to train weights (based on local information) for minimizing overall error. The instantaneous error of neuron i at the nth training iteration is:
 
 e   i ( n )= d   i ( n )= y   i ( n )
 
     where e i (n) is the instantaneous error, d i (n) is the desired output and y i (n) is the neuron output (see for example “Modelling the Effect of Carbon Content on Hot Strength Steels Using a Modified Artificial Neural Network,”  ISIJ International, Vol.  38 , No.  10, pp. 1121-1129, 1998 by L. X. Kong et al.). 
     Weights are trained from the iteration n+1 in gradient descent:
 
 W   ij ( n+ 1)= W   ij ( n )+γδ i ( n ) x   j ( n )
 
     where W ij (n) is the weight between nodes i and j at iteration n, x j (n) is the present input, δ i (n) is the local gradient which pointed to the required change in the weight and γ is the learning rate. This method was improved by incorporating ‘momentum learning’ to decrease noise and increase convergence using:
 
 Ŵ   ij ( n+ 1)= W   ij ( n+ 1)+α( W   ij ( n )− W   ij ( n− 1))
 
     where Ŵ ij (n+1) and W ij (n) are the weights between nodes i and j at iteration n for momentum learning and gradient descent respectively and α is the momentum factor. 
     The stopping criterion was chosen at maximum epoch (iterations through the patterns represented in the input) of 1000 and at a mean square error (MSE)≦0.01: 
     
       
         
           
             MSE 
             = 
             
               
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       0 
                     
                     p 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       m 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             d 
                             ij 
                           
                           - 
                           
                             y 
                             ij 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
                 ⁢ 
                 
                     
                 
               
               
                 m 
                 · 
                 p 
               
             
           
         
       
     
     where m is the size of the training dataset, p is the total number of neurons and y ij  and d ij  are the network and desired output for data series i at neuron j respectively. 
     Input data for analysis of the fin motions of the knifefish ( Xenomystus nigri ) are derived from a simplified bulk momentum approach based on elongated body theory (see for example “Hydromechanics of Aquatic Animal Propulsion,”  Annual Review of Fluid Mechanics, Vol.  1, pp. 413-446, 1969 by M. J. Lighthill, incorporated herein by reference). 
     In this approach, the mean thrust power  P  is given by subtracting the mean rate at which kinetic energy is wasted in the wake  P k    from the total mean rate of working  P t   :
 
   P   =    P   t   −    P   k   = U (   MwW   )−0.5 U (    Mw   2   )
 
     where M is the added mass at the trailing edge of the undulatory anal fin: 
     
       
         
           
             M 
             = 
             
               
                 1 
                 4 
               
               ⁢ 
               πρ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 d 
                 s 
                 2 
               
               ⁢ 
               β 
             
           
         
       
     
     where ρ, d s  and β are water density, the depth of a propulsive section and the shape factor respectively. The shape factor β is considered to be approximately equal to 1 (see for example “Aquatic Animal Propulsion of High Hydromechanical Efficiency,”  Journal of Fluid Mechanics, Vol.  44 , Part  2, pp. 265-301, 1970 by M. J. Lighthill, incorporated herein by reference). 
     The propulsive efficiency η p  is given by:
 
η p =1−(    P   t   −   P   )/    P   t   
 
     The values for W, w,  P ,  P t   ,  P k    and η p  as a function of the swimming velocity U were taken from Table 1 of “Swimming in the Electric Eels and Knifefishes,”  Canadian Journal of Zoology, Vol.,  61, pp. 1432-1441, 1983 by R. W. Blake, incorporated herein by reference.). This table is illustrated below: 
     
       
         
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 U 
                 R L   
                 U 
                 W 
                 w  
                 {circumflex over (T)} 
                 
                   P 
                 
                 
                   P 
                   kin 
                 
                 
                   P 
                   tot 
                 
                 η p   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0.413 
                 6.2 
                 2.75  
                 0.516 
                 0.125 
                 8.62 
                 35.6 
                 6.46  
                 42.0  
                 0.85 
               
               
                 0.306 
                 4.5 
                 2.00 
                 0.390 
                 0.108 
                 5.50 
                 16.5 
                 3.48 
                 19.9  
                 0.83 
               
               
                 0.370 
                 5.5 
                 2.46  
                 0.478 
                 0.115 
                 7.36 
                 27.2 
                 3.74 
                 31.0 
                 0.88 
               
