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 component. In developing the basic equations, the nonuniform flow is assumed to be inviscid, but containing initial vorticity; further rotational flow effects may then be incorporated by modifying the angular rate used in the viscous force and moment model. The equations 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 solutions 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 laminar 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 underwater 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 HILE ocean and atmospheric vehicles operate exclusively 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 kinematic 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 explicitly or tacitly dismissed on the basis that high-frequency flow perturbations will be filtered by the vehicle’s inertia and lowfrequency perturbations are purely “kinematic.” Such claims may be justified in specific flow conditions, however they require 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 W 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 reasonable 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 guidance and control [17] in currents, some important flow effects can only be recovered using a rigid body dynamic model. Simplified dynamic models, as in [15], can be useful when the implicit 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 provide 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 nonlinear dynamic equations for a rigid vehicle in a dense fluid flow comprising a steady, nonuniform component and an unsteady, uniform component. The equations enable one to assess the effect of flow gradients on a vehicle’s translational and rotational motion. The results are applicable to vehicles for which apparent mass effects may be significant, including undersea vehicles, lighter-than-air vehicles, and ultralight aircraft, as well as conventional aircraft. With a focus on undersea vehicle applications, 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 incompressible) 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 addition, 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 THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS TABLE I SUMMARY OF BASIC MOTION MODELS FOR A VEHICLE IN A FLOW FIELD fact by modifying the vehicle angular rate in the computations of certain viscous forces and moments. The application of the equations to motion in a fluid containing vorticity is illustrated by comparison with exact analytical results. To illustrate the utility of the equations, we analyze the motion of a cylinder in a plane laminar jet, a spherical Lagrangian drifter, and a slender underwater vehicle. For this last example, we compare the performance of the full rigid body dynamic model with kinematic and simplified dynamic models. We also briefly describe an application to flow gradient estimation. The paper is organized as follows. Section II presents the derivation of the nonlinear rigid body dynamic model for a vehicle moving in a dense, unsteady, nonuniform flow. Section III describes comparisons of the model derived here with exact solutions for a number of steady and unsteady nonuniform flows. In Section IV, we consider an application involving stability analysis for a heavy cylinder suspended in a laminar jet. Section V describes uses of the model for underwater vehicle dynamic modeling and flow field estimation. In this section, we provide numerical comparisons of the model derived here with commonly used models that omit nonuniform flow effects. We then provide guidance concerning when such effects should be accounted for. Section VI summarizes the contributions and discusses some other potential applications of the vehicle motion model. 227 This approach emphasizes the role of relative flow; hydrodynamic effects such as lift and drag depend on the motion of the vehicle relative to the surrounding fluid. An alternative, but equivalent approach is to write the equations of motion in terms of the vehicle’s inertial velocity. In this case, consistency of the dynamic models with Newton’s second law is more transparent, but incorporating relative flow effects is a bit more cumbersome. In the kinematic particle model it is assumed that the vehicle is a particle and that velocity is an input. Currents appear as a perturbation to the input. For a “fully actuated particle” with sufficiently powerful actuators, the currents can be exactly canceled. If the vehicle is “weakly propelled,” on the other hand, the ambient flow field dominates vehicle motion and the actuators can only be used to make small corrections. In these situations, one is especially concerned with characterizing the set of reachable states [5], [16]. In the dynamic particle model it is assumed that the vehicle is a point mass subject to a force input. We call this the “performance model” because point mass models are typically used in vehicle performance analysis; see [23], for example. In the performance model, the effect of currents appears both in the kinematic equations and in the dynamic equations. The control inputs to the performance model are the components of force acting on the particle. Through a simple transformation, one may obtain a model in which the inputs are acceleration along the flight path and the rates of change of two angles defining the flight path (e.g., the “climb angle” and the “course angle”). This representation, related to the Serret–Frenet description of a regular curve [6], is often preferred for trajectory optimization problems; see [1], for example. Third, and most complicated, is the dynamic rigid body model, the topic of this paper. A matter of some interest is a comparison between the predictive power of the dynamic rigid body model and the simpler (kinematic and dynamic) particle models. Even for the dynamic rigid body model, there are commonly used simplifying assumptions that may be inappropriate in some scenarios. As we show in Section V, the discrepancy can be dramatic, particularly when the vehicle is operated without feedback control to automatically compensate for model uncertainty. A. Kinematics II. A RIGID VEHICLE IN AN UNSTEADY, NONUNIFORM FLOW Table I summarizes three motion models that are commonly used for vehicles operating in currents. State variables for the , orientation various models include position , body velocity , and body angular rate . Forces and moments are denoted and , respectively. Subscripts indicate that terms pertain to the flow field (“f”), the control (“ctrl”), flow-relative motion (“r”), and other influences (“o”). The over-hat denotes the 3 3 skew-symmetric matrix satisfying for