IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY

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1 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY Approximate Analytical Turning Conditions for Underwater Gliders: Implications for Motion Control Path Planning Nina Mahmoudian, Jesse Geisbert, Craig Woolsey, Member, IEEE Abstract This paper describes analysis of steady motions for underwater gliders, a type of highly efficient underwater vehicle which uses gravity for propulsion. Underwater gliders are winged underwater vehicles which locomote by modulating their buoyancy their attitude. Several underwater gliders have been developed have proven their worth as efficient long-distance, long-duration ocean sampling platforms. Underwater gliders are so efficient because they spend much of their flight time in stable, steady motion. Wings-level gliding flight for underwater gliders has been well studied, but analysis of steady turning flight is more subtle. This paper presents an approximate analytical expression for steady turning motion for a realistic underwater glider model. The problem is formulated in terms of regular perturbation theory, with the vehicle turn rate as the perturbation parameter. The resulting solution exhibits a special structure that suggests an efficient approach to motion control as well as a planning strategy for energy efficient paths. Index Terms Motion control, path planning, steady motion, underwater glider. I. INTRODUCTION F OR decades, marine scientists engineers have envisioned the use of autonomous underwater vehicles (AUVs) for long-term, large-scale oceanographic monitoring. However, the propulsion systems power storage limitations of conventional AUVs do not allow for long-term deployments, at least not without a significant investment in undersea infrastructure to enable recharging. Conventional, battery-powered, propeller-driven AUVs can only operate on the order of a few hours before their power is depleted. On the other h, buoyancy-driven underwater gliders have proven to be quite effective for long-range, long-term oceanographic sampling. Gliders are highly efficient winged underwater vehicles which locomote by modifying their internal shape. In a typical configuration, a buoyancy bladder modulates the glider s net weight while one or more moving mass actuators modulate its center Manuscript received June 07, 2007; revised March 31, 2009; accepted December 14, First published February 02, 2010; current version published February 10, This work was supported by the U.S. Office of Naval Research under Grant N Associate Editor: A. Pascoal N. Mahmoudian C. Woolsey are with the Department of Aerospace Ocean Engineering, Virginia Polytechnic State University, Blacksburg, VA USA ( ninam@vt.edu; cwoolsey@vt.edu). J. Geisbert is with the Naval Surface Warfare Center Carderock Division, West Bethesda, MD USA ( jesse.geisbert@navy.mil). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JOE of mass. By appropriately cycling its actuators, the vehicle can propel itself with great efficiency. The exceptional endurance of underwater gliders is due to their reliance on gravity (weight buoyancy) for propulsion attitude control. The first generation of underwater gliders includes Seaglider [1], Spray [2], Slocum [3]. These legacy gliders were designed with similar functional objectives [4], [5] so they are similar in weight, size, configuration. Although each legacy glider has demonstrated significantly greater endurance than conventional AUVs, it was recognized in [5] that a change in vehicle configuration could further increase glider efficiency. A prototype of the blended wing-body glider proposed in [5] has been developed jointly by the Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego the Applied Physics Laboratory, University of Washington, Seattle. Potential applications of this vehicle, dubbed the Liberdade/XRay, include long-term ocean sampling as well as persistent undersea surveillance. With the primary emphasis on locomotive efficiency, glider motion control has been a secondary concern. However, applications such as undersea surveillance may require more careful consideration of guidance control requirements. This paper builds on the preliminary work in [6] [7] to provide a better understing of glider maneuverability, particularly with regard to turning motions. Outcomes will include more effective maneuvering behaviors for existing gliders improved design guidelines for future underwater gliders. As a preliminary step, we develop an approximate analytical expression for steady turning motion for a realistic glider model. Analytical results, approximate or otherwise, are important for motion planning also for vehicle design, as they may provide guidelines for sizing actuators stabilizers. The conditions for steady turning flight of an underwater glider differ significantly from those for an aircraft. Deriving a closed-form expression for the steady turn is quite challenging. Instead, we begin by considering wings-level equilibrium flight as a nominal motion consider turning motion as a perturbation. Given a desired equilibrium speed glide path angle, one may determine the center of gravity (CG) location the net weight required. The resulting longitudinal gliding equilibrium is the nominal solution to a regular perturbation problem in which the vehicle turn rate is the perturbation parameter. The derivation of analytical conditions for steady turning motions requires a vehicle dynamic model. Nonlinear dynamic models presented in [6] [8] provided the basis for investigations of longitudinal gliding flight. Although the emphasis was /$ IEEE