               
                 0.333 
                 5.0 
                 2.22  
                 0.433 
                 0.108 
                 6.22  
                 20.6  
                 2.96 
                 23.6 
                 0.87 
               
               
                 0.260 
                 3.9 
                 1.73 
                 0.369 
                 0.084 
                 4.20 
                 10.92 
                 1.40 
                 12.32 
                 0.88 
               
               
                 0.200 
                 3.0 
                 1.33 
                 0.260 
                 0.069 
                 2.22 
                 4.44 
                 0.74  
                 5.18 
                 0.86 
               
               
                 0.178 
                 2.6 
                 1.15 
                 0.221 
                 0.069 
                 1.96  
                 3.34  
                 0.62  
                 3.96 
                 0.84 
               
               
                 0.170 
                 2.55 
                 1.13 
                 0.210 
                 0.068 
                 1.84 
                 3.12 
                 0.60 
                 3.72 
                 0.84 
               
               
                 0.153 
                 2.3 
                 1.02 
                 0.201 
                 0.065 
                 1.66 
                 2.54  
                 0.50  
                 3.04 
                 0.84 
               
               
                 0.150 
                 2.25 
                 1.00 
                 0.191 
                 0.061 
                 1.46 
                 2.20  
                 0.42 
                 2.62 
                 0.84 
               
               
                 0.136 
                 2.05 
                 0.91 
                 0.162 
                 0.053 
                 1.08  
                 1.48  
                 0.28  
                 1.76 
                 0.84 
               
               
                 0.133 
                 2.0 
                 0.89 
                 0.153 
                 0.052 
                 1.02  
                 1.22 
                 0.28 
                 1.50 
                 0.81 
               
               
                 0.130 
                 1.95 
                 0.87 
                 0.142 
                 0.052 
                 0.92 
                 1.20 
                 0.26  
                 1.46 
                 0.82 
               
               
                 0.117 
                 1.75 
                 0.78 
                 0.149 
                 0.049 
                 0.94  
                 1.10 
                 0.22 
                 1.50 
                 0.83 
               
               
                 0.103 
                 1.55  
                 0.67 
                 0.127 
                 0.049 
                 0.76  
                 0.78  
                 0.18  
                 0.98  
                 0.80 
               
               
                 0.094 
                 1.41 
                 0.63 
                 0.130 
                 0.044 
                 0.72  
                 0.68 
                 0.14 
                 0.82  
                 0.82 
               
               
                 0.090 
                 1.35  
                 0.60 
                 0.131 
                 0.043 
                 0.72 
                 0.66  
                 0.12  
                 0.78  
                 0.83 
               
               
                 0.087 
                 1.30  
                 0.58 
                 0.126 
                 0.044 
                 0.70 
                 0.60  
                 0.12  
                 0.74  
                 0.82 
               
               
                 0.085 
                 1.27  
                 0.57 
                 0.111 
                 0.045 
                 0.60 
                 0.52  
                 0.12  
                 0.64  
                 0.80 
               
               
                 0.082 
                 1.23  
                 0.55 
                 0.109 
                 0.044 
                 0.58  
                 0.46  
                 0.12 
                 0.58  
                 0.78 
               
               
                 0.080 
                 1.20 
                 0.53 
                 0.107 
                 0.039 
                 0.52  
                 0.41  
                 0.10 
                 0.52  
                 0.81 
               
               
                 0.079 
                 1.19 
                 0.52 
                 0.103 
                 0.040 
                 0.50 
                 0.40  
                 0.10 
                 0.50 
                 0.80 
               
               
                 0.077 
                 1.16  
                 0.51 
                 0.092 
                 0.039 
                 0.44 
                 0.32  
                 0.08 
                 0.40  
                 0.80 
               
               
                 0.076 
                 1.12 
                 0.50 
                 0.083 
                 0.035 
                 0.34 
                 0.28  
                 0.08 
                 0.34 
                 0.78 
               
               
                 0.073 
                 1.11 
                 0.49 
                 0.076 
                 0.037 
                 0.32  
                 0.24  
                 0.08  
                 0.32 
                 0.75 
               
               
                 0.067 
                 1.00 
                 0.45  
                 0.061 
                 0.038 
                 0.24  
                 0.16  
                 0.08  
                 0.24  
                 0.67 
               
               
                 0.066 
                 0.91 
                 0.44 
                 0.058 
                 0.036 
                 0.22 
                 0.14  
                 0.06 
                 0.20 
                 0.70 
               