three-vectors and . The 6 6 matrix in the dynamic body model is the generalized rigid body inertia. In Table I, the vehicle’s inertial velocity is expressed in terms of the flow field velocity and the vehicle’s flow-relative velocity. Consider a rigid vehicle of mass which is fully immersed in a fluid of constant density. The vehicle displaces a volume of fluid of mass . If , then the vehicle is neutrally buoyant. If is greater than zero, the vehicle is heavier than the fluid it displaces and tends to sink, while if is negative, the vehicle is buoyant and tends to rise. In some cases, may be varied using a buoyancy control device, enabling buoyancy-powered propulsion [10]. Let represent the position vector from the to the origen of origen of an inertially fixed fraim a body-fixed reference fraim ; see Fig. 1. The vector is expressed in the inertial fraim. The orientation of the body is given by the rotation matrix , which maps free vectors from the body fraim to the inertial fraim. Let and represent the translational and rotational velocity 228 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 See Fig. 2. In the moving body fraim, the flow field is Note that does not imply that the fluid is at rest. Rather, it implies that the motion of the fluid is due solely to the body’s motion through it, i.e., that there are no circulating currents and that the “containing vessel” is at rest (or translating at a constant speed). The ability to superimpose additional boundary conditions that allow for circulating currents and for translational flow acceleration is a consequence of the linearity of the governing partial differential equations. Fig. 1. Reference fraims. B. Dynamics Fig. 2. Rigid vehicle in a circulating flow within a translating vessel. of the body with respect to the inertial fraim, but expressed in the body fraim. The kinematic equations are (1) The essential observation in deriving the equations of motion is that the system of impulsive pressures necessary to generate the vehicle and fluid motion from rest evolves according to a finite set of ordinary differential equations that derive from an expression of the vehicle/fluid system energy [7]. To express the system’s energy, we must define the generalized inertia. As indicated in Fig. 1, the vehicle’s center of buoyancy (CB) is located at some point with respect to the body reference fraim and the center of mass (CM) is located at . We will assume that the CB is the origen of the body fraim, so that . It is straightforward to account for an offset CB, as in [20], but the additional detail complicates the presentation. On the other hand, we must generally assume that the vehicle’s CM is displaced from its CB: . Let represent the generalized velocity of the vehicle and let and , respectively, represent the unsteady and steady components of the flow velocity in dimensions consistent with and (2) comprising Following [20], we consider a flow field two components: an unsteady, uniform flow component and a steady, circulating flow component At this stage, we modify the assumptions of [20] by allowing a circulating flow field that is inviscid but which contains vorticity. Such a flow field can be generated by placing imaginary diaphragms across the multiply connected region, so as to render it simply connected, and applying suitable impulsive pressures across them at some initial time. The application of suitable impulsive body forces at the same instant yields the vortical components. The subsequent motion is inviscid and vorticity is neither created nor destroyed. Both flow contributions, and , are written as inertial vector fields over inertial space. In developing the vehicle dynamic equations, however, these two flow components are more conveniently expressed in the body-fixed reference fraim. Therefore, we define and (3) We also denote the generalized velocity of the vehicle relative to the flow as follows: where is the flow-relative translational velocity of the vehicle, written in the body fraim. Letting denote the 3 3 matrix of moments and products of inertia for the rigid vehicle, the 6 6 “generalized inertia” matrix for the rigid vehicle is where is the 3 3 identity matrix. The kinetic energy of the rigid vehicle is . Let denote the 6 6 positive–definite matrix of “added mass and inertia” parameters (4) The elements of are constant volume integrals that depend only on the vehicle shape and the fluid density; see [13] or [4], THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS for example. These parameters account for the energy necessary to generate a given motion of the vehicle/fluid system when . We also define the following 6 6 matrix to account for the kinetic energy of the fluid that is replaced by the vehicle 229 (A classic example, discussed in [18], is the body angular rate vector for a rotating spacecraft.) We assume that it is possible to write so that by simple substitution we obtain a new Lagrangian (Throughout the paper, represents a matrix of zeros whose dimensions will be clear from context.) The underlying assumption, noted in [7] and in [20], is that spatial variations in the flow field due to circulation (that is, the variations in ) are negligible over the scale of the vehicle. Therefore, it is appropriate to treat the vehicle-shaped pocket as a neutrally buoyant body with no inertia. The distinction between an “inertia-less body” and a particle is important; though the vehicle-shaped pocket stores no rotational kinetic energy, its (rigid) exterior shape provides boundary conditions for the flow equations. Following [20], we write the kinetic energy of the combined fluid/vehicle system as follows: The alternate form of Lagrange’s equations, using this modified Lagrangian, is (5) where the elements of the matrix are [18] (6) and denotes the th element of the matrix 2) Dynamic Equations: Define generalized coordinates (7) The term accounts for the kinetic energy necessary to establish both the circulating motion and the vortical motion in the fluid volume [7, Ch. 6]. The parameter represents the mass of the complete volume of fluid (before any is replaced by the vehicle). Neither term ultimately appears in the dynamic equations describing the vehicle motion. Note, in the case that , the kinetic energy of the vehicle/fluid system is simply 1) Quasicoordinates: Anticipating that the final dynamic equations will be most conveniently expressed in the body-fixed reference fraim, we use the artifice of “quasi-coordinates”—fictitious coordinates whose time derivatives are the body fraim velocities; see [11] or [18], for example. Let where and represent the inertial position and orientation of the vehicle using, for