Feedback-Linearizing Control for Velocity and Attitude Tracking of an ROV with Thruster Dynamics Containing Input Dead Zones

Transcription

1 Feeback-Linearizing Control for Velocity an Attitue Tracking of an ROV with Thruster Dynamics Containing Input Dea Zones Joran Boehm 1, Eric Berkenpas 2, Charles Shepar 3, an Derek A. Paley 4 Abstract This paper presents a ynamics an control framework to accomplish six egree-of-freeom tracking of attitue, velocity, an rotational rate setpoints for a remotely operate vehicle with nonlinear thruster ynamics. The thruster ynamics contain input ea zones that complicate linear state feeback control esign, an are compensate with nonlinear control strategies, specifically feeback linearization. Moeling the thruster ynamics in the control esign mitigates the input ea zones. Simulations with experimentally obtaine thrust parameters show improve reference setpoint tracking when compensating for the thruster ynamics. I. INTRODUCTION Remotely operate vehicles (ROVs) are wiesprea an versatile, being applicable to eep-sea exploration an mining [1], marine research [2], hull inspection [3], an wreckage surveying [4]. To accomplish these tasks, ROV control is typically accomplishe through a variety of methos ranging from irect human-in-the-loop control to autonomous, logicriven control [5]. Controllers for autonomous or semiautonomous operation have been esigne through a variety of feeback frameworks, incluing feeback linearization [6], [7], robust control [8], [9], an aaptive control [9], [1]. Most ROV operations are accomplishe by semiautonomous or full human control, whereby irect commans from an operator are either processe by a controller or fe irectly to iniviual thrusters [5]. Direct-controlle ROVs typically have orthogonal thruster configurations that allow for intuitive translations from commans to thrusts, but such actuator placement can complicate the vehicle esign. As a result, fewer thrusters are often use, thus limiting maneuverability of the ROV [5]. To maintain generality, we analyze an ROV that has a specific thruster placement configuration to accomplish fully actuate control. An autostabilizing control system is assume to process user commans into setpoints. J. Boehm is supporte by a grauate research fellowship from the National Geographic Society. 1 Joran Boehm is a grauate stuent in the Department of Aerospace Engineering at the University of Marylan, College Park, MD, 2742, USA. jboehm77@um.eu 2 Eric Berkenpas is the irector of the Exploration Technology Laboratory at the National Geographic Society, Washington, DC, 236, USA. eberkenp@ngs.org 3 Charles Shepar is the lea mechanical system esigner of the Exploration Technology Laboratory at the National Geographic Society, Washington, DC, 236, USA. mshepar@ngs.org 4 Derek A. Paley is the Willis H. Young Jr. Professor of Aerospace Engineering Eucation in the Department of Aerospace Engineering an the Institute for Systems Research, University of Marylan, College Park, MD, 2742, USA. paley@um.eu This work is presente with relevance to the application of ROVs to aquatic imaging, the primary function of an ROV uner evelopment by the National Geographic Society (NGS) shown in Fig. 1. Unerwater filmmaking requires smooth setpoint tracking with human-in-the-loop operations. Reference setpoint attitues an velocities are typically generate through user input an, for complicate thruster configurations, controllers are capable of effectively tracking commane trajectories. Often ROVs maintain only active close-loop control of three or four egrees of freeom, while allowing roll an pitch parameters to be passively stabilize by relying on the natural stability of the vehicle ue to the relative locations of the centers of gravity an buoyancy [5], [7], [8], [11]. However, for the purposes of eep-sea imaging, it is useful to have full user control of all attitue parameters, similar to a multi-rotor aerial rone, in orer to obtain the esire cinematic effects. To enhance controller performance an reuce limit-cycle behavior, actuator ynamics are accounte for in the control esign [8], [12]. A variety of methos for moeling thrusters for unerwater vehicles have been evelope in previous