JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 3, No., January February 27 Engineering Notes ENGINEERING NOTES are short manuscripts describing new deelopments or important results of a preliminary nature. These Notes should not exceed 25 words (where a figure or table counts as 2 words). Following informal reiew by the Editors, they may be published within a few months of the date of receipt. Style requirements are the same as for regular contributions (see inside back coer). Global Directional Control of a Slender Autonomous Underwater Vehicle Hye-Young Kim Angstrom, Inc., Seongnam, Republic of Korea Craig A. Woolsey Virginia Polytechnic Institute State Uniersity, Blacksburg, Virginia 246-23 DOI:.254/.2475 I. Introduction CONVENTIONAL streamlined autonomous underwater ehicles (AUVs) are underactuated by design. Typically, they include a propulsor torque actuators, such as control planes. In this Note, it is assumed that the 6 degrees of freedom ehicle is actuated in 4 degrees of freedom: surge, roll, pitch, yaw. The control objectie is to globally asymptotically stabilize longitudinalaxis translation of the ehicle in a desired inertial direction. A proably conergent, global directional controller could enhance the operation of agile ehicles in dynamic enironments, such as riers, tidal basins, or other coastal areas. Energy-based tools are appealing for designing such controllers because they proide natural cidate Lyapuno functions for stability analysis. As an example, a passiity-based technique known as interconnection damping assignment was used to stabilize the unstable dynamics of a slender, underactuated AUV in []. The paper assumed an idealized (iniscid) dynamic model did not consider the attitude kinematics. A nonlinear directional controller was deeloped in [2] using potential energy shaping, although that model also omitted iscous effects. This Note treats the directional stabilization problem for a ehicle model which requires minimal assumptions about the iscous effects. The result therefore allows a broad range of iscous force moment models. Directional stabilization for AUVs may be considered a natural, intermediate step between waypoint naigation general, threedimensional path following, as described in [3], for example. Seeral approaches hae been proposed to address three-dimensional path following for underactuated, 6 degrees of freedom AUVs. These approaches include adaptie feedback [4], switching theory [5], backstepping [3,6]. Here, we consider the more fundamental problem of directional stabilization for a slender, underactuated AUV we approach the problem from an energy perspectie. The Receied October 25; accepted for publication August 26. Copyright 26 by the American Institute of Aeronautics Astronautics, Inc. All rights resered. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $. per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drie, Daners, MA 923; include the code $. in correspondence with the CCC. Optical System Engineer, Former Post-Doctoral Research Assistant, Virginia Tech s Aerospace Ocean Engineering Department; gpdud@yahoo.com. Associate Professor, Department of Aerospace Ocean Engineering; cwoolsey@t.edu. 255 stabilization result, which is adapted from [7], is based on the method of feedback passiation described in [8]. In this approach, a gien system is transformed into a feedback interconnection of two passie subsystems. It follows from passie system theory that the interconnection is passie therefore that the system is stable. Asymptotic stability may be shown through further analysis, with suitably defined feedback dissipation included, if necessary. As is the case in this Note, exploiting intrinsic properties such as passiity often allows one to derie control algorithms which are globally effectie which work with the natural dynamics rather than dominate or supplant them. II. AUV Equations of Motion The AUV is modeled as a slender, axisymmetric rigid body whose mass m equals the mass of the fluid which it displaces; thus, the ehicle is neutrally buoyant. The ehicle is equipped with a single propulsor, which is aligned with the axis of symmetry, with torque actuators that proide independent control moments about all three axes. Vehicle kinematics: The principal axes of the displaced fluid define a body-fixed reference frame represented by the unit ectors b, b 2, b 3 in Fig.. Another reference frame, denoted by the unit ectors i, i 2, i 3,isfixed in inertial space. The location of the body frame with respect to the inertial frame is gien by the inertial ector x. The body s orientation with respect to inertial space is gien by the unit quaternion q q ; q T T. The unit quaternions proide a global, if redundant, parameterization of ehicle attitude; see [9] for a general reiew of attitude parameterizations [] for an early application of quaternions for underwater ehicle control. Let ; 2 ; 3 T represent the translational elocity let!! ;! 2 ;! 3 T represent the rotational elocity of the AUV with respect to inertial space, where! are both expressed in the body frame. Also, define the operator ^ such that for any 3-ectors a b, ^ab a b. The 3 3 skew-symmetric matrix ^a is sometimes called the cross-product equialent matrix corresponding to the ector a. The kinematic equations are _x R IB q () _q Qq! 2 (2) where R IB q2q ^q ^q q Qq q ^q where is the 3 3 identity matrix. Vehicle dynamics: Haing assumed, without further loss of generality, that the -axis is the axis of symmetry, define the diagonal matrix M diagm ;m 2 ;m 2 as the sum of m the added mass matrix. Similarly, let M 22 represent the sum of the rigid body inertia the diagonal added inertia matrix. Finally, suppose that the center of mass is located by the body frame ector r cm let M 2 m^r cm. Then! can be related to the body/fluid linear momentum p the body/fluid angular momentum h as follows: p h M M 2 M T 2 M 22! (3)