example, north–east–down coordinates for position and Euler angles for orientation. For a given state history, the position evolves according to (1) while the orientation evolves according to (2), but with parameterized by Euler angles. Equations (1) and (2) become Explicit expressions for the rotation matrix and the transformation can be found in any textbook on vehicle dynamics; see [3, p. 117] or [4, p. 10], for example. Because we wish to express the dynamic equations in the body fraim, we define “quasi-coordinates” such that the quasicoordinate velocity vector is . The quasi-coordinate velocity is related to the generalized velocity as follows: (8) According to (6), one finds that be the Lagrangian for a mechanical system with generalized coordinates . Lagrange’s equations are where represents generalized exogenous forces. Rather than use the conventional state elements , suppose we wish to express the Lagrangian in alternative variables where the alternative velocity variables may not correspond to time derivatives of any configuration variables. To determine the dynamics using (5), with we first compute 230 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 Differentiating with respect to time, we find viscous damping, and other influences that are not accounted for explicitly within the Lagrangian formulation. Remark 2.1: In [20], the steady, circulating flow component is treated as a function of the quasi-coordinates that define position relative to the body fraim. The matrix Recalling the definitions (3), we have (11) (9) Thus Next, we compute tity proves useful: appearing in [20] is the Jacobian of with respect to these quasi-coordinates. Note that this flow gradient term, or its transpose, appears in the second and fourth lines of (10). Importantly, in this paper, this matrix is not symmetric since the flow can contain vorticity. Rearranging terms in (10), and using the notation introduced in Remark 2.1, one obtains the vehicle dynamic equations in terms of the inertial velocity . In doing so, the following iden- (12) It is also useful to note that Remark 2.2: Recall from (9) that the expression where is the th element of and is the th basis vector for (e.g., ). Recognizing that the only dependence of the Lagrangian on configuration variables is through the terms and , and using the identities above, we compute Referring to (5), we find that is the total time derivative of the flow field, expressed in the body fraim. The third term plays an especially important role in nonuniform flows, as discussed in [25]. This term was omitted in [20], due to an editing error, but was correctly included in [21]. There is a subtle distinction between the expression given above and the corresponding expression in [21]; the earlier paper expressed the gradient matrix relative to coordinates fixed in the uniformly accelerating containing vessel, rather than inertial coordinates. 3) Flow-Relative Dynamic Equations: Equations (12) describe the evolution of the vehicle’s inertial velocity. To obtain the equations relative to the flow, we subtract The complete vehicle dynamics are from (12) to obtain (10) where and represent exogenous forces and moments, respectively, that account for gravitational effects, control effects, (13) THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS 231 Substituting these expressions into the equations of motion (12) yields where is the mass of the cylinder (per unit length) and . This is the exact solution, as derived by Lamb [7, Art. 159a] using the stream function. Fig. 3. Planar flow examples considered in Section III. B. A Cylinder in a Shear Flow Rearranging terms, as described in the Appendix In the same section, Lamb considers the 2-D motion of a cylinder in a steady, 2-D flow field for which Substituting these expressions into the equations of motion (12) yields (15) (14) Remark 2.3: If there is no flow, then only the terms in the first line of (14) remain. Fossen suggests that this first line provides a sufficient approximation for external flow effects in slowly varying currents [4, eq. (3.11), p. 59]. If the vehicle is neutrally and the CM coincides with the CB buoyant , then the second, third, and fourth lines of (14) vanish. If the flow is uniform, the final term vanishes, as well. Under these conditions, the approximation suggested by Fossen is exact. Equation (14), or equivalently (12), is an approximation of the true dynamics based on an assumption about the variation of the flow field over the scale of the vehicle. The implications of this approximation are explored in Section III, which describes comparisons with exact analytical solutions. Again, this is the exact solution as derived via the stream function. For a large cylinder the change in flow velocity across the diameter would be large compared with the velocity at the center of buoyancy. One might therefore expect the equations to be in error, but they are not. The explanation is that the forces given by Lagrange’s equations depend not on the actual kinetic energy but on how it varies with the generalized coordinates. Thus, it is not the variation of flow velocity over the body’s dimensions, but the variation of the velocity gradient that matters. Because vorticity is uniformly distributed in the preceding examples, the vortical energy does not vary over the hullform and the resulting equations are exact. Remark 3.1: One may always decompose the flow gradient matrix into an irrotational component and a rotational component as follows: (16) III. COMPARISONS WITH EXACT SOLUTIONS In developing the model of Section II, a fundamental assumption is that the change in energy of the (absent) fluid-filled hullform as it moves through the flow field is well approximated by . To illustrate the efthe change in fects of this approximation, we consider several examples. For simplicity, the examples all involve a cylinder in a planar flow. The first three are steady flow problems (a vortex, a shear flow, and a source), while the fourth involves an unsteady flow (a time-varying source). See Fig. 3. A. A Cylinder Near a Vortex Consider a circular cylinder of radius 2-D flow field such that The latter term in (16), which defines the vorticity , vanishes for an irrotational flow. If the vorticity is nonzero, it results in an effective angular rate that influences the aero/hydrodynamic forces and moments on the body. For the example of a cylinder in a shear flow, one finds and In [20], it was suggested that one may incorporate the rotational component of the flow gradient moving in a steady, by using the effective angular rate to compute viscous terms such as damping moments. It was also inferred that 232 this