work. A two-state axial flow ynamic moel [13] [15] accounts for thrust overshoot but is limite to uniirectional flow characterization. A two-state rotational flow moel [16] has no more moel accuracy than the axial flow moel. Lastly, a multi-irectional axial flow moel [17] requires a large number of parameters to be ientifie with extensive system testing. This paper expans upon a singlestate voltage-riven thruster moel presente in [8]. We consier an analog voltage signal (throttle) as the control input for the thruster ynamics, which also exhibit a ea zone nonlinearity. A single-state ynamic thruster moel is vali for low-spee movement [8], [14], [15]. In previous work, robust an aaptive control techniques have been use for ea zone compensation in the absence of well-ientifie moel parameters [18], [19]. This paper utilizes feeback linearization to compensate for nonlinearities in thruster ynamics, because high-quality propeller spee, thrust, an torque ata obtaine from a six-axis Gough-Stewart platform loa cell (Fig. 2) are available [2]. Other techniques [18], [19] for improving the robustness of feeback-linearizing methos are out of the scope of this paper. The contributions of this paper are (1) a nonlinear control law for throttle-controlle thruster ynamics with input ea zones using experimentally obtaine parameters; an (2) implementation of a feeback-linearizing an eazone-compensating thruster controller for the six egree-of-

2 Y O O X O Z O 2 1 cm Fig. 1. Computer renering of the ROV uner evelopment by the National Geographic Society. Boy-fixe reference frame axes are marke. freeom (DOF) attitue an spee setpoint tracking of an ROV with throttle-controlle thruster ynamics. In aition, we illustrate the close-loop control results through simulation an compare these to an alternative thruster controller base on a lookup table that oes not account for actuator ynamics. Ongoing work seeks to experimentally emonstrate the results on an ROV testbe currently uner evelopment (see Fig. 1). The organization of this paper is as follows. Section II presents the full six DOF equations of motion for a rigiboy ROV an a feeback-linearizing thrust control law to stabilize the setpoint-tracking ynamics of the system. Section III presents the rotor-spee ynamics of the thrusters an proposes a nonlinear analog throttle signal control law to achieve stability of setpoint tracking. Section IV combines the control methos of the previous sections into the full state moel. Close-loop performance is illustrate through simulation an comparison to a controller with uncompensate thruster ynamics. Section V summarizes the paper an iscusses ongoing work. II. ROV DYNAMICS AND CONTROL The rigi-boy ynamics of an unerwater vehicle incluing hyroynamic rag an ae mass parameters efine in a boy-fixe reference frame are [21] M ν + C(ν)ν + D(ν)ν + g(η) = τ (1) η = J(η)ν, (2) where the portion of the state vector escribing boy frame linear velocities (u, v, an w) an angular velocities (p, q, an r) is given by ν = [u, v, w, p, q, r] T, an the terms escribing position an orientation of the boy frame with respect to the Earth-fixe frame are η = [x, y, z, φ, θ, ψ] T, where x, y, an z are Earth-fixe position coorinates an φ, θ, an ψ are the Euler angles of roll, pitch, an yaw, respectively [11]. If absolute position or orientation are not relevant, some or all of the Earth-relative states may be Fig. 2. Six-axis Gough-Stewart platform testbe setup instrumente with six loa cells use for system ientification of an ROV thruster [2]. omitte from the full state vector. We consier the task of setpoint control, where the Earth-relative coorinates x, y, an z are not inclue in the state feeback control. Therefore, we use the attitue-only state vector, η = [φ, θ, ψ] T. M enotes the iagonal mass an inertia matrix incluing ae mass an inertia parameters, C(ν) is the nonlinear Coriolis an centripetal matrix, D(ν) is the iagonal linear an quaratic hyroynamic rag matrix, J(η) is the transformation matrix escribing attitue rate of the vehicle boy-fixe frame relative to the Earth-fixe frame, an g(ν) is the restoring force an moment vector that combines gravitational an buoyancy effects. Aitionally, the external