2 132 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY 2010 on wings-level flight, turning motions were also discussed in [6] [7] examples were shown for the given vehicle models with chosen parameter values. Bhatta [7] also presented the results of a numerical parametric analysis. However, no analytical expressions were provided, so it is difficult to make general conclusions about the relationship between parameter values turning motion characteristics. Here, following a more analytical approach as in [9], we study the existence stability of steady turning motions for general parameter values. Section II develops a dynamic model for an underwater glider. Section III gives the conditions for wings-level gliding flight. These provide the nominal conditions for a regular perturbation analysis in Section IV, by which approximate conditions for steady turning flight are derived. Section V introduces the problem of optimal motion planning for underwater gliders. Here it is recognized that, by exploiting the special structure of the approximate solution given in Section IV, one may apply existing optimal path planning results obtained for planar mobile robots. A brief summary discussion is given in Section VII. II. VEHICLE DYNAMIC MODEL The glider is modeled as a rigid body of mass. The vehicle displaces a volume of fluid of mass.if, then the vehicle is neutrally buoyant. If the excess mass is greater than zero, the vehicle is heavy in water tends to sink. If is negative, the vehicle is buoyant in water tends to rise. For underwater gliders, is typically modulated by an inflatable bladder, which changes the value of by changing the displaced volume. Define a body-fixed, orthonormal reference frame centered at the geometric center of the vehicle (the center of buoyancy) represented by the unit vectors,,. The vector is aligned with the longitudinal axis of the vehicle, points out the right wing, completes the right-hed triad; see Fig. 1. Define another orthonormal reference frame, denoted by the unit vectors,,, which is fixed in inertial space such that is aligned with the force due to gravity. The relative orientation of these two reference frames is given by the proper rotation matrix, which maps free vectors from the bodyfixed reference frame into the inertial reference frame. Let represent the stard basis vector for,, let the character denote the skew-symmetric matrix satisfying for vectors. Then, in terms of conventional Euler angles (roll angle, pitch angle, yaw angle ), we have The location of the body frame with respect to the inertial frame is given by the inertial vector. Let represent the translational velocity let represent the rotational velocity of the AUV with respect to inertial space, where are both expressed in the body frame. The kinematic equations are Fig. 1. Reference frames. The linear momentum of the body/fluid system is denoted by the body vector. The angular momentum of the body/fluid system about the body frame origin is denoted by the body vector. The vectors are the conjugate momenta corresponding to, respectively. Explicit expressions relating to require a number of additional definitions. A stard reference on marine vehicle modeling is [10]. For simplicity, we assume that, so that the vehicle center of mass (less the contribution of ) is located in the - plane. We assume that the lateral mass particle is located along the -axis; its position in the body frame is denoted. (See Fig. 1.) Because this paper concerns steady motions, we do not consider the internal dynamics of the moving mass actuator; the position of is treated as a parameter. A multibody dynamic model is considered in [11], for example. The mass matrix is the sum of two matrices. First, the added mass matrix accounts for the energy necessary to accelerate the fluid around the body as it translates. If the underwater glider s external geometry is such that the - - planes are planes of symmetry, then the added mass matrix is diagonal The second term in is the rigid body mass matrix, where is the identity matrix The inertia matrix is the sum of three components. First, the added inertia matrix accounts for energy necessary to accelerate the fluid as the body rotates. Assuming that the - - planes are planes of symmetry, the added inertia matrix is diagonal, like the added mass matrix The second term in, the rigid body inertia matrix, accounts for the distribution of fixed mass relative to the body reference frame. This matrix takes the form (1) (2)

3 MAHMOUDIAN et al.: APPROXIMATE ANALYTICAL TURNING CONDITIONS FOR UNDERWATER GLIDERS 133 where the off-diagonal terms in arise, for example, from the rigid body s offset center of mass located at the point, in the body frame. Third, the matrix accounts explicitly for the inertia contribution of the laterally adjustable point mass In addition to the added mass the added inertia, generally, there will be potential flow inertial coupling between translational rotational kinetic energy. This coupling is characterized by the matrix For the class of underwater gliders considered here The terms appearing in may be significant [12], though they are often ignored for simplicity. In terms of the kinetic energy metric the velocities, the momenta are The dynamic equations, which relate external forces moments to the rate of change of linear angular momentum, are (4) (5) The terms represent external moments forces which do not derive from scalar potential functions. These moments forces include control moments, such as the yaw moment due to a rudder, viscous forces, such as lift drag. The viscous force moment are most easily expressed in the current reference frame. Let denote the vehicle s angle of attack let denote the sideslip angle. The current frame is related to the body frame through the proper rotation matrix (3), through the dynamic pressure, as well as the Reynolds number the vehicle s geometry. We make the following common assumptions concerning the viscous forces. The side force is linear in its arguments vanishes when are zero. The lift force static pitch moment are linear in vanish when is zero. In addition, we assume