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Research Papers

Trajectory Optimization for DDE Models of Supercavitating Underwater Vehicles

[+] Author and Article Information
Carlo L. Bottasso

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, Milano 20156, Italycarlo.bottasso@polimi.it

Francesco Scorcelletti

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, Milano 20156, Italy

Massimo Ruzzene, Seong S. Ahn

School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA 30332-0150

J. Dyn. Sys., Meas., Control 131(1), 011009 (Dec 08, 2008) (18 pages) doi:10.1115/1.3023117 History: Received August 17, 2007; Revised August 20, 2008; Published December 08, 2008

In this study we first develop a flight mechanics model for supercavitating vehicles, which is formulated to account for the dependence of the cavity shape from the past history of the system. This mathematical model is governed by a particular class of delay differential equations, featuring time delays on the states of the system. Next, flight trajectories and maneuvering strategies for supercavitating vehicles are obtained by solving an optimal control problem, whose solution, given a cost function and general constraints and bounds on states and controls, yields the control time histories that maneuver the vehicle according to a desired strategy, together with the associated flight path. The optimal control problem is solved using a novel direct multiple shooting approach, which is formulated to properly handle conditions dictated by the delay differential equation formulation governing the dynamic behavior of the vehicle. Specifically, the new formulation enforces the state continuity line conditions in a least-squares sense using local interpolations, which supports local time stepping and drastically reduces the number of optimization unknowns. Examples of maneuvers and resulting trajectories demonstrate the effectiveness of the proposed methodology and the generality of the formulation. The results are also compared with those obtained from a previously developed model governed by ordinary differential equations to highlight the differences and demonstrate the need for the current formulation.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic of the vehicle configuration

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Figure 2

Inertial and body frames

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Figure 3

Schematic of cavity configurations predicted through different cavity models and with the inclusion of time delay (a) Cavity configuration without memory effects (b) Cavity configuration with memory effects

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Figure 6

Body frame and cavitator frame

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Figure 7

3D view of a fin

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Figure 9

Force coefficients with respect to angle of attack for several relative immersions (dF={0.1,0.3,0.5,0.7,0.9})

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Figure 10

Fin immersion evaluation

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Figure 11

Schematic of vehicle/cavity interaction

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Figure 12

Line continuity conditions of a single component of the state vector

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Figure 14

Dive trajectories for the ODE and DDE models

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Figure 15

Pitch angle variation during the minimum time dive maneuvers for the ODE and DDE models

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Figure 16

Control time histories for the minimum time dive for the ODE and DDE models

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Figure 17

Fin relative immersions for the minimum time dive for the ODE and DDE models

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Figure 18

Family of turns

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Figure 19

Euler angles for representative minimum time turns

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Figure 20

Controls for representative minimum time turns

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