256 J. GUIDANCE, VOL. 3, NO. : ENGINEERING NOTES i i 2 i 3 2 c 2 c 3 c µ 3 α b b 2 r cm Fig. 2 Hydrodynamic angles for an axisymmetric body. Fig. b 3 Spheroidal underwater ehicle. To express f, we introduce two hydrodynamic angles. Let 8 < arctan 2 3 2 =or 3 (6) : 2 3 More precisely, p h represent the translational rotational impulse required to instantaneously set the body into motion through the fluid at elocity angular rate!; see []. In any case, p h eole according to a finite system of ordinary differential equations which describe the body s motion. Let f represent the iscous force moment, respectiely, let f c c represent thrust the control moment. Defining ; ; T, the AUV dynamic equations are _p p! f f c (4) _h h! p r cm mgr T IB i 3 c (5) Some remarks on the dynamic equations (4) (5) are in order. ) Because M is not a scalar multiple of the identity, the linear momentum of an underwater ehicle is generally not collinear with its linear elocity. This is an important distinction from heaier-thanair flight ehicles spacecraft which accounts, for example, for the Munk moment appearing in the term p in (5); see [2]. 2) While there is no net graitational force, because the ehicle is assumed to be neutrally buoyant, a graitational moment r cm mgr T IBi 3 appears on the right side of Eq. (5). This restoring moment tends to keep the body s center of mass below its center of buoyancy, unless the moment is countered by the control moment c. The terms f ;! ;! represent the iscous force moment, respectiely, that act on the ehicle as a consequence of its motion through the fluid. We assume quasisteady flow, as is stard for air marine ehicle dynamic models. Een under the quasisteady flow assumption, howeer, f are difficult to express explicitly oer the full range of motion. Rather than restrict the model s alidity to a small neighborhood of the nominal motion, we make some simple relatiely general modeling assumptions. Assumption 2..! when! when!; ; 8 2 R In words, we assume that the iscous moment opposes the angular rate, in general, that it anishes for pure translation along the longitudinal axis. We impose a bit more structure on the model of the iscous force. By definition, drag opposes ehicle elocity lift acts orthogonally to the elocity ector. Considering axial symmetry, we assume that the lift force acts in the plane determined by the elocity ector the ehicle s longitudinal axis. This assumption fails to capture out-of-plane forces due to asymmetric fluid flow, howeer, it is consistent with stard modeling assumptions. We also neglect the dependence of f on!, but we note in Remark 3.2 that our approach could accommodate such dependence, in certain cases. where the 4-quadrant arctangent is used. Identifying with, we hae 2;. Rotating the body frame through the angle about the b axis defines an intermediate reference frame in which the elocity ector has components only in the intermediate frame s 3 plane. Moreoer, the 3-axis component of elocity in this intermediate frame is nonnegatie. Let 8 p < arctan 2 2 2 3 (7) : where, once again, the 4-quadrant arctangent is used. Note that 2;. Rotating through the angle about the intermediate 2-axis yields a new reference frame, defined by unit ectors c, c 2, c 3,in which the -axis is aligned with the elocity ector as shown in Fig. 2. Following the terminology of [3], we refer to this reference frame as the current frame. The proper rotation matrix R BC ; e ^ e ^ 2 cos sin @ sin sin cos sin cos A cos sin sin cos cos transforms free ectors from the current frame to the body frame. Remark 2.2: The hydrodynamic angles are not the stard angle of attack sideslip angle; howeer, the gien definitions are more conenient for this work. Note that is discontinuous when that is discontinuous when. Thus, R BC ; is discontinuous under these conditions. Of course, the discontinuities are an artifact of the definition of the current frame do not represent any physical effect. Still, special care must be taken in the stability analysis; in fact, the discontinuities are easily treated using Filippo s theory [4] some extensions described in [5]. It is stard practice to express lift drag in terms of nondimensional coefficients. To nondimensionalize the forces, we define f as the product of dynamic pressure a reference area S: f 2 kk2 S where is the fluid density. The iscous force takes the form C D f f R BC ; @ C L A (8) We make the following assumptions about the form of the drag lift coefficients. Assumption 2.3. ) C D is a continuous, een function which is positie for all. 2) C L is a continuous, odd function which is positie (negatie) when e i lies in the first or third (second or fourth) quadrant of the complex plane.
