226 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 Vehicle Motion in Currents Peter G. Thomasson and Craig A. Woolsey Abstract—In this paper, we present a nonlinear dynamic model for the motion of a rigid vehicle in a dense fluid flow that comprises a steady, nonuniform component and an unsteady, uniform com- ponent. In developing the basic equations, the nonuniform flow is assumed to be inviscid, but containing initial vorticity; further ro- tational flow effects may then be incorporated by modifying the an- gular rate used in the viscous force and moment model. The equa- tions capture important flow-related forces and moments that are absent in simpler models. The dynamic equations are presented in terms of both the vehicle’s inertial motion and its flow-relative motion. Model predictions are compared with exact analytical so- lutions for simple flows. Applications of the motion model include controller and observer design, stability analysis, and simulation of nonlinear vehicle dynamics in nonuniform flows. As illustrations, we use the model to analyze the motion of a cylinder in a plane lam- inar jet, a spherical Lagrangian drifter, and a slender underwater vehicle. For this last example, we compare predictions of the given model with those of simpler models and we demonstrate its use for flow gradient estimation. The results are applicable to not only un- derwater vehicles, but also to air vehicles of low relative density such as airships and ultralights. Index Terms—Fluid flow, nonlinear systems, vehicle dynamics. I. INTRODUCTION W HILE ocean and atmospheric vehicles operate exclu- sively in time-varying, nonuniform currents, the effect of the ambient flow on the vehicle dynamics is typically ignored in motion models. In the simplest case, the vehicle dynamics are ignored entirely and the vehicle is considered to be a massless particle moving at a prescribed flow-relative velocity—the kine- matic particle model. Another common model—the dynamic particle or “performance” model—treats the vehicle as a mass particle, ignoring the attitude dynamics including any moment effects due to the flow field. The effects of nonuniform flow on the six-degree-of-freedom motion of a rigid vehicle are rarely considered in engineering analysis. These effects are explic- itly or tacitly dismissed on the basis that high-frequency flow perturbations will be filtered by the vehicle’s inertia and low- frequency perturbations are purely “kinematic.” Such claims may be justified in specific flow conditions, however they re- quire validation. When the effects of a flow field on a vehicle’s rigid body motion are considered, a conventional approach is to substitute the flow-relative velocity for the inertial velocity Manuscript received March 13, 2012; revised September 09, 2012; accepted October 10, 2012. Date of publication February 11, 2013; date of current version April 10, 2013. The work of C. A. Woolsey was supported by the U.S. Office of Naval Research under Grants N00014-08-1-0012 and N00014-10-1-0185. Associate Editor: F. S. Hover. P. G. Thomasson is with the College of Aeronautics, Cranfield University, Cornwall TR20 9SL, U.K. (e-mail: p.g.thomasson@pgthomasson.co.uk). C. A. Woolsey is with the Department of Aerospace and Ocean Engineering, Virginia Polytechnic and State University (Virginia Tech), Blacksburg, VA 24061 USA (e-mail: cwoolsey@vt.edu). Digital Object Identifier 10.1109/JOE.2013.2238054 in the dynamic equations developed for calm conditions [4, p. 59]. This approach approximates the dynamics of a vehicle in a steady, uniform flow, however that may or may not be a reason- able approximation depending on the actual flow characteristics. There is one area of vehicle dynamic modeling in which careful attention has been given to nonuniform flow effects: the flight of aircraft in turbulence. Dobrolenskiy [2] and Etkin [3, Ch. 13, pp. 529–563] provide thorough treatments which represent the standard modeling approach for this scenario. These treatments incorporate two essential assumptions, each entirely appropriate in the context of conventional aircraft motion: apparent mass effects are negligible and the vehicle motion is well described by a linear (small perturbation) model. Vehicles operating in nonuniform flow fields are subject to forces that are not captured by kinematic particle models and moments that are not accounted for in dynamic particle models. These forces and moments are even stronger when apparent mass effects are significant, as occurs for maritime vehicles, lighter-than-air vehicles, and ultralight aircraft, for example. While the simplicity of particle models makes them attractive for flow field estimation [14], [8], path planning [19], and guid- ance and control [17] in currents, some important flow effects can only be recovered using a rigid body dynamic model. Sim- plified dynamic models, as in [15], can be useful when the im- plicit assumption of a steady, uniform flow is appropriate, but analysis for more dynamic environments may require a higher fidelity model. In small perturbation applications, such as gust response analysis, linearized rigid body dynamic models pro- vide sufficient accuracy and allow the use of classical frequency analysis tools. In cases where the flow field varies significantly and the resulting vehicle motion violates the small perturbation assumption, however, the proposed model is more appropriate. In this paper, we provide a careful development of the non- linear dynamic equations for a rigid vehicle in a dense fluid flow comprising a steady, nonuniform component and an un- steady, uniform component. The equations enable one to assess the effect of flow gradients on a vehicle’s translational and ro- tational motion. The results are applicable to vehicles for which apparent mass effects may be significant, including undersea ve- hicles, lighter-than-air vehicles, and ultralight aircraft, as well as conventional aircraft. With a focus on undersea vehicle ap- plications, the dynamics are presented in notation familiar to the ocean engineering community. We review and amend the development in [20] which follows Lamb’s treatment [7] of a rigid body moving through a volume of perfect (inviscid and in- compressible) fluid that is itself in motion. The volume of fluid may be accelerating and the treatment allows for flow gradients due to cyclic flow through a multiply connected region. In ad- dition, we extend the analysis used in [20] to the more general case of motion in an inviscid stream containing vorticity. As in [20], further rotational flow effects are incorporated after the 0364-9059/$31.00 © 2013 IEEE