force/moment vector is treate as the control input, efine as [21] τ = K t T, (3) where K t is the thruster configuration matrix that escribes the orientation of each thruster an T is the vector of input thrusts. Martin an Whitcomb [6] efine the following feebacklinearizing control law, assuming perfect knowlege of vehicle states: T = Kt 1 [C(ν)ν + D(ν)ν + g(η) + M( ν K P (η) η K D ν)], where K t is assume to be invertible (or at least has a pseuo inverse). The NGS six-thruster ROV is amenable to this framework. Aitionally, let ν = ν ν an η = η η convert the state-space equations into error coorinates relative to known reference attitue an velocity setpoints ν an η obtainable from user inputs. Assume ν is reaily known an continuous. The proportional gain matrix K P (η) is a 6 3 matrix varying with vehicle orientation relative to the Earth-fixe (4)

3 frame. The erivative gain matrix K D is a constant positiveefinite symmetric matrix. The control law (4) yiels the close-loop ynamics [6] t ( ν) = K P (η) η K D ν (5) ( η) = J(η) ν, (6) t which asymptotically stabilize the origin ν = an η = [6]. III. THRUSTER DYNAMICS AND CONTROL A. Dynamic Thruster Moel with an Input Dea Zone The control law in (4) efines a esire set of actuator thrusts that stabilize the close-loop setpoint-tracking ynamics of the ROV. Thrust can be relate to the propeller angular velocity of an ROV thruster by a ea zone function [8] k T 1 (n n δ T 1 ), n n δ T 1 T (n) =, δ T 1 < n n < δ T 2 (7) k T 2 (n n δ T 2 ), n n δ T 2, where n represents propeller angular velocity, the constants k T 1, k T 2, an δ T 2 are positive, an δ T 1 is negative. In orer to etermine the esire propeller angular velocity n for a esire thrust T, inverting the ea zone function (7) yiels [8] sgn(t ) T k T 1 + δ T 1, T < n =, T = (8) sgn(t ) T k T 2 + δ T 2, T >. The esire propeller angular velocity n is fe back into a control scheme for the actuator ynamics. Bessa et al. [8] propose the following voltage-riven ynamic moel for an ROV thruster: ṅ = k 1 n k 2 n n + k v u, (9) where u is the input motor voltage an the constants k 1, k 2, an k v are positive. Equation (9) is a single-state thruster moel that is vali at low propeller spees [8]. We consier an alternate version of (9) that, instea of being riven by a irect motor voltage, is controlle by an analog voltage throttle signal with a ea zone aroun zero volts. The new moel is ṅ = k 1 n k 2 Q(n) + γ(u), (1) where Q(n) is the reaction torque on the propeller, an the function γ(u) relates throttle signal u to motor torque by a ea zone function k v1 (u δ v1 ), u δ v1 γ(u) =, δ v1 < u < δ v2 (11) k v2 (u δ v2 ), u δ v2. The effects of the nonlinear function (11) on (1) are presente in Fig. 3 by plotting steay-state propeller spee Propeller Spee [rpm] Experiment Moel Throttle [V] Fig. 3. Steay-state propeller angular velocity ata from [2] resulting from constant throttle inputs show the nonlinear ea zone behavior of the moel (1). ata as a function of the constant throttle voltages that rive the system to those operating points. A fit base on (1) is also plotte to valiate the accuracy of the moel. Note that (11) can be inverte as kv1 1 α + δ v1, α < γ 1 (α) =, α = (12) kv2 1 α + δ v2, α >, for any generic commane motor torque α. In (1), Q represents the collecte inertial an hyroynamic reaction torque enacte on the thrusters, which is often a quaratic function of n, an may be efine with a ea zone similar to (7), i.e., k Q1 (n n δ Q1 ), n n δ Q1 Q(n) =, δ Q1 < n n < δ Q2 (13) k Q2 (n n δ Q2 ), n n δ Q2, with positive constants k Q1, k Q2, an δ Q2, an negative constant δ Q1. Steay-state thrust an torque ata of a thruster [2] were use to ientify the parameters in the moels (7) an (13). The moels were fit to these ata as shown in Fig. 4. B. Feeback-Linearizing Control Design To rive the ynamics (1) to a known setpoint n we use the inverse ea zone function (12) an feeback linearization to erive a control law that compensates for nonlinearities in the thruster ynamics. We show below that this framework exponentially stabilizes n = n. Theorem 1: Assuming u can change instantaneously, the ynamics (1) exponentially stabilize the setpoint n = n n = using the control law where γ 1 (α) is efine in (12) an u = γ 1 (α), (14) α = ṅ + k 1 n + k 2 Q(n) k u n, (15)