J. GUIDANCE, VOL. 3, NO. : ENGINEERING NOTES 257 In general, the dimensionless coefficients C D C L depend on Reynolds number as well as angle of attack. This effect can be incorporated with little impact on the assumptions the conclusions, but we ignore Reynolds number effects in this presentation. For a slender, axisymmetric rigid body, the assumption that C D is een positie is an empirical fact. The assumption regarding the form of C L is consistent with intuition for a slender body in a steady flow. In reality, at large angles of attack, such a body is subject to complicated, unsteady forces that are not captured by simple models; see [6], for example. While these effects certainly impact the dynamics, they are difficult to model accurately. Such effects are typically ignored in control design with the understing that welldesigned model-based feedback can proide suitable system performance een when the model is imperfect. III. Global Asymptotic Directional Stabilization The main objectie is to stabilize the steady motion q eq q d ;! eq ; eq d where q d is a unit quaternion representing some constant, desired attitude d d where d > is a constant desired speed. We will first transform the problem, so that the equilibrium is at the origin. Let q d represent the quaternion conjugate of q d define the attitude error quaternion e e e q d q where * denotes quaternion multiplication; see [9], for example. When q q d, we hae e e d ; ; ; T. To shift the equilibrium to the origin, define the attitude error ector ~e e e d Also, define the elocity corresponding momentum error ectors ~ eq d ~!!! eq! ~p h ~ M M 2 ~ M T 2 M 22 ~! The complete error dynamics are _ ~e Q~e e 2 d ~! (9) _ ~p ~p pd ~! f f c () _ h ~ h ~ h d ~! ~p p d ~ d r cm mgr T IBi 3 c () The reised control objectie is to stabilize the equilibrium at the origin for Eqs. (9 ). By defining the thrust control moments appropriately, Eqs. () () may be transformed so that they form a passie map from the control moment (the passie input) to the body angular rate (the passie output). The rotational kinematics (9) are well known to be passie. Moreoer, the complete equations (9 ) may be written in a nonlinear cascade form which is amenable to the method of feedback passiation described in [8]. Consider the feedback control law f c f cos C D sin C L k ~ (2) c r cm mgr T IBi 3 p d u (3) where k > is a control gain u is a control moment which remains to be determined. The first term in (2) requires that thrust balance the longitudinal component of iscous force the second term ensures conergence to the desired speed. In the language of passie systems theory [8], consider the storage function S dyn ~p T M M 2 ~p~h 2 h ~ M T (4) 2 M 22 Under the gien choice of feedback, _S dyn ~ f c f ~! u ~! (5) The last term is nonpositie by Assumption 2.. The first term satisfies T C D B C B ~ f c f f @ ~ 2 A R BC ; @ ~ 3 C L C A 2 T 3 C D 6 B C B ~ 4f c f @ A R BC ; @ C L C7 A5 ~ 2 sin ~ 3 cos sin C D cos C L f ~ ff c f cos C D sin C L g (6) Consider the first term in (6). By definition of the angle, q ~ 2 sin ~ 3 cos ~ 2 2 ~2 3 Also, gien the assumptions on the form of C D C L the fact that 2;, we hae sin C D cos C L [ strictly positie for 2;]. Therefore, the first term in (6) is nonpositie. Now consider the second term in (6) define f c according to (2). The rate of change of S dyn becomes q _S dyn ~ 2 2 ~2 3sin C D cos C L f k ~ 2 ~! u ~! ~! u The dissipation inequality aboe suggests that the subsystem () () has been rendered passie with input u output y ~!.Of course, recalling Remark 2.2, the notion of passiity for systems defined by discontinuous equations requires some care. Regardless, one may conclude from Theorem on p. 53 of [4] that the origin is a stable equilibrium of the system () () when u. Moreoer, one may use an extension of Lasalle s inariance principle presented in [5] to conclude asymptotic stability. At this point, we hae completed the first step of a two-step design procedure in which we first stabilize the ehicle dynamics then the attitude kinematics. Remark 3.: Although Assumption 2.3 concerning the form of C D C L is quite mild, the thrust control law (2) assumes precise knowledge of these functions. One can improe robustness to uncertainty in these coefficients by instead choosing where f c sgn ~ f k ~ >supj cos C D sin C L j Of course, the signum function introduces another discontinuity into the system dynamics. Such discontinuities appear in certain robust nonlinear control methods, such as sliding mode control.