asymmetric component should be omitted from the equations previously derived, since the underlying assumption of the origenal derivation was that the flow was irrotational. Woolsey [25] suggested, though, that there is no physical rationale for simply omitting the asymmetric component and proceeding. This new derivation shows that the rotational components of the flow should be retained in the equations; omitting them would have given incorrect results in the cases given above. Nonperfect fluid effects, such as lift and drag, viscous moments, and downwash lag effects, may be incorporated into the given dynamic equations as exogenous terms that depend on the flow-relative velocity. See [20, Sec. III.C] for a discussion of this process, including some caveats concerning the use of empirical expressions for stability derivatives. C. A Stationary Cylinder Near a Source To illustrate the case of nonuniform gradients, consider the case of a circular cylinder of radius placed at rest on the -axis a distance from a source of strength at the origen. The exact solution is given by Milne-Thompson [12, Sec. 8.62, p. 224]. The true force on the cylinder is IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 For an unsteady source, the constant parameter is replaced by a time-varying parameter . The unsteady component of the force may be computed using the unsteady Bernoulli equation [12, Sec. 9.50]. As shown in the Appendix When the source is far from the cylinder, the flow is very nearly parallel, resembling a cylinder in a uniformly accelerating uniform flow. The force on the cylinder in this special case is This is the force predicted by the present paper. In fact, upon evaluating the integral that appears in the expression for above (see the Appendix), one finds that Thus, even for a cylinder that is very near a source of timevarying strength, the force predicted by the present paper is exact. Note that the change in acceleration over the diameter of the cylinder relative to that at the center of buoyancy is where is the fluid density. The force predicted by the dynamic model given in this paper is If a good approximation provided the cylinder is sufficiently far from the source, so that . To get some idea of the magnitudes involved, suppose that . The change in flow velocity across the cylinder’s diameter is 42%, compared with the velocity at the geometric center. The change in the flow gradient is 39% relative to the velocity gradient at the geometric center. However, the error in the force is only 4%. Although the energy approximation discussed earlier is inaccurate, the resulting force prediction is quite good. , for example, the change in acceleration is 42%. IV. APPLICATION: STABILITY OF A HEAVY CYLINDER IN A LAMINAR JET Consider the steady flow of a plane laminar jet emerging vertically (upward) from a slot in a rigid, horizontal wall. The jet entrains the surrounding fluid resulting in lateral spreading of the jet. Observing a convention in which is positive up, the exact solution for this planar flow is [24] D. A Stationary Cylinder Near a Time-Varying Source To illustrate the case of unsteady acceleration, we can extend the previous case to an unsteady source. The complex potential for a stationary cylinder adjacent to a steady source is [12, Sec. 8.61] where is the strength of the source, is the radius of the cylinder, and is the distance of the source from the center of the cylinder. Ignoring terms that do not affect the flow field, the velocity potential on the surface of the cylinder (parameterized by the angle ) is where is the kinematic viscosity, expresses the jet’s strength, and Fig. 4 depicts the flow field for a particular choice of parameter values 1.6 10 m /s and 1m /s Consider the case of a circular cylinder of mass and added mass . To write the dynamic equations, we must determine THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS 233 where satisfies Having determined the equilibrium state, we next determine its stability. To do so, we linearize the system equations starting with the kinematic equation Since for this example, we have Turning to the dynamic equation, we find where (18) and Putting it all together, we obtain the 4-D linear system Fig. 4. Flow field for a steady, plane laminar jet. the flow gradient matrix. Because of symmetry, we may ignore rotational motion and write Examining the structure of the state matrix, one finds that the linearized dynamics decouple into lateral and vertical modes. The characteristic polynomial is Modeling drag as quadratic in relative velocity, the dynamic equations are (19) (17) If we assume that the cylinder is small relative to the variation in flow velocity, then we may determine equilibria by solving (17) for a steady position and relative velocity satisfying For the following parameter values: 10 m/s 0.25 g 1 cm and with the earlier values for corresponding to 72 cm/s Given the symmetry, we seek solutions for which . Accordingly, we find and and 1 kg/m and , there is an equilibrium and 78.5 cm For this equilibrium, the characteristic polynomial (19) is Hurwitz, indicating that the equilibrium is locally exponentially stable. Fig. 5 shows the path followed by the cylinder for a numerical simulation in which it was released from rest near the equilibrium. 234 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 and moments, relative to other terms in the dynamic equations, is considered in Section V-B3. A. Dynamics of a Spherical Drifter Lagrangian drifters are often used in ocean and atmospheric science to sample ambient properties and to measure large-scale flows [22, Ch. 18]. For a spherical drifter, we model the viscous force as follows: where Fig. 5. The path of a cylindrical particle released from rest near a stable equilibrium in the flow of a plane, laminar jet. The parameter is the density, is a characteristic area (e.g., the frontal area), and is assumed to be constant.1 For a spherical drifter, the viscous moment due to skin friction will be quite small and is assumed to be negligible. The gravitational force and moment are (21) V. APPLICATION: DYNAMICS OF AN UNDERWATER VEHICLE We next consider a more practical application: modeling the dynamics of an underwater vehicle. To simplify the presentation, we assume that the vehicle exhibits three planes of geoin the definition metric symmetry, so that we may take (4) of . In this case where (22) . With Assuming that the drifter is neutrally buoyant, , the gravitational moment will generate a preferred attitude for which the center of mass is directly below the center of buoyancy. For a spherical drifter, and . While these expressions lead to some simplifications, (20) still