4 6 4 Experiment Moel Experiment Moel i.e., u max. Since u cannot change instantaneously, it may pass through the ea zone. The maximum amount of time the motor spens in the ea zone while transitioning to a thruster operating point outsie the ea zone is Thrust [N] Propeller Spee [rpm] Fig. 4. Steay-state thruster moel compare to experimental ata from [2] for thrust an torque. for k u >. Proof: Consier the scenario where the system is operating uner the first conition in (12), i.e., α <. Therefore, (1) becomes ṅ = k 1 n k 2 Q(n) + ṅ + k 1 n + k 2 Q(n) k u n = ṅ k u n, Torque [Nm] (16) which implies t ( n) = k u n. (17) Equation (17) is a scalar Hurwitz linear system in error coorinates relative to the setpoint n. The same steps yiel ientical results for the thir conition of (12), so operating on either en of the ea zone yiels the system (17). In the case that α =, substituting (15) into (1) also yiels (17), which completes the proof. If perfect knowlege of ṅ is not available in practice, a piecewise constant estimate may be use in its place. We observe the following result. Corollary 1.1: Let δ >. Using the estimate ñ = ṅ +ɛ, where the estimation error ɛ satisfies ɛ < δ, the solution to the close-loop ynamics (17) using the control law (14) is boune by n δ/k u. Proof: With the estimate ñ = ṅ + ɛ, (17) becomes t ( n) = k u n + ɛ. (18) The time-erivative of the quaratic Lyapunov function V = ( n) 2 /2 along solutions of (18) satisfies V k u ( n) 2 + n δ, (19) which implies the close-loop ynamics (17) converge to n δ/k u. In practice, physical thrusters have a maximum ramp spee, i.e., u cannot change instantaneously, which limits the convergence rate to the esire setpoint. We moel this limitation as a maximum allowable throttle change rate, t max = δ v2 δ v1 u max >. (2) During this time, the ynamics (1) will be unforce, requiring aitional analysis of the system in this scenario. Theorem 2: Consier the ynamics (1). When u is within the throttle ea zone, the zero-input ynamics ṅ = k 1 n k 2 Q(n) (21) exponentially stabilize the origin n =. Proof: We analyze the stability properties of the unforce system (21) with the quaratic Lyapunov function which varies accoring to V = 1 2 n2, (22) V = k 1 n 2 k 2 Q(n)n. (23) Equation (23) can take one of three forms epening on the value of n. Because the constants k 1 an k 2 are positive, V is negative efinite if Q(n)n is positive semi-efinite for all n. Accoring to (13), Q(n) either has the same sign as n or is zero for δ Q1 < n n < δ Q2, because k Q1, k Q2, an δ Q2 are all positive an δ Q1 is negative. We therefore conclue Q(n)n is positive semi-efinite, an V is negative efinite for all n. Note that k a n 2 V k b n 2 an V k 1 n 2 for k b >.5 > k a >, which implies that the unforce system (21) exponentially stabilizes the origin. The thruster motor operates in the ea zone in one of only three scenarios: uring startup, win-own, or a transition between forwar an reverse thrust. In all of these scenarios, convergence to zero propeller spee is either avantageous or inconsequential (as in the case of motor startup). Typically t max in (2) is on the orer of tens of millisecons, whereas the win-own time for an ROV thruster was experimentally observe to be as much as half a secon [2]. As a result, crossing the throttle ea zone is not preicte to estabilize the physical system uring regular operation. Fig. 5 epicts a simulation of the ea-zone-compensating controller (14) successfully riving the actuator ynamics from multiple operating points of n to a positive setpoint value of n, taking into account the limitation u u max. IV. FULL MODEL AND SIMULATION RESULTS With analysis of the capabilities of the thruster control esign complete, we further verify its performance with full system simulations. The systems in (1), (2), an (1) represent the full system ynamics of the ROV, incluing rigi-boy ynamics an uncouple actuator ynamics for each thruster, i.e., ẋ = f(x) + g(u), (24)