               
                 0.056 
                 0.84 
                 0.37 
                 0.056 
                 0.036 
                 0.20 
                 0.12  
                 0.06 
                 0.18 
                 0.67 
               
               
                   
               
             
          
         
       
     
     Sensitivity Analysis 
     Sensitivity analysis was performed on the trained neural network to determine the relative importance of each variable using weights derived from the training process and measuring the change in the predicted output for every 50 divisions of 1 SD of the mean input. The optimality and accuracy was tested based on the cross-validation scheme using 50% of the dataset for training and 50% for MLP-NN performance testing. The sensitivity analysis showed that the swimming speed U and fin lateral speed W were the major determinants of η p  and this suggested the use of these parameters for implementation into circuitry. In alternative embodiments, the parameter w could also have been included as an input, as well as other parameters. 
       FIG. 1  illustrates sensitivity for the predicted efficiency η p  (defined as the change in the mean output for every 50 divisions within 1 SD of the mean input), swimming speed U, lateral velocity of the fin W, lateral velocity of pushing on the water slice w, thrust T, mean thrust power P, kinetic energy wasted in the wake P kin  and mean total power P tot . The numerical results of the sensitivity analysis are shown in the table below: 
     
       
         
               
               
               
             
           
               
                   
                   
               
               
                   
                 Parameter 
                 Sensitivity to η p   
               
               
                   
                   
               
             
             
               
                   
                 U (m s − 1) 
                 0.356122 
               
               
                   
                 W (m s − 1) 
                 1.114775 
               
               
                   
                 w (m s − 1) 
                 0.061856 
               
               
                   
                 T (×110 − 3N) 
                 0.011263 
               
               
                   
                 P (×10 − 4W) 
                 0.000644 
               
               
                   
                 P kin  (×10 − 4W)  
                 0.019360 
               
               
                   
                 P tot  (×10 − 4W) 
                 0.000158 
               
               
                   
                   
               
             
          
         
       
     
     The neural network model was then constructed based on the cross-validation scheme (75% of the data were used to train the neural network and 25% was used to test its performance) and good agreement was found between neural network predictions and actual values (P&gt;0.05).  FIG. 2  shows a three-dimensional representation of the output from the trained neural network based on U, W and η p . The output of the trained neural network is shown in the table below: 
     
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                   
                 W 
               
             
          
           
               
                 U 
                 0.05 
                   
                 0.10 
                 0.15 
                 0.20 
                   
                 0.25 
                 0.30 
                 0.35 
                   
                 0.40 
                 0.45 
                   
                 0.50 
               
               
                   
               
               
                 0.05 
                 0.69 
                   
                 0.79 
                 0.84 
                 0.86 
                   
                 0.86 
                 0.87 
                 0.88 
                   
                 0.88 
                 0.88 
                   
                 0.88 
               
               
                 0.10  
                 0.70 
                   
                 0.79 
                 0.83 
                 0.85 
                   
                 0.86 
                 0.87 
                 0.87 
                   
                 0.88 
                 0.88 
                   
                 0.88 
               
               
                 0.15 
                 0.70 
                   
                 0.79 
                 0.83 
                 0.84 
                   
                 0.85 
                 0.86 
                 0.87 
                   
                 0.87 
                 0.88 
                   
                 0.88 
               
               
                 0.20  
                 0.71 
                   
                 0.79 
                 0.82 
                 0.84 
                   
                 0.85 
                 0.86 
                 0.86 
                   
                 0.87 
                 0.87 
                   
                 0.88 
               
               
                 0.25 
                 0.71 
                   
                 0.79 
                 0.81 
                 0.83 
                   
                 0.84 
                 0.85 
                 0.86 
                   
                 0.86 
                 0.87 
                   
                 0.87 
               
               
                 0.30 
                 0.72 
                   
                 0.78 
                 0.81 
                 0.82 
                   
                 0.83 
                 0.84 
                 0.85 
                   
                 0.86 
                 0.86 
                   
                 0.87 
               
               
                 0.35 
                 0.73 
                   
                 0.78 
                 0.81 
                 0.82 
                   
                 0.83 
                 0.84 
                 0.85 
                   
                 0.85 
                 0.86 
                   
                 0.87 
               
               
                 0.40  
                 0.74 
                   
                 0.78 
                 0.80 
                 0.81 
                   
                 0.82 
                 0.83 
                 0.84 
                   
                 0.85 
                 0.86 
                   
                 0.86 
               
               
                 0.45 
                 0.74 
                   
                 0.78 
                 0.80 
                 0.81 
                   
                 0.82 
                 0.83 
                 0.84 
                   
                 0.84 
                 0.85 
                   
                 0.86 
               
               
                 0.50 
                 0.75 
                   
                 0.78 
                 0.80 
                 0.80 
                   
                 0.81 
                 0.82 
                 0.83 
                   
                 0.84 
                 0.85 
                   
                 0.85 
               
               
                   