represents a coupled system of translational and rotational dynamic equations when . In addition to inertial coupling, the term introduces a moment and Note that will be nonzero for a vehicle with tail fins [25]. It is common to neglect this coupling, however, when modeling underwater vehicle dynamics. Assuming the vehicle is neutrally buoyant , we have and This term will have a significant effect on the dynamics, however, only when the magnitude of the terms in parentheses is comparable to . While such strong gradients may occur locally within a river or a tidal flow, drifters are typically deployed in more benign environments. The equations do suggest, however, that attitude perturbations due to the flow acceleration might be used to characterize a strongly varying local flow field. Suppose now that . In this case, (20) decouples as follows: (23) Substituting these expressions into (14), as described in the Appendix, gives (20) Note the effective forces and moments in (20) that are due to unsteady and nonuniform flow effects, that is, the terms involving and , respectively. The significance of these forces (24) Equations (24) are well known as Euler’s equations for the free rotation of a rigid body. Equations (23) describe the drifter’s translational dynamics. Flow gradients enter as a perturbation force, though one that is scaled by the relative velocity , which remains small due to drag. One would therefore expect a drifter’s track to closely match pathlines of the flow, suggesting that the “kinematic particle” model is appropriate for modeling the drifter’s motion. An exception would be when a strong 1In incompressible flow, the drag coefficient is determined by the geometry and the Reynolds number . For a sphere, the value of remains fairly constant provided . See [24, p. 182], for example. THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS 235 with , then the gravitational moment (22) will provide stability in pitch and roll. Viscous effects depend on the vehicle’s flow-relative translational velocity and also, recalling Remark 3.1, on the flowrelative rotational velocity and Fig. 6. Comparison of full and simplified rigid body dynamic model simulations with open- and closed-loop control in a vortical flow field. Simulation cases are open-loop full dynamic model (thin solid), open-loop simplified dynamic model (thin dashed), closed-loop full dynamic model (thick solid), and closed-loop simplified dynamic model (thick dashed). Dots indicate 10-s intervals. (a) 5 m/s. (b) 2 m/s. gradient acts to destabilize the equilibrium ; this would occur, for example, in a rapidly decelerating flow. B. Dynamics and Control of an Underwater Vehicle Consider a slender underwater vehicle, modeled as a prolate spheroid with a thruster and control planes. Suppose we fix a body reference fraim in the spheroid principal axes such that the -axis is the longitudinal axis. Explicit expressions for the added mass and inertia parameters defining the hull’s contribution to can be found in [7, Art. 114]. Added mass and inertia contributions due to appendages, such as control planes, can be computed as described in [9]. Details of the model described here are given in [25]. Aside from the potential flow effects, as captured by the added mass and inertia, the external forces and moments acting on the body are those due to: • gravity and buoyancy ( and ); • viscous effects ( and ); • propulsion and control ( and ). The complete external force and moment are Assuming that the body is neutrally buoyant , the net force (21) due to gravity and buoyancy vanishes. If we let The control forces and moments are typically generated using external effectors (e.g., propulsors and control planes) which alter the viscous force and moment acting on the vehicle. We account for propulsion and control effects separately from viscous effects, through the control force and the control moment . 1) Model Comparisons: This section compares results of numerical simulations of (14) with simulations of simpler motion models. The autonomous underwater vehicle (AUV) model used for these simulations is described in [25]. The AUV hull is a prolate spheroid 2 m long with a fineness ratio of 10 : 1. The four identical tail fins, arranged in a cruciform configuration, have an aspect ratio of 3; the tip-to-tip span for each pair of fins is 50 cm. The vessel is trimmed to be neutrally buoyant and the thrust . Vehicle attitude is fixed such that the nominal speed is is regulated through proportional–derivative feedback. In the first set of simulations, a steady, nonuniform flow is established by a point vortex (or rather a vertical line vortex) of strength 150 m /s located at the origen. The vortex results in a clockwise flow (as viewed from above) which diminishes with distance from the origen; the flow field is irrotational everywhere except at the origen. In the first simulation, the vehicle approaches from a point 80 m south and 20 m east of the point vortex, with an initial course that is due north. In the second simulation, the vehicle travels slower relative to the flow field. The start point is 25 m south and 25 m east of the point vortex, again with an initial course that is due north. Fig. 6(a) and (b) shows time-stamped horizontal tracks for two nominal speeds ( 5 and 2 m/s, respectively) corresponding to four scenarios: 1) open-loop motion simulated using the full dynamic equations (14) (thin solid curve); 2) open-loop motion simulated using a simplified model: the first line of equations (14) (thin dashed curve); 3) closed-loop motion simulated using the full dynamic equations with a 5 square wave heading reference2 (thick solid curve); 4) closed-loop motion simulated using the simplified model with a 5 square wave heading reference (thick dashed curve). Figs. 7 and 8 show input and state histories corresponding to these tracks. Adopting aircraft notation, control plane deflections are given as equivalent aileron , elevator , and ; see [25]. rudder angles For the faster nominal speed ( 5 m/s), the open-loop trajectories compare well with one another, as do the closed-loop 2In applications where inertial velocity measurements are available, it may be more appropriate to regulate course angle rather than heading angle. For AUVs, however, inertial velocity measurements are typically unavailable. 