5 Propeller Spee [rpm] Simulation Fig. 5. Simulations of the ea-zone-compensating controller riving a thruster from multiple operating points to a positive setpoint with consieration for u max. where x = [η T ν T n T ] T an n is the vector of the (six) thruster states of the ROV. The vector fiels f(x) an g(u) are J(η)ν f(x) = M 1 [K t T (n) C(ν)ν D(ν)ν g(η)], K n n K Q Q(n) (25) an g(u) = [ γ(u)] T, where K n an K Q are iagonal matrices containing the parameters of the ynamics from (1) for each iniviual thruster. The close-loop ynamics take the form t η ν = n J(η) ν K P (η) η K D ν K u n (26) where K u is the iagonal matrix of feeback gains. Theorem 3: The full close-loop ynamics for the ROV (26) asymptotically stabilize the origin x = x x =. Proof: The system (26) combines the ynamics in (1) an (2), which asymptotically stabilize the origin ν = an η =, with the ynamics of the thrusters (1) which are fully uncouple an each exponentially stabilize the origin n =. Therefore, because each sub-system is asymptotically stable, the full close-loop system asymptotically stabilizes the origin x =. Simulations of the system (24) verify the setpoint-tracking performance of the ea-zone-compensating controller (14) when implemente in a full-vehicle scenario. Tracking performance of the control scheme compares favorably to a lookup table that is base solely on ata presente in Fig. 3. The lookup table etermines the control input that rives a thruster to the esire steay-state propeller angular velocity. This metho oes not account for actuator ynamics, an assumes rapi convergence to steay-state propeller spee. Fig. 6 inicates superior performance of the ea-zonecompensating controller over the lookup-table metho. The tracking task involves varying setpoints in forwar velocity u, pitch rate q, an pitch θ. Parameter values use in simulation of the thruster ynamics are reporte in Table I. Physical parameters for the thrusters were etermine via system ientification methos using a six-axis Gough- Stewart platform loa cell [2]. To further compare performance of the control methos, the root-mean-square tracking error (RMSE) was calculate for each egree of freeom over the course of the simulation time perio an normalize by the average value of each respective egree of freeom. Normalize RMSE results are presente in Fig. 7. The ea-zone-compensating controller isplays comparable tracking performance in linear velocities when compare to the lookup-table metho, but significant improvement comes in tracking attitue an angular velocity setpoints, with tracking errors being reuce by as much as 5%. Divergence from the reference setpoint isplaye in the performance of the lookup-table controller is attribute to the trae-off of tracking forwar velocity with the other states, as multiple setpoints were processe simultaneously. V. CONCLUSION This paper presents a moeling an control framework for using feeback-linearizing control methos on throttle-riven ROV thrusters that have input ea zones. Compensating the nonlinearities in the actuator ynamics achieves better setpoint tracking performance in simulation when compare to lookup-table control methos that o not account for thruster ynamics. Overall the use of feeback linearization to stabilize both the rigi-boy vehicle ynamics an the thruster behavior is effective as long as sufficient knowlege of moel parameters is available. Ongoing work seeks to exten this framework to a two-state moel of thruster ynamics with the inclusion of axial flow as an unmeasure state. Current an future control schemes will support upcoming fiel trials with the NGS ROV. TABLE I THRUSTER MODEL PARAMETER VALUES Parameter Value Units k T N rpm -2 k T N rpm -2 δ T rpm 2 δ T rpm 2 k Q N m rpm -2 k Q N m rpm -2 δ Q rpm 2 δ Q rpm 2 k v rpm (V s) -1 k v2 677 rpm (V s) -1 δ v V δ v V k s -1 k rpm (N m s) -1 u max 1 V s -1