               
             
          
         
       
     
     Experimental Implementation 
     Three circuit modules were designed: (1) fin lateral speed versus swimming speed giving propulsive efficiency ( FIG. 3 ), (2) propulsive efficiency versus fin lateral speed giving swimming speed ( FIG. 4A ) and (3) propulsive efficiency versus swimming speed giving fin lateral speed ( FIG. 4B ). The experimental implementation (summarized in  FIGS. 3 ,  4 A and  4 B and consisting of 1918, 2318 and 2362 gates respectively) utilizes inverters as well as logic circuits of two, three and four 6-bit and 7-bit inputs. Each module is composed of combinatorial logic and is activated by two inputs and produces one output. Each module represents the entire array of swimming efficiencies, swim speeds and fin speed efficiencies illustrated in  FIG. 2 . 
     In  FIG. 3 , swimming speed (U) is input at  32  into combinatorial circuit  30  and fin lateral speed (W) is input at  34 . Propulsive efficiency (η p ) is output at  36  from combinatorial circuit  30 . In an embodiment, inputs  32  and  34  consist of 6-bit inputs that encode the input parameters as binary values. Other widths of input and other forms of encoding the input parameters are possible. In an embodiment, output  36  contains an individual signal for each value in the efficiency curve. In other embodiments, output  36  can be encoded as a binary value. In some embodiments, circuit  30  may be replaced by a sequential circuit or may be implemented as a software routine in the control unit of a vehicle. In some embodiments interpolation between points on the efficiency curve may be performed to provide an output for all input values. 
     In  FIG. 4A , fin lateral speed (W) is input at  41  into combinatorial circuit  40  and propulsive efficiency (η p ) is input at  42 . Swimming speed (U) is output at  43  from combinatorial circuit  40 . In an embodiment, inputs  41  and  42  consist of 6-bit and 7-bit inputs respectively that encode the inputs as binary values. Other widths of input and other forms of encoding the inputs are possible. In an embodiment, output  43  contains an individual signal for each value of swimming speed. In other embodiments, output  43  can be encoded as a binary value. In some embodiments, circuit  40  may be replaced by a sequential circuit or may be implemented as a software routine in the control unit of a vehicle. In some embodiments interpolation between points on the efficiency curve may be performed to provide an output for all input values. 
     In  FIG. 4B , swimming speed (U) is input at  46  into combinatorial circuit  45  and propulsive efficiency (η p ) is input at  47 . Fin lateral speed (W) is output at  48  from combinatorial circuit  45 . In an embodiment, inputs  46  and  47  consist of 6-bit and 7-bit inputs respectively that encode the inputs as binary values. Other widths of input and other forms of encoding the inputs are possible. In an embodiment, output  48  contains an individual signal for each value of fin lateral speed. In other embodiments, output  48  can be encoded as a binary value. In some embodiments, circuit  45  may be replaced by a sequential circuit or may be implemented as a software routine in the control unit of a vehicle. In some embodiments interpolation between points on the efficiency curve may be performed to provide an output for all input values. 
     A variety of control scenarios are possible with the circuits illustrated in  FIGS. 3 ,  4 A and  4 B or with alternative embodiments of these circuits. Note that the outputs of circuits  30 ,  40  and  45  are equal to the results of the neural network model. The values of propulsive efficiency are at least 0.69 for all inputs and achieve 0.88 at higher forward velocities. By having a model of efficiency over a range of forward velocities, it is possible for the control unit of an underwater vehicle to utilize such information to improve propulsive efficiency given the other constraints imposed by the overall system. 
     SUMMARY 
     A basic design for the major design units of biomimetic MUVs based on oscillatory propulsion employing a boxfish model are described in Deng et al. (2005). In this design, there are five main units: locomotory, sensory, power, communications and control. In this design, the MUV is propelled by an electromechanical actuated-fin system. Two side fins (for steering and moving upward and downward) and a plate-like caudal fin (ostraciiform tail) for propulsion are driven by a PZT bimorph actuator with motion amplification from four bar mechanisms. The fins are powered by electric energy (e.g. a lithium battery) and the power supply, communications (e.g. ultrasonic transmitters), sensory (e.g. flow velocity detection) units feed into the control (CPU) unit. 