the stard quadratic-in-lift model for the drag force. The matrix appearing in the viscous moment contains terms which characterize viscous angular damping (such as pitch yaw damping). Equations (1), (2), (4), (5) completely describe the underwater glider motion in inertial space, with actuators fixed. In studying steady motions, we typically neglect the translational kinematics (1). Moreover, the structure of the dynamics (4) (5) is such that we only need to retain a portion of the rotational kinematics (2). Define the tilt vector, which is simply the body frame unit vector pointing in the direction of gravity. Referring to (2), it is easy to see that. Therefore, we consider the following, reduced set of equations: (6) (7) (8) The steady-state flight conditions are determined by solving the nonlinear state equations (6) (8) for the state control vectors that make the state derivatives identically zero. Because of the complexity involved in computing an analytical solution, numerical algorithms for computing trim conditions are common [13]. Here we take an analytical approach to find (approximate) solutions for the steady-state flight in terms of the model parameters. First, the case of wings-level gliding flight is reviewed. Then, steady turning (helical) flight is analyzed. III. WINGS-LEVEL GLIDING FLIGHT The conditions for wings-level gliding flight are that,,. The second condition implies that therefore that. The third condition implies that. Also, we require that that. Inserting these conditions into (7) (8) solving for the remaining equilibrium conditions gives (9) The viscous force moment are (10) Following the analysis in [9], one may use (10) to show that (11) where can be any real-valued scalar where denotes the rudder angle, if a rudder is present. The viscous force moment depend on the vehicle s speed

4 134 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY 2010 Note that system is a particular solution to the linear algebraic Having obtained values for ( for, therefore ), one may solve the third component of (12) for the required net weight for a given glide speed obtained from (10) for which. The null space of is. The free parameter is a measure of how bottomheavy the vehicle is in a given, wings-level flight condition. This parameter plays an important role in determining longitudinal stability of the gliding equilibrium. Next, one may solve (9) for,, given a desired speed a desired glide path angle. Expressed in the inertial frame, (9) gives (14) Thus, one may independently assign the glider s equilibrium attitude, by moving the center of mass according to (11), its speed, by changing the net weight according to (14). For the minimum drag flight condition, for example (12) Equation (12) states that there is no net hydrodynamic force in the -direction that net weight is balanced by the vertical components of the lift drag forces. The lift drag force components, expressed in the current frame, are where where is the reference area used in the definitions of the nondimensional coefficients. The first component of (12) may be rewritten as which implies that (13) Given values of, the best possible glide path angle is This glide path maximizes range (in still water) corresponds to minimum drag flight For turning flight, write IV. STEADY TURNING FLIGHT must be parallel to. Therefore, one may where is the turn rate. A steady turn is an asymmetric flight condition, so we no longer assume that are zero. Moreover, to effect maintain such an asymmetric flight condition requires that or or both be nonzero. Turning flight for an underwater glider is considerably different than turning flight for an aircraft. For an underwater glider, the center of mass is typically not the origin of the body reference frame angular linear momenta are coupled through inertial asymmetries. Linear momentum is not parallel to linear velocity, because added mass is directional because of coupling between linear angular velocity introduced by the offset center of mass. Propulsion is provided not by a thruster but by the net weight of the vehicle (weight minus buoyant force). In fact, the problem of finding analytical steady turning solutions for underwater gliders is quite challenging. Instead, we formulate the problem as a regular perturbation problem in the turn rate seek a first-order approximate solution. To argue that the higher order solutions are small corrections requires some well-founded notion of small so we begin by nondimensionalizing the dynamic equations. We choose the reference parameters length: mass: time: where is a characteristic length scale for the vehicle (such as length overall) is the nominal speed. With these definitions, the nondimensional momenta are related to the nondimensional velocities through the nondimensional generalized inertia matrix as follows: These conditions provide an upper bound on achievable performance, but operational considerations may dictate a steeper glide path angle. where

5 MAHMOUDIAN et al.: APPROXIMATE ANALYTICAL TURNING CONDITIONS FOR UNDERWATER GLIDERS 135 where where,,,,, are nondimensional stability derivatives. Note that The nondimensional dynamic equations are (15) (16) (17) where the overdot represents differentiation with respect to nondimensional time where where As we have stated, remains constant in turning flight; equivalently, remain constant. We seek equilibrium solutions for which the perturbed value of takes the following form: By construction, the perturbed equilibrium turning motion will have the same pitch angle as the corresponding, unperturbed wings-level flight condition. Using the definitions observations above, the equilibrium equations may be written as To express the viscous forces moments explicitly, we also define (18) To simplify the analysis, we assume that where,, are nondimensional stability derivatives representing rotational damping. The assumption that the