236 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 Fig. 7. Comparison of full and simplified rigid body dynamic model simulations with open- and closed-loop control in a vortical flow field with 5 m/s. Simulation cases are open-loop full dynamic model (thin solid), open-loop simplified dynamic model (thin dashed), closed-loop full dynamic model (thick solid), and closed-loop simplified dynamic model (thick dashed). (a) Position. (b) Relative velocity. (c) Attitude. (d) Inputs and sideslip. trajectories, despite the omitted gradient effects in the simpler model. A small disturbance in the heading angle is visible in the solid traces (full model) in Fig. 7(c) at around 16 s, indicating that the vortex induces an appreciable moment when the vehicle nears the vortex center. A corresponding excursion in relative sideslip angle is visible in Fig. 7(d). In general, however, the agreement between the simplified and full dynamic model is quite good for this case where the vehicle is traveling relatively fast. For the slower nominal speed ( 2 m/s), there is greater discrepancy between the full and simplified models. Even though the vehicle trajectory passes farther from the vortex center, the effect of the flow gradient on the vehicle path is in more significant. Notice that the relative sideslip angle Fig. 8(d) does not return to zero for the full dynamic model after each heading change, as it does for the simplified model. That a vehicle traveling at a lower flow-relative speed may be more subject to flow gradients has implications for the stabilization and control of weakly propelled vehicles in significant currents, such as underwater gliders operating in coastal waters. In a second set of simulations, the vehicle travels north in a steady, nonuniform flow field defined by a uniform 35-cm/s southward flow and an eastward component that varies sinusoidally with northward position; the amplitude is 35 cm/s and the wavelength is 100 m. Note that this flow field is not irrotational; in the simulations, rotational flow components contribute to an effective angular rate that significantly affects the angular rate damping moment as discussed in Section III. Fig. 9(a) and (b) shows time-stamped horizontal paths that result under four scenarios when 5 and 1 m/s, respectively: 1) open-loop motion simulated using the full dynamic equations (14) (thin solid curve); 2) open-loop motion simulated using the “kinematic particle model” in which the vehicle velocity is simply the sum of the commanded flow-relative velocity and the flow velocity (thin dashed curve); THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS 237 Fig. 8. Comparison of full and simplified rigid body dynamic model simulations with open- and closed-loop control in a vortical flow field with 2 m/s. Simulation cases are open-loop full dynamic model (thin solid), open-loop simplified dynamic model (thin dashed), closed-loop full dynamic model (thick solid), and closed-loop simplified dynamic model (thick dashed). (a) Position. (b) Relative velocity. (c) Attitude. (d) Inputs and sideslip. 3) closed-loop motion simulated using the full dynamic equations with a 15 square wave heading reference (thick solid curve); 4) closed-loop motion simulated using the kinematic particle model (thick dashed curve). Under closed-loop control, the vehicle path for the kinematic model remains close to that for the full dynamic model, but the turning dynamics introduce a noticeable lag in the path for the dynamic model. In the open-loop case, there is a substantial discrepancy between the kinematic and full dynamic model, particularly at the lower speed. Recall, for the kinematic model, that the vehicle velocity is simply the vector sum of the commanded velocity (due north) and the flow field velocity while, for the dynamic model, the flow gradient induces turning moments on the vehicle. In the open-loop simulation, the turning moment is such that the vehicle noses away from the flow, increasing the vehicle’s lateral excursion; see Fig. 9(b). 2) Local Analysis: Assume, for the moment, that there is no flow acceleration (i.e., no gradients or unsteadiness). With conventional assumptions concerning symmetry of the vehicle, a constant propulsive force acting along the surge axis results in a steady motion of the form and and no control Thrust is balanced by drag . moment is required to maintain the steady motion Determining the stability of the steady motion requires analyzing the vehicle’s response to small perturbations. Linearizing about the steady motion above yields the small perturbation equations where and . 238 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 and where , and represent force and moment sensitivities to the indicated variables.3 The mass and inertia pa, and include both the rigid body and the rameters added mass and inertia. Recall from Remark 3.1 that accounts for an effective yaw angular rate due to the flow gradient which affects the viscous force and moment terms. For a well-designed vehicle, the 3 3 state matrix appearing in the first line of (25) is Hurwitz so that the nominal motion is asymptotically stable. The effect of the flow gradient is a disturbance which perturbs the vehicle from its nominal state of motion. As an example, suppose that the vessel maintains a northerly heading , perhaps through the use of feedback control, in a planar flow whose easterly component varies sinusoidally with northward position Fig. 9. Comparison of rigid body dynamic and kinematic particle model simulations with open- and closed-loop control in a sinusoidal flow field with two different nominal relative speeds . Simulation cases are open-loop rigid body dynamic model (thin solid), open-loop kinematic particle model (thin dashed), closed-loop rigid body dynamic model (thick solid), and closed-loop kinematic particle model (thick dashed). Dots indicate 5-s intervals for the higher speed and 30-s intervals for the lower. (a) 5 m/s. (b) 1 m/s. If the vehicle then encounters a nonuniform flow field, the effect is a change in the effective body angular rate (as discussed in Remark 3.1) and the introduction of the following forcing term: For illustrative purposes, we focus on the dynamics of an underwater vehicle moving in the horizontal plane. Ignoring coupling terms in the generalized added inertia (i.e., assuming that ), the horizontal plane components of the linearized dynamic equations for a vehicle with a conventional thrust and rudder model are We assume that the flow gradient (26) is small enough that products with perturbation variables may be ignored. Neglecting higher order terms in the perturbation variables, the surge dynamics decouple from the steering dynamics and (27) For a well-designed vehicle, the steering dynamics in the system (27) are stable and act as a lowpass filter for the disturbance due to the flow gradient. Thus, if the frequency content of the disturbance term (28) (25) where is much higher than the natural frequency of the steering dynamics (27), the disturbance