A biomimetic approach to efficient underwater propulsion
As the applications of underwater robots grow, finding efficient propulsion techniques is of the utmost importance. For the most part, this effort has concentrated on the hydrodynamic screw, which has many inherent problems, such as cavitation and efficiency. To overcome these shortcomings, current research has focused on the use of biomimetic propulsion, which simulates the undulation of fish tails, or the sinusoidal oscillation of the stingray or cuttlefish. Our objective is to mimic the propulsion technique the manta ray uses to swim efficiently over large distances at impressive speeds.
The locomotion of animals—be it crawling, walking, swimming, or flying—is produced by oscillatory motion of the limbs, wings, or other appendages. Muscles contract in a periodic manner, typically controlled by a central pattern generator (CPG).1 A CPG is a subnetwork of interconnected neurons that provide the fundamental control mechanism for rhythmic motion. The neurons accomplish this by autonomously producing specific phase, amplitude, and frequency relationships between the muscle groups they govern.2 Feedback loops enable the CPG to find the most effective or efficient output for a given environment or task.3 An added level of autonomy such as this would allow an underwater vehicle to traverse a path in the same way as a manta ray would, regardless of the cross currents or obstacles.
Our study combined a synthesized CPG and a four-truss morphing structure actuated by three linear stepper motors to produce the oscillatory output necessary for locomotion. The CPG modeled the adaptation, or fatigue, response characteristic of actual neurons. We used LabVIEW and MatLab's Simulink to actuate the structure with the generator's output.
The principle behind the analysis of statically and kinematically determinate morphing truss structures is based upon Maxwell's necessary conditions.4,5 By replacing passive members with active, load-bearing actuators, significant shape changes can be achieved under substantial restraining loads. Figure 1 illustrates this principle with a simple five-member structure. Here it is clear that the only strain energy, ε, that is present in the actuated structure under no external loading is the active member. These characteristics make morphing structures a prime candidate for the structural design of an autonomous underwater vehicle mimicking the manta ray. A high-authority, shape-changing structure would be capable of reproducing the undulations necessary for oscillatory underwater propulsion.
Keeping simplicity in mind for this preliminary study, we created a 2D triangular-truss structure featuring four cells and three actuators shown schematically in Figure 2. This layout allows easy actuator integration from truss peak to truss peak. The structure has exceptional bending stiffness that enables it to withstand restraining forces and moments, and employs pin joints to avoid rotational resistance that would cause strain energy within the passive members.6
To ensure the structure could handle the manta application, it was analyzed statically to determine the resultant forces, with hydrodynamic forces approximated by point loads on the truss nodes. Using the method outlined by Lu et al.,6 the deflection δo of the truss system's end tip was determined as a function of actuation strains, structure geometry, and external loading. Tests of this analytic model demonstrated the morphing structure's suitability.7
The actuators were controlled by the output of ‘neurons’ in the synthesized CPG. Each neuron's activity is mapped from its input u to its output v, and each output is, in turn, connected to the input of the others. Together, these create a neural network with a connectivity matrix that describes the neuron interactions.
In solving the equations that represent a CPG, finding an appropriate connectivity matrix becomes the driving problem. We used the harmonic balance method to linearize the equation system into an eigenvalue/eigenvector problem with prescribed frequency, phase, and amplitude.8
This approach assumes that neuron activity is oscillatory over time, and that the transfer function describing the neural dynamics acted as a low pass filter, attenuating all but the first harmonic of the signal. The result is a numerical approximation for a connectivity matrix producing an oscillatory CPG.
The neural dynamics are framed in a state-space model, as diagrammed in Figure 3, with the signal vector q(t) shown in Figure 4 These signals are correlated to actuator position, and output to our physical four truss system using LabVIEW.
The four-truss system, controlled by the synthesized CPG, behaved as expected, undulating as would a fish tail. These results illustrate that morphing structures, coupled with a biologically inspired central pattern generator control system, can mimic the manta ray's efficient propulsion.
Future work will integrate feedback systems to provide the robot with a mechanism to find the most effective or efficient outputs for a given environment or task. We will fully integrate the structures and controls to provide a dynamic model relating the forces borne by the members and actuators while moving against hydrodynamic loads. This study will shed light on resonance frequencies within the structure that, if appropriate, could offer an efficient operating point for the controls, structure, and propulsion output.
We gratefully acknowledge the support of the National Science Foundation through Awards 0348448 and 0237708 and the David and Lucille Packard Foundation through the Packard Fellowship for Science and Engineering.