damping moments are decoupled is reasonable for a vehicle with two planes of external geometric symmetry. Recall that for a steady turn. Define a characteristic frequency let denote its nondimensional value. Let where is a small, nondimensional parameter. One may treat the problem of solving for steady turning flight conditions as an algebraic regular perturbation problem in. When, the vehicle is in wings-level equilibrium flight. If, then either or or both must be nonzero. (Recall that remains fixed at its nominal value, which corresponds to the nominal wings-level flight condition when are zero.) Having nondimensionalized the terms appearing in the dynamic equations, we simplify notation by omitting the overbar; in the sequel, all quantities are nondimensional unless otherwise stated. The nondimensional equilibrium equations are (19) To obtain the regular perturbation solution in, substitute the following polynomial expansions for,,,,, : Here in the sequel, we suppress the subscript to simplify notation. Let ; the rudder deflection will appear as a free parameter in the solution to the regular perturbation problem. Substituting these polynomial expansions into

6 136 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY 2010 (18) (19) collecting powers of gives a regular perturbation series in. Solving the coefficient equation for gives the nominal, wings-level flight conditions. Solving the coefficient equation for gives approximate values for,,,,,, accurate to first order in. The first-order solution to the regular perturbation problem defined by (18) (19) is (20) (21) (22) (23) (24) (25) where represents the sine function represents cosine. These explicit analytical expressions provide insight concerning the role of design parameters such as wing sweep angle, vertical stabilizer size, moving mass actuator size, rudder size in determining a vehicle s turning capability. They also exhibit an interesting structure, which is discussed in Remark 4.1. Remark 4.1: That,, remain constant to first order in suggests that lateral mass deflections rudder deflections have no first-order effect on speed or angle of attack. Thus, the problem of controlling longitudinal motion (speed glide path angle) decouples from the problem of controlling directional motion (turn rate) to first order in. As discussed in Section V, this observation suggests a natural approach to motion control path planning for underwater gliders. Remark 4.2: Note that the rudder deflection appears as a free parameter in (20) (25). For a rudderless vehicle, one simply sets. However, including a rudder provides additional freedom to control the lateral directional dynamics. For example, using (23), one could adjust the rudder angle so as to zero the sideslip angle, to first order in, thereby reducing the total drag. [Equation (23) gives guidance for sizing a rudder for this purpose.] Drag reduction is especially critical for underwater gliders, whose primary operational advantage is efficiency. On the other h, a rudder is an external actuator which is subject to damage or fouling which introduces an additional failure mode into the system. Equations (20) (25) provide a first-order approximation for steady turning motions. To assess stability of the true, neighboring turning motion, one may linearize about the approximate equilibrium condition compute the eigenvalues. Recognizing that the eigenvalues of the resulting time-invariant state matrix depend continuously on its parameters, stability properties of the true equilibrium may be inferred from stability properties of the approximate equilibrium provided that: 1) the equilibrium is hyperbolic 2) is small relative to the real part of every eigenvalue. See [14, Sec. 1.7] for a brief discussion or [15, Ch. 9] for more details. V. MOTION CONTROL AND PATH PLANNING Having characterized steady wings level turning flight, as discussed in Sections III IV, respectively, one can formulate a motion control strategy that relies on the analytical results. Given feasible values for desired speed, glide path angle, turn rate, for example, one may compute feedforward actuator comms to adjust the net weight CG to achieve the given flight condition. An illustration of such a control system is shown in Fig. 2. The equation in the feedforward block is a notional expression indicating the first step in the motion control scheme: obtain the parameter values for (that is, for,, ) that correspond to the

7 MAHMOUDIAN et al.: APPROXIMATE ANALYTICAL TURNING CONDITIONS FOR UNDERWATER GLIDERS 137 Fig. 2. Steady motion-based feedforward/feedback control system. desired steady motion (characterized by,, ). However, since the turning motion results are only approximate, a feedback compensator is included to compensate for approximation error other uncertainties. The feedback-compensated parameter comms are then realized within the vehicle dynamics through an appropriately designed servocontrol system. The design analysis of such a feedforward/feedback motion control system requires a more sophisticated dynamic model, one which incorporates the buoyancy moving mass actuator dynamics servocontrol laws, for example. See [11] for details. Beyond its advantages for general motion control, as described above, the approximate solution for steady turning flight described in Section IV exhibits remarkable properties that suggest a procedure for planning energy efficient paths. Referring to Remark 4.1, recall that both the speed the angle of attack remain constant to first order in as the turn rate varies. Referring to (14), the equilibrium speed for wings-level flight varies with angle of attack net mass. In practice, one would choose maintain an angle of attack for efficient flight, such as the angle for minimum drag. One may also assume that ( therefore ) remains constant, as changes in net weight are relatively costly. (The assumption that remains