6 Surge Spee [m/s] Dea-Zone Compensator (a) Pitch Spee [eg/s] Dea-Zone Compensator (b) Pitch Angle [eg] Dea-Zone Compensator Fig. 6. tracking performance comparison between the propose ea-zone-compensating controller an thrust-table-base control for (a) linear forwar velocity, (b) angular pitch velocity, an (c) pitch angle. (c) ACKNOWLEDGMENT The authors woul like to thank Arthur Clarke, Alan Turchik, an Coy Golhahn of the National Geographic Society for their assistance with thruster testing an ata collection. REFERENCES [1] M. Luvigsen, F. Søreie, K. Aasly, S. Ellefmo, M. Zylstra, an M. Parey, ROV base rilling for eep sea mining exploration, in OCEANS Abereen, June 217, pp [2] R. Nian, B. He, J. Yu, Z. Bao, an Y. Wang, ROV-base unerwater vision system for intelligent fish ethology research, International Journal of Avance Robotic Systems, vol. 1, no. 9, p. 326, 213. [3] S. Negaharipour an P. Firoozfam, An ROV stereovision system for ship-hull inspection, IEEE Journal of Oceanic Engineering, vol. 31, no. 3, pp , July 26. [4] S. M. Nornes, M. Luvigsen, Ø. Øegar, an A. J. Sørensen, Unerwater photogrammetric mapping of an intact staning steel wreck with ROV, IFAC-PapersOnLine, vol. 48, no. 2, pp , 215. [5] R. D. Christ an R. L. Wernli, The ROV Manual: A User Guie for Remotely Operate Vehicles, 2n e. Waltham, MA, USA: Elsevier, 214. [6] S. C. Martin an L. L. Whitcomb, Preliminary experiments in fully actuate moel base control with six egree-of-freeom couple ynamical plant moels for unerwater vehicles, in IEEE International Conference on Robotics an Automation, May 213, pp Normalize RMSE Dea-Zone Compensator Linear Velocity Angular Velocity Attitue Degrees of Freeom Fig. 7. Tracking errors for each egree of freeom involve in setpoint tracking. [7], Nonlinear moel-base tracking control of unerwater vehicles with three egree-of-freeom fully couple ynamical plant moels: Theory an experimental evaluation, IEEE Transactions on Control Systems Technology, vol. 26, no. 2, pp , March 218. [8] W. M. Bessa, M. S. Dutra, an E. Kreuzer, Dynamic positioning of unerwater robotic vehicles with thruster ynamics compensation, International Journal of Avance Robotic Systems, vol. 1, no. 9, p. 325, 213. [9] L. G. Garca-Valovinos, T. Salgao-Jimnez, M. Banala-Snchez, L. Nava-Balanzar, R. Hernnez-Alvarao, an J. A. Cruz-Leesma, Moelling, esign an robust control of a remotely operate unerwater vehicle, International Journal of Avance Robotic Systems, vol. 11, no. 1, p. 1, 214. [1] P.-M. Lee, B.-H. Jeon, S.-W. Hong, Y.-K. Lim, C.-M. Lee, J.-W. Park, an C.-M. Lee, System esign of an ROV with manipulators an aaptive control of it, in Proceeings of the International Symposium on Unerwater Technology, May 2, pp [11] H. D. Nguyen, S. Malalagama, an D. Ranmuthugala, Design, moelling an simulation of a remotely operate vehicle Part 1, Journal of Computer Science an Cybernetics, vol. 29, no. 4, pp , 213. [12] L. L. Whitcomb an D. R. Yoerger, Development, comparison, an preliminary experimental valiation of nonlinear ynamic thruster moels, IEEE Journal of Oceanic Engineering, vol. 24, no. 4, pp , Oct [13] A. J. Healey, S. M. Rock, S. Coy, D. Miles, an J. P. Brown, Towar an improve unerstaning of thruster ynamics for unerwater vehicles, in Proceeings of IEEE Symposium on Autonomous Unerwater Vehicle Technology, July 1994, pp [14] M. Blanke, K.-P. Linegaar, an T. I. Fossen, Dynamic moel for thrust generation of marine propellers, IFAC Proceeings Volumes, vol. 33, no. 21, pp , 2. [15] T. I. Fossen an M. Blanke, Nonlinear output feeback control of unerwater vehicle propellers using feeback form estimate axial flow velocity, IEEE Journal of Oceanic Engineering, vol. 25, no. 2, pp , April 2. [16] R. Bachmayer, L. L. Whitcomb, an M. A. Grosenbaugh, An accurate four-quarant nonlinear ynamical moel for marine thrusters: theory an experimental valiation, IEEE Journal of Oceanic Engineering, vol. 25, no. 1, pp , Jan 2. [17] J. Kim, Thruster moeling an controller esign for unmanne unerwater vehicles (UUVs), in Unerwater Vehicles, A. V. Inzartsev, E. Rijeka: IntechOpen, 29, ch. 13. [18] X. Wu, X. Wu, an X. Luo, Aaptive control of nonlinearly parameterize system with unknown ea-zone input, in 8th Worl Congress on Intelligent Control an Automation, July 21, pp [19] N. F. Jasim an I. F. Jasim, Robust control esign for spacecraft attitue systems with unknown ea zone, in IEEE International Conference on Control System, Computing an Engineering, Nov 211, pp [2] J. Boehm, E. Berkenpas, B. Henning, M. Roriguez, C. Shepar, an A. Turchik, Characterization, moeling, an simulation of an ROV thruster using a six egree-of-freeom loa cell, in OCEANS 218 MTS/IEEE Charleston, Oct 218, pp [21] T. Fossen, Guiance an control of ocean vehicles. Wiley, 1994.