258 J. GUIDANCE, VOL. 3, NO. : ENGINEERING NOTES Implementation difficulties, such as chattering behaior, can be aoided by replacing the discontinuous signum function with a smooth approximation. See [7], for example. Remark 3.2: Referring to (5), note that one could accommodate angular rate dependency in the iscous force f, proided! appears as follows: f ;!f F! ;!! where f takes the form (8) where the components of the 3 3 matrix F! ;! are known. The rotational error kinematics (9) are passie with input ~!, output e, storage function S kin ~e 2 k e ~e ~e where k e > ; see [8], for example. Note that S kin ~e 2 k e e 2 e e k e e The feedback passiation technique described in Sec. 4.3 of [8] suggests choosing @Skin T u w @~e 2 Q~e e d w 2 k ee Keeping in mind the technical implications of Remark 2.2, the control law gien aboe would appear to render the complete closedloop system passie from the new input w to the output y ~! with respect to the positie definite storage function S~e; ~p; ~ hs kin ~es dyn ~p; ~ h To see this, obsere that q _S ~ 2 2 ~2 3sin C D cos C L f k ~ 2 ~! y w y w Apart from any concern oer passiity for discontinuous systems, one may recognize that S is a Lyapuno function that choosing w K ~! with K > ensures that _S. To conclude global asymptotic stability, note that S is radially unbounded, apply Lasalle s inariance principle, as extended for nonsmooth systems [5]. Define the set E f~e; ~p; ~ hj _S g let M be the largest inariant set contained in E. By Lasalle s principle, trajectories conerge to the set M as t!.ifmfg, then the equilibrium is globally asymptotically stable. Now, _S implies that d that!. This implies that p p d h h d. Thus, within the set M, _ h ~ k 2 ee It follows that e which implies that e ~e. Theorem 3.3: The feedback control law f c f cos C D sin C L k ~ (7) c r cm mgr T IBi 3 p d K! 2 k ee (8) globally asymptotically stabilizes the steady motion d e B C @ A ; @ A;! @ A with d > proided k e > ; k > ; K > Remark 3.4: One may modify the control law proposed in Theorem 3.3, using a simple line-of-sight scheme, to drie the ehicle to a desired linear path in inertial space. This is a special case of the more general ( more difficult) problem of threedimensional path following. Recall that the control law gien by (7) (8) stabilizes steady longitudinal translation in a particular inertial direction. To stabilize motion along a line, one might ary the desired orientation, expressed by q d, in a way that causes the ehicle s trajectory to conerge to the line. Considering only linear paths, one may assume without further loss of generality that the desired path coincides with the positie inertial i axis. Let be the ector (expressed in the inertial frame) from the body frame origin to a point on the i axis which is some look-ahead distance k x L further along the line: xk x L @ k x L A x @ x2 x3 The ector proides coordinates for the translational kinematics normal to the i axis. [Note that the along-track position x is ignored]. The translational kinematics are A _ T R IBq (9) The desired orientation, which depends on the translational error, may be obtained by using the ector to define the desired attitude matrix R IB q d. (In this case, if the ehicle assumes the desired attitude, the b axis is directed toward a point some distance k x L ahead on the i axis.) Of course, this modification of the directional stabilization algorithm iolates the stability analysis supporting Theorem 3.3, since q d now aries with the system state. A more inoled stability analysis is required to show global asymptotic stability to the desired linear path. For a simple AUV model, spectral analysis erifies that the approach works locally, proided the conditions of Theorem 3.3 are met proided k x >. Simulations suggest that the proposed line-following algorithm proides global asymptotic stability, but this claim has not been proen. IV. Conclusions Feedback passiation was used to deelop a nonlinear feedback control law which globally asymptotically stabilizes longitudinalaxis translation of a slender AUV in a desired inertial direction. Because the hydrodynamic angles which transform the iscous force into the body reference frame are discontinuous at the desired equilibrium, the analysis required the use of extended ersions of Lyapuno s direct method Lasalle s inariance principle to systems with discontinuous equations. The stability result was proen using ery general assumptions about the iscous force moment, allowing for a broad range of global, iscous models. Acknowledgements This work was supported in part by NSF grant CMS-332, ONR grant N4---588, ONR grant N4-5--56. The authors gratefully acknowledge Naira Hoakimyan Vahram Stepanyan for seeral enlightening discussions. References [] Astolfi, A., Chhabra, D., Ortega, R., Asymptotic Stabilization of Selected Equilibria of the Underactuated Kirchhoff s Equations, Proceedings of the Americal Control Conference, IEEE, Piscataway, NJ, 2, pp. 4862 4867. [2] Leonard, N. 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