constant ignores small changes in buoyant force due to hydrostatic variations in the ambient density; here, we assume that water is incompressible.) Given that remain constant to first order in, recalling that the pitch angle was constrained to be constant in the problem formulation for steady turning flight, it follows that the vertical component of velocity remains constant to first order in. To see this, compute Note that is precisely the vertical component of velocity in unperturbed, wings-level flight. Since both the magnitude the vertical component of velocity remain constant to first order in, so does the horizontal component of velocity. Viewing the glider s motion from above (i.e., projecting trajectories onto the horizontal plane), underwater glider equilibrium motions appear as constant-speed straight line circular paths. Moreover, the circular paths must satisfy physical bounds on turn radius. Assume then that the vehicle operates in quasi-steady motion, passing smoothly along a manifold of steady turning motions that all correspond to a single nominal speed pitch angle (characterizing the nominal wings-level gliding motion). Viewed from above, this quasi-steady motion is characteristic of a simple model for a wheeled mobile robot: the constant-speed kinematic car. One essential distinction between the kinematic car the underwater glider is that the kinematic car drives in the direction of its heading angle while an underwater glider experiences sideslip. To first order in, the sideslip angle is, where is given in (23). Therefore, we have an approximate one-to-one relationship between the course angle (the azimuthal direction of the velocity vector) the vehicle turn rate. Accounting explicitly for the variation in sideslip angle with turn rate, one may model the horizontal component of the quasi-steady underwater glider motion as a constant-speed kinematic car. Of course, this model is based on an approximation any motion planning control system based on such a model would require feedback to eliminate the effect of approximation error other uncertainties; see [11], for example. One approach to planning efficient paths for underwater gliders is to concatenate steady motions to drive the vehicle from a given initial point, with a given initial heading, to a desired final point with a desired heading. In implementation, one would need to design a compensator to manage the transient behavior at the switching points. A question of reachability also arises, since an underwater glider must ascend or descend to locomote. A glider cannot progress between two points at the same depth, for example, without concatenating at least one ascending one descending motion. Here, we restrict our attention to situations where the final point is strictly below (or strictly above) the initial point can be reached in a single descending (or ascending) flight without exceeding the vehicle s physical limitations (such as the shallowest glide slope). With this assumption, noting that the sink rate the glide path angle remain constant to first order in,we project the vehicle path onto the horizontal plane simply ignore the vertical component of motion. Returning to the path planning problem, a meaningful objective would be to choose

8 138 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY 2010 a path that minimizes the total change in depth. Underwater gliders use potential energy for propulsion; minimizing change in depth minimizes the amount of work required to reinflate the buoyancy bladder. Because a glider s horizontal speed remains constant, whether the vehicle is turning or flying straight, because the descent rate remains constant, it follows that the optimal path must be the one of minimum length. Equivalently, given that the vehicle s horizontal component of speed is constant, the optimal path is the minimum time path. Thus, considering only motion in the horizontal plane, the path planning problem reduces to the following: choose the turn rate, within specified bounds, to minimize the time of transit from a given initial point to a given final point with a specified initial final heading. A. Dubins Car Viewing the glider motion from above, the vehicle moves in the manner of a planar vehicle which drives forward at constant speed which may turn, in either direction, at any rate up to some maximum value. Such a vehicle is often referred to as a Dubins car. Dubins [16] showed that the minimum time control policy which brings the car from a given point to another, with specified initial final directions, is a concatenation of three motions: a left or right turn at maximum rate, a straight transit or a second turn at maximum rate, a final turn at maximum rate. Note that a constant speed turn at maximum turn rate corresponds to a turn of minimum radius. (Actually, Dubins considered the problem in terms of minimizing the length of a continuous curve with bounded curvature, but the two problems are equivalent.) Variations of Dubins problem have enjoyed renewed attention in recent years, in part because of increasing interest in mobile robotics. Reeds Shepp [17] characterized the family of optimal trajectories for a variation of the Dubins car in which the vehicle could move in reverse, as well as forward. Sussmann Tang [18] generalized further by convexifying the nonconvex control set defined in [17], managing to sharpen the results presented by Reeds Shepp by Dubins. Parallel studies, as outlined in a series of the Institut National de Recherche en Informatique et en Automatique (INRIA) technical reports papers [19] [22], also investigated controllability optimal path planning for Dubins Reeds Shepp mobile robots. Anisi s thesis [23] reviews some of the recent results provides some historical context. For the purpose of explaining the Dubins car problem, let represent the vehicle s state (i.e., its position heading in the horizontal