     Other types of propulsion system have been studied based on different biomimetic models besides the ostraciiform propulsion of the boxfish. For example anguilliform propulsion is described in Ostrowski et al. (1998), carangiform propulsion (where approximately two thirds of the body undulates) is described in Morgansen et al. (2001), thunniform propulsion is described in Triantafyllou et al. (1995), incorporated herein by reference, and an undulatory fin model is described in “Biomimetric Compliant System for Smart Actuator-Driven Aquatic Propulsion: Preliminary Results,”  Proceedings of  203  ASME International Mechanical Engineering Congress  &amp;  Exposition  ( IMECE &#39; 03), November 2003 by B. P. Trease et al., incorporated herein by reference, and “Kinematics and Force Characterization of a Knifefish-Inspired Mechanical Propulsor,”  Proceedings of Biological Approaches to Engineering Conference , March 2008 by K. Collins et al., incorporated herein by reference. In the case of undulatory fins, Trease et al. (2003) describes sinusoidally undulating flexible fins with distributed compliance based on a rib structure and Collins et al. (2008) describes such a design incorporated into a ‘WaveDrive’ actuating mechanism with preliminary measurements on thrust production. 
     Nevertheless, currently the ‘ostraciiform model’ is favored with respect to the design and function of small, highly maneuverable and stable AUVs (see for example Gordon et al. (2000), “Hydrodynamic Stability of Swimming In Ostraciid Fishes: Role of the Carapace in the Smooth Trunkfish  Lactophyrs triqueter  (Teleostei: Ostraciidae),”  The Journal of Experimental Biology, Vol.  206, pp. 725-744, 2003 by I. K. Bartol et al., “Fish Functional Design and Swimming Performance,”  Journal of Fish Biology, Vol.  65, pp. 1193-1222, 2004 by R. W. Blake, and Deng et al. (2005), each of which is incorporated herein by reference). 
     While these approaches to AUV function have certain advantages as far as body design, maneuverability and stability are concerned, the current focus on an oscillating plate as a basis for propulsion has significant disadvantages. In particular, it can be shown that the maximum Froude efficiency of a low aspect ratio ‘plate-like’ caudal fin propeller (ostraciiform tail) has a relatively low upper value of approximately 0.5 (see for example “Mechanics of Ostraciiform Propulsion,”  Canadian Journal of Zoology, Vol.  59, pp. 1067-1071, 1981 by R. W. Blake). In fact, boxfish are not propelled by the reciprocating motions of their caudal fin during routine activity. Rather, they swim through the action of undulatory median and paired fins and, in rectilinear swimming, the caudal fin is often collapsed presumably to reduce drag. It has been observed that the caudal fin is mainly employed as a rudder for steering and in bouts of unsteady swimming. 
     A more efficiently propelled underwater vehicle can be achieved by using an undulatory fin and can use a control unit that utilizes a model of swimming efficiency over a range of velocities. Undulatory fin based propulsion results in increased efficiency over a range of velocities and is particularly well adapted to vehicles utilizing low velocities and lower power utilization. In an embodiment of such a vehicle, part of a control unit could be based on the three circuit modules described above to optimize propulsive efficiency relative to the flow velocity. External information on flow velocity could be conveyed by sensor signals to the control unit which in turn could instruct the locomotor unit to appropriately match fin frequency, and hence W, to achieve optimum propulsive efficiency. Under these conditions, the energy drain on the power unit would be minimized at any given U. This power saving could protract the operational life of the vehicle and/or provide more power to other functions such as sensors and communications. 
     Undulatory median and paired fin swimming is an adaptation for propulsion for high hydromechanical efficiency at low forward speeds (see for example Blake (2004)). Given that biomimetically inspired engineering designs need not be constrained by the limitations imposed by phylogenetic (historical) or ontogenetic (developmental) factors, optimal structural and functional solutions can be found by selecting appropriate design features from a variety of ‘fish models’. Specifically, the relatively low drag, high stability and maneuverability of the boxfish carapace can be combined with an undulatory fin-based propulsion system rather than an oscillating fin design. 
     The foregoing descriptions of embodiments of the present invention have been presented for purposes of illustration and description only. They are not intended to be exhaustive or to limit the present invention to the forms disclosed. Accordingly, many modifications and variations will be apparent to practitioners skilled in the art. The scope of present invention is defined by the appended claims.