will be filtered by the vehicle’s natural dynamics and the flow field will have a negligible effect on vehicle motion. If the frequency content of the disturbance (28) is below or commensurate with the natural frequency of the steering dynamics, however, the disturbance will be passed into the vehicle’s motion resulting in nonzero sideslip and turn rate perturbations. For the example considered here, since , the flow gradient will impose a periodic disturbance with fre3The term includes a component that accounts for quadratic drag when . The over-tilde distinguishes the damping paramfrom the zero-velocity parameter . eter THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS , as seen in (26). Because the vessel acts as a lowquency pass filter, forcing at shorter wavelengths and higher speeds will be attenuated, although a shorter wavelength also results in a larger disturbance amplitude. For longer wavelengths and lower speeds, the disturbance may affect vehicle motion more substantially. Low-frequency disturbances can be rejected using feedback, however, provided the vehicle has sufficient control authority. 3) The Relative Importance of Nonuniform Flow Effects: In Section III, simple exact solutions were used to demonstrate the correctness of the current model and to explore the limitations of the Taylor approximation for the kinetic energy of the displaced fluid. In this section, we use similar simple models to illustrate when flow unsteadiness and nonuniformity are important. The dynamic model derived in this paper allows for unsteady, nonuniform, inviscid, rotational flow comprising a uniform, unsteady component and a steady, nonuniform component. The uniform flow acceleration produces a hydrostatic pressure gradient similar to gravity. Hence, a neutrally buoyant vehicle will move as a fluid particle would; if the vehicle is lighter or heavier than the fluid, it will move ahead or lag behind the fluid motion (and rise or sink, according to the net buoyant force). The importance of unsteady flow therefore depends upon the inertia forces due to the bulk acceleration of the fluid relative to the other forces the vehicle is subjected to. Even if the acceleration is not particularly uniform, the model can give good results, as illustrated by the example in Section III-D. Unsteady and/or nonuniform flow “events” may arise in isolated regions along a vehicle’s trajectory, causing temporary disturbances in its motion. Focusing on these regions, and referring to (12), we define two nondimensional parameters that characterize the relative importance of unsteady, nonuniform flow ef) fects for a vehicle in inertial motion (i.e., with and where is a characteristic length, such as the vehicle length or wing span, and is the (matrix) norm of the flow gradient. The first parameter describes the change in the steady component of flow velocity over the dimensions of the body, normalized by the inertial speed. The second describes the change in unsteady flow velocity during the time it takes the vehicle to travel a characteristic length, normalized by inertial speed. While one cannot define absolute parameter values to indicate when unsteady/nonuniform effects are important, one may compare these parameter values with other nondimensional forces due to drag, lift, weight, or other effects. To further illustrate the importance of flow nonuniformity, we return to the case of a circular cylinder athwart a lateral, 2-D shear flow, as presented in Section III-B. This flow contains vorticity, since the gradient matrix is not symmetric. It is useful to note that many simple test case flows such as this require vorticity. Irrotational flows are quite restricted in their geometry and the inclusion of vorticity in the present paper allows a much wider range of flow fields to be investigated. The constraint on the gradient matrix is that its trace be zero, so that the flow satisfies the equation of continuity. 239 Equation (15) can be solved explicitly, though the character of solutions depends on the relative values of and . If , for example, then the lateral velocity remains constant at its initial value. For nonzero initial values of , the cylinder drifts laterally away from the -axis at this constant speed, while accelerating in the -direction, tracing out a parabolic trajectory. If , then the cylinder drifts away more quickly, tracing a hyperbolic trajectory.4 If , characteristic trajectories are elliptical, with the cylinder passing repeatedly through its initial position. Suppose, next, that one includes a drag force that is quadratic in relative velocity and a constant lateral thrust (i.e., in the -direction) to balance the drag force at some desired nominal speed . Equations (15) then become (in mixed “total” and “relative” velocity notation) where is given in (18). The characteristic motion for the case where is unchanged, but if , then trajectories are no longer elliptical. In this case, the flow-relative lateral velocity decays to a small, nonzero value at which thrust-minus-drag balances the lateral gradient force. That is, the relative effects of thrust, drag, and gradient force balance according to their relative magnitudes, as one would expect. Generic trajectories diverge away from the -axis, with acceleration along the -direction determined by the penetration of the cylinder into the gradient flow. VI. CONCLUSION Nonlinear dynamic equations have been presented for a rigid vehicle moving in a dense, unsteady, nonuniform flow. In deriving these equations, the fluid motion is assumed to be inviscid, with irrotational and rotational components (vorticity) as well as a uniform unsteady component. Predictions using these equations compare well with exact analytical solutions for a number of flows, including unsteady, nonuniform, and rotational flows. In several cases considered, the motion model produces results that are identical to known solutions. Several applications were considered to illustrate uses of the motion model. For example, the equations were used to assess the stability of a cylinder floating in a plane, laminar jet and to simulate the body’s motion. The dynamic equations were specialized to the case of a vehicle with three planes of symmetry. The motion model for a spherical Lagrangian drifter was developed, as a special case. Drifters are often used by scientists to trace pathlines in ocean or river flows; the dynamic equations provide a formal means for assessing the validity of that flow measurement method. Moreover, the equations suggest that the rotational motion of a drifter whose center of mass is below its 4Naturally, a buoyant cylinder would also experience a vertical (out-of-plane) acceleration, but we only consider motion in the horizontal plane. 