plane) let the turn rate be the input, which satisfies the inequality constraint. The vehicle moves at some constant, nonzero forward speed. Then, the equations of motion are where (26) The problem is to find an input history which brings the system from a specified initial state to a specified final state while minimizing. Note that, since the speed is constant, this minimum time problem is equivalent to Dubins Fig. 3. Solid model of the underwater glider Slocum (generated using Rhinoceros 3.0). For this model, the mass is 50 kg, the wing span is 1 m, the outer diameter is 0.2 m, the overall length is 1.5 m. minimum arclength problem. As shown in [16], the optimal history contains at most three distinct segments (i.e., two switches of the control among its three possible values). These results are sharpened in [18], where the control histories are shown to be of the type BSB (for bang singular bang ) or BBB (for bang bang bang ). Having characterized the family of cidate optimal input histories as a small set of geometrically feasible paths, one simply compares their lengths selects the shortest. B. Dubins Car With Control Rate Limits The classical Dubins car problem assumes that turn rate can be treated as an input with magnitude limits but no rate limits. (Equivalently, the arclength minimization problem imposes limits on the curvature but not on its derivative.) The assumption may or may not be appropriate for wheeled robotic vehicles, but it is certainly not appropriate for underwater gliders. For these vehicles, turn rate is controlled indirectly by shifting the CG to effect a banked turn, a dynamic process which may have a large time constant. To explore the effect of control rate limits on the Dubins optimal path result, one may augment the state vector given in Section V-A as follows:. Let the turn acceleration be the input: where satisfies the state inequality constraint satisfies the input inequality constraint. The equations of motion are where Although we now consider turn acceleration as an input, we still assume that the underwater glider state varies in a quasi-steady manner. That is, we assume that the vehicle state varies along the continuum of (approximate) equilibrium states, as parameterized by the turn rate. Under this assumption, as shown in Section IV, the vehicle s speed remains constant to first order in turn rate. In fact, this problem has been treated in some detail by Scheuer [24], as summarized in [25]. Her work extends that of Boissonnat et al. [26] of Kostov Degtiariova-Kostova [27], who considered the case where the derivative of the turn rate (equivalently, the derivative of curvature) is constrained, but not the magnitude of the turn rate. In [26], it was shown

9 MAHMOUDIAN et al.: APPROXIMATE ANALYTICAL TURNING CONDITIONS FOR UNDERWATER GLIDERS 139 Fig. 4. Wings-level equilibrium glide characteristics for the Slocum model. The plot on the left shows the glide path angle () pitch angle () corresponding to given values of angle of attack (). The plot on the right shows corresponding values of for steady, descending glide motions. that time-optimal paths exist that they consist of straight segments clothoids at maximum turn acceleration. It was also shown that the minimum time curves can be quite complicated, possibly including infinitely many clothoidal segments. Independently, Kostov Degtiariova-Kostova [27] proposed a method for constructing suboptimal paths from clothoids straight segments. The term suboptimal, as used in [27], means that the amount by which the transit time exceeds the minimum time is bounded by a function of the turn acceleration limit. In [25], Scheuer Laugier suggest an intuitive approach for approximating Dubins paths by concatenating straight, circular, clothoidal segments. They also show that the resulting paths are suboptimal, in the sense described above. The technique described in [25] can be easily implemented quickly executed with respect to the time scale of typical underwater glider motions. VI. NUMERICAL CASE STUDY: SLOCUM To verify our steady turn predictions for a realistic vehicle model, we have applied the results to the model for Slocum, shown in Fig. 3 for which hydrodynamic model parameters are given in [7]. We consider perturbations from a wings-level equilibrium flight condition at speed 1.5 kn angle of attack 4.3, the angle which corresponds to the maximum lift to drag ratio. We assume that, noting that neither Liberdade/XRay nor the deep-water ( thermal ) version of Slocum uses a rudder. Fig. 4 shows the wings-level equilibrium glide characteristics for the Slocum glider. The lift drag parameters are 2.04 rad Other important parameters include 1.5 m 40 kg. Once one has computed the conditions for equilibrium flight, one may examine stability. The simplest approach is spectral analysis. Using the Slocum model described in [7], we linearize about the wings-level, equilibrium flight condition corresponding to the following parameter values: m/s with net mass 0.61 kg. As mentioned in Section III, the free parameter provides a measure of how bottom-heavy the vehicle is, in a given flight condition. This parameter plays an important role in determining longitudinal stability of wingslevel gliding equilibria. The effect of varying on the stability of wings-level turning equilibria has been investigated numerically. The results show that the equilibrium condition mentioned above is stable provided, which agrees with the analysis in Section 8.3 of Bhatta s dissertation [7]. Here, the value of is fixed at 0.117, as in [7]. The eigenvalues of the state matrix corresponding to the given equilibrium condition are All eight eigenvalues of the linearized system have negative real part, so the flight condition is stable. With stability of wings-level equilibrium flight confirmed, one may next compute the first-order solution for,,, as described in Section IV. The approximate equilibrium velocity angular velocity are When, the values above correspond to the given steady, wings-level flight condition. For small, nonzero values of, the values correspond (approximately) to a steady turning motion; see Fig. 5. To determine the range of stable turning motions that can be obtained using this approximation, the equations of motion are linearized about the approximate turning motion, parameterized by. When, all eight eigenvalues of the linearization have negative real part, with being closest to the imaginary axis. As increases, this critical eigenvalue moves left along the real axis, coalesces with another real eigenvalue, then breaks away into a complex conjugate pair. One may infer that the system has a locally unique, stable fixed

10 140 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY 2010 Fig. 5. Wings-level ( =0) turning ( =0:01) flight paths for the Slocum model. Fig. 6. Eigenvalue plots for actual approximate equilibria for 0 <<0:1. (A closer view of the dominant eigenvalues is shown on the right-h side.) TABLE I APPROXIMATE AND ACTUAL STEADY MOTION CONDITIONS FOR V = 1.5 kn AND = 4.3 point provided remains smaller than magnitude of the real part of the critical eigenvalue(s) [15]. Fig. 6 shows the movement of eigenvalues for the approximate equilibrium as well as those for the true equilibrium (computed numerically) as increases from zero. The true approximate root loci agree very closely for. The true system exhibits a stable turning motion for. Table I gives approximate actual values (obtained from numerical simulations) for key variables for various values of. Note that, as increases in value, so does the error between the approximate true equilibrium values. Regardless, the system does converge to a steady turning motion for all values. Remark 6.1: Note in Table I that the actual turn radius is minimum around. Since further increases in (or equivalently in ) fail to lower the turn radius, there is no point in moving the particle beyond this critical location. Such an observation may provide guidelines for actuator sizing

11 MAHMOUDIAN et al.: APPROXIMATE ANALYTICAL TURNING CONDITIONS FOR UNDERWATER GLIDERS 141 TABLE II APPROXIMATE AND ACTUAL STEADY MOTION CONDITIONS FOR V = 2.0 kn AND = 4.3 TABLE III APPROXIMATE AND ACTUAL STEADY MOTION CONDITIONS FOR V = 1 kn AND = 4.3 Fig. 7. Dubins suboptimal paths for Slocum. Open circles denote initial values; open triangles denote final values. in future glider designs. There is no reason, for example, to provide moving mass control authority which does not yield greater turning ability. Comparing the results for speeds of 1.0, 1.5, 2.0 kn (illustrated in Tables I III, respectively), one may observe several trends. For example, in every case, actual speed increases with increasing turn rate. (Recall that the approximation suggests that speed remains relatively constant, for small.) Roll angle sideslip angle (approximate actual) increase more rapidly with turn rate at lower nominal speeds than at higher nominal speeds. Moreover, the discrepancy between the approximate actual values is greatest (for given ) at the lowest speed. Because the relative stability of the nominal flight condition decreases with decreasing speed (i.e., the critical eigenvalues move closer to the imaginary axis), one should expect poorer agreement between the approximation reality at these lower speeds. Revisiting the topic of path planning motion control, as discussed in Section V, Fig. 7 shows the results of a Dubins-type path planning application for the Slocum glider considered in this section. The objective is to construct an input sequence that brings the glider from a given initial position heading to a desired final position heading. The time histories to the left in Fig. 7 show the turn rate history the corresponding heading angle history. Shown on the right-h side is the resulting path, viewed from above. In these simulations, the vehicle travels at a nominal speed of 1.5 kn. The input is the lateral mass location. In the first simulation, the input is subject to magnitude limits, effectively resulting in turn rate limits; the corresponding

12 142 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 1, JANUARY 2010 path in Fig. 7 is labeled as the Dubins path. In the second simulation, the input is subject to both magnitude rate limits, effectively resulting in both turn rate turn acceleration limits; the corresponding path is labeled as the suboptimal path. Note that the time histories corresponding to the suboptimal path are slightly longer than those corresponding to the Dubins path, as one should expect. It must be emphasized that there is no feedback control in these simulations; here, the dynamics are simply evolving under a sequence of open-loop control comms for. A more sophisticated approach, which incorporates actuator dynamics the feedback control system illustrated in Fig. 2, is discussed in [11]. VII. CONCLUSION An approximate solution for steady turning motions of underwater gliders has been derived using a sophisticated dynamic model. The problem was formulated as a regular perturbation problem using wings-level, equilibrium flight as the nominal state turn rate as the small perturbation parameter. As an illustration, the result was applied to an existing model of the Slocum underwater glider. The analytical result, though approximate, is valuable because it gives better insight into the effect of parameters on vehicle motion stability. This insight can, in turn, lead to better usage guidelines for current vehicles design guidelines (e.g., actuator sizing) for future vehicles. An important observation concerning the structure of the approximate solution is that, to first order in turn rate, the glider s horizontal component of motion matches that of the Dubins car, a classic example in the study of time-optimal control of mobile