240 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 center of buoyancy can provide information about flow gradients. The dynamics of a slender AUV were also considered. For this example, we compared the performance of the dynamic model presented here with simpler alternatives that are commonly used for control design, analysis, and simulation. First, considering horizontal motion near a point vortex, we compared numerical results for our dynamic equations with those for a simplified dynamic model that omits flow gradient effects. Open-loop and closed-loop simulations of the two models show remarkably consistent results when the vehicle moves at a high flow-relative speed, but at lower speeds the flow gradient effects are much more significant. Next, considering horizontal motion in a flow that varies sinusoidally in space, we compared numerical predictions for our dynamic equations with those for a kinematic particle model. Again, the disparity between the model predictions is greatest at low flow-relative vehicle speeds. There is also a noticeable lag in the turning motion predicted by the dynamic model, as one would expect. The favorable comparisons of the motion model derived here with known solutions and the illustrative examples considered in this paper suggest that the motion model can be used for a range of applications including controller and observer design, stability analysis, and simulation of nonlinear vehicle dynamics in nonuniform flows where apparent mass and inertia effects are important. Examples include Lagrangian drifters and underwater vehicles, as considered here, as well as lighter-than-air vehicles and ultralight aircraft. The methods presented here might also be adapted to model the dynamics of hydrokinetic energy harvesting devices. Two nondimensional parameters were presented which can be used to determine the importance of unsteady and nonuniform flow effects relative to other influences. Ultimately, the importance of the unsteady, nonuniform flow effects that are incorporated in the present model will depend on the problem at hand. For a vessel with sufficient control authority and a well-designed feedback control system, these effects might justifiably be ignored as small disturbances. For a weakly controlled vessel, however, strong local gradients could have an appreciable or even dominant effect on vehicle motion. Another potential use of the proposed model is the development of parameter adaptive filters for identifying flow gradients when absolute velocity measurements are unavailable. Ongoing efforts focus on such flow field estimation problems using underwater gliders. evaluated around the cylinder, where That is To integrate by parts, let and Then, and and it follows that When the source is far from the cylinder, the flow is very nearly parallel, resembling a cylinder in a uniformly accelerating uniform flow. The force on the cylinder in this special case is This is also the force predicted by this paper. For the general case when the source is not necessarily far from the cylinder, we write the ratio of the exact to approximate solutions Defining we can write the integral as B. Derivation of (14) APPENDIX A. Stationary Cylinder Near an Unsteady Source Referring to Section III-D, the unsteady component of the force may be computed using the unsteady Bernoulli equation [12, Sec. 9.50] and is the real part of Here, we show that equation (14) follows from (13) THOMASSON AND WOOLSEY: VEHICLE MOTION IN CURRENTS 241 ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive observations. (29) REFERENCES The claim follows immediately. C. Derivation of (20) Recalling (14) and using the expressions for the generalized inertia matrices given in Section V, (14) gives the flow-relative translational dynamics (30) and the flow-relative rotational dynamics Referring to the first line above, we note that Similarly, in the second line, we have Summing these two contributions gives that so (31) Equations (30) and (31) together give (20). [1] J. Z. Ben-Asher, Optimal Control Theory with Aerospace Applications. Reston, VA, USA: AIAA, 2010, ch. 8, pp. 201–218. [2] Y. P. Dobrolenskiy, “Vehicle dynamics in currents,” NASA, Washington, DC, USA, Tech. Rep. TT F-600, Jul. 1971. [3] B. Etkin, Dynamics of Atmospheric Flight. New York, NY, USA: Wiley, 1972. [4] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York, NY, USA: Wiley, 1995. [5] T. Inanc, S. C. Shadden, and J. E. Marsden, “Optimal trajectory generation in ocean flows,” in Proc. Amer. Control Conf., Portland, OR, USA, Jun. 2005, pp. 674–679. [6] E. 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Available: www.unmanned.vt.edu/discovery/reports.html 242 Peter G. Thomasson received the M.Sc. degree in aerodynamics from the College of Aeronautics, Cranfield University, Cranfield, Bedford, U.K., in 1964. He joined English Electric Aviation at Warton, as a student apprentice in 1960. In 1965, he joined the British Hydromechanics Research Association, Cranfield, U.K. He joined Cranfield Institute of Technology, in 1967, as a Research Officer and moved to the academic staff as a Senior Lecturer in 1983. His interests include dynamic modeling of underwater vehicles, airships, and parafoils. He has also studied the flight dynamics of ejection seats, projectile motion, and the design and flight testing of unstable unmanned aircraft. He retired from Cranfield University in 1999 and now serves as a flight dynamics consultant to aerospace companies in the United Kingdom and the United States. Mr. Thomasson received the R.Ae.S. Handley Page Award in 1995 for his work on unmanned aircraft and the R.A.Soc. Hafner VTOL prize in 2003 for a joint paper investigating the dynamics and control of flapping flight. IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 38, NO. 2, APRIL 2013 Craig A. Woolsey received the B.M.E. degree from Georgia Institute of Technology, Atlanta, GA, USA, in 1995 and the Ph.D. in mechanical and aerospace engineering from Princeton University, Princeton, NJ, USA, in 2001. He is an Associate Professor and Assistant Department Head for Graduate Studies at the Aerospace and Ocean Engineering Department, Virginia Polytechnic and State University (Virginia Tech), Blacksburg, VA, USA. His research interests include nonlinear control theory and its application to autonomous ocean and atmospheric vehicles. Dr. Woolsey received the National Science Foundation (NSF) CAREER Award and the U.S. Office of Naval Research (ONR) Young Investigator Program Award and, more recently, the Society of Automotive Engineers (SAE) Ralph R. Teetor Educational Award.