robots. Moreover, these minimum time Dubins paths yield glider paths which minimize change in depth, therefore the change in potential energy. Because gliders use potential energy for propulsion, Dubins paths are inherently energy efficient. An obvious concern, with regard to the Dubins path planning approach, is that the solution for steady turning motion is only approximate. A motion plan based on this solution will introduce heading error which, integrated over time, may lead to a significant navigation error. One may incorporate feedback to compensate for the error in the approximation, although feedback corrections incur additional energy cost. Design evaluation of a combined feedforward/feedback motion control algorithms is described in [11]. Ocean currents can also significantly influence a glider s motion, even at depth. Because underwater gliders move quite slowly, relative to conventional AUVs, operate over much longer time spans, even light currents can have a large, cumulative effect on vehicle motion. The Dubins path planning procedure has recently been adapted to the case of a constant ambient current in [28] [29]. In the former paper, the Dubins path is planned relative to the (moving) ambient fluid with suitably redefined endpoint conditions. The latter paper provides a geometric procedure that generates a subset of these time-optimal convected Dubins paths. The question of optimal gliding flight conditions was addressed in [30] for sailplanes in ambient winds. In this study, the authors obtained experimentally verified conditions for optimal gliding flight (i.e., for the minimum glide path angle). Combining energy-optimal flight conditions with the path planning procedure described in [29], with suitable modifications to allow for turn acceleration limits, could provide a constructive, energy-efficient approach to underwater glider guidance in currents. ACKNOWLEDGMENT The authors would like to thank P. Bhatta, G. D Spain, J. Graver, S. Jenkins, N. Leonard, J. Luby, B. 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13 MAHMOUDIAN et al.: APPROXIMATE ANALYTICAL TURNING CONDITIONS FOR UNDERWATER GLIDERS 143 [19] J.-D. Boissonnat, A. Cérézo, J. Leblond, Shortest paths of bounded curvature in the plane, in Proc. IEEE Int. Conf. Robot. Autom., Nice, France, May 1992, pp [20] P. Souéres, J.-Y. Fourquet, J.-P. Laumond, Set of reachable positions for a car, IEEE Trans. Autom. Control, vol. 39, no. 8, pp , Aug [21] J.-D. Boissonnat X.-N. Bui, Accessibility region for a car that only moves forwards along optimal paths, Institut National de Recherche en Informatique et en Automatique (INRIA), Sophia Antipolis Cedex, France, Tech. Rep. 2181, [22] X.-N. Bui, J.-D. Boissonnat, P. Souéres, J.-P. Laumond, Shortest path synthesis for Dubins non-holomic robot, in IEEE Int. Conf. Robot. Autom., San Diego, CA, May 1994, vol. 1, pp [23] D. A. Anisi, Optimal motion control of a ground vehicle, Swedish Defence Research Agency, Stockholm, Sweden, Tech. Rep. FOI-R SE, [24] A. Scheuer, Planification de Chemin à Courbure Continue Pour Robot Mobile Non-Holonome, Ph.D. dissertation, Institut National Polytechnique de Grenoble, Grenoble, France, [25] A. Scheuer C. Laugier, Planning sub-optimal continuous-curvature paths for car-like robots, in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., Victoria, BC, Canada, Oct. 1998, pp [26] J.-D. Boissonnat, A. Cérézo, J. Leblond, A note on shortest paths in the plane subject to a constraint on the derivative of the curvature, Institut National de Recherche en Informatique et en Automatique (INRIA), Sophia Antipolis Cedex, France, Tech. Rep. 2160, [27] V. Kostov E. Degtiariova-Kostova, The planar motion with bounded derivative of the curvature its suboptimal paths, Institut National de Recherche en Informatique et en Automatique (INRIA), Sophia Antipolis Cedex, France, Tech. Rep. 2189, [28] T. G. McGee J. K. Hedrick, Optimal path planning with a kinematic airplane model, J. Guid. Control Dyn., vol. 30, no. 2, pp , [29] L. Techy C. A. Woolsey, Minimum time path planning for UAVs in winds, J. Guid. Control Dyn., vol. 32, no. 6, pp , [30] S. A. Jenkins J. Wasyl, Optimization of glides for constant wind fields course headings, J. Aircraft, vol. 27, no. 7, pp , Nina Mahmoudian received the B.A.E degree from Tehran Polytechnic, Tehran, Iran, in 1999 the Ph.D. degree in aerospace engineering from Virginia Polytechnic State University (Virginia Tech), Blacksburg, in She is a postdoctoral associate with vessel dynamics group in Virginia Tech s Aerospace Ocean Engineering Department. Her research interests include nonlinear dynamics control of mechanical systems. Jesse Geisbert received the B.S. degree in aerospace ocean engineering the M.S. degree in ocean engineering from Virginia Polytechnic State University (Virginia Tech), Blacksburg, in , respectively. He is an employee of the Naval Surface Warfare Center, Carderock Division, Bethesda, MD. He focuses on model research, design, testing, evaluation of Navy concepts working in the experimental test team under the Resistance Propulsion Hydrodynamics Department, Code Craig Woolsey (M 95) received the B.M.E. degree from Georgia Institute of Technology, Atlanta, in 1995 the Ph.D. in mechanical aerospace engineering from Princeton University, Princeton, NJ, in He is an Associate Professor Assistant Department Head for Graduate Studies at the Aerospace Ocean Engineering Department, Virginia Polytechnic State University (Virginia Tech), Blacksburg. His research interests include nonlinear control theory for mechanical systems, particularly energy-based control methods, applications to autonomous ocean atmospheric vehicles. Dr. Woolsey received the National Science Foundation (NSF) CAREER Award the U.S. Office of Naval Research (ONR) Young Investigator Program Award, more recently, the Society of Automotive Engineers (SAE) Ralph R. Teetor Educational Award.