ROBUST ATTITUDE STABILIZATION OF AN UNDERACTUATED AUV K. Y. Pettesen and O. Egeland Depatment of Engineeing Cybenetics Nowegian Univesity of Science and Technology N- Tondheim, Noway Fax: + 9 99 E-mail: fkistin.yttestad.pettesen, Olav.Egelandg@itk.ntnu.no Keywods : Nonlinea contol, Feedback stabilization, Undeactuated systems. Abstact Robustness to actuato failues may be obtained by changing to a contol law that contols the vehicle using only the emaining actuatos, when an actuato failue is detected. Such a contol law is in this pape poposed fo attitude stabilization of an AUV with only two contol toques. To this end, a stability esult fo a class of homogeneous timevaying systems is pesented. The poposed contol law is a continuous, peiodic time-vaying feedback law. It povides exponential stability that does not depend on exact knowledge of the model paametes, and the exponential stability of the contolled system is thus obust to uncetainties in the model paametes. Intoduction Contol of undeactuated vehicles, i.e. vehicles with fewe independent contol actuatos than degees of feedom to be contolled, is a eld of inceasing inteest. It is a continuation of the eseach on nonholonomic systems. The nonholonomic systems have a constaint on the velocity, while the undeactuation leads to a constaint on the acceleation. Undeactuated vehicles have been studied by e.g. Bynes and Isidoi [] who gave esults on stabilizability of a class of undeactuated vehicles. Leonad [, ] showed how open-loop small-amplitude peiodic time-vaying focing can be used to contol both undeactuated spacecaft and undewate vehicles. Moin and Samson [] developed a continuous time-vaying feedback law that exponentially stabilizes an undeactuated spacecaft, using the homogeneity popeties of the system. Contol of autonomous undewate vehicles has many inteesting applications. The AUVs may pefom envionmental suveying, inspect undesea cables and oshoe oil installations, and nd sunken ships, aicaft and othe lost atifacts. As they ae untetheed, they may opeate unde ice, opening up vast, lagely unexploed Actic aeas that ae inaccessible to any othe kind of eseach vessel, and opeate at depths too deep fo tetheed vehicles. They ae also of militay inteest. Citical to thei success is thei capability fo accuate and eliable autonomous contol. Robustness to actuato failues may be obtained by equipping the vehicle with edundant actuatos. A less costly option would be obtained by, when detecting an actuato failue, changing to a contol law that contols the vehicle using only the emaining actuatos. This epesents a softwae solution to fault handling in the event of an actuato failue. This is a cost-educing altenative to fault handling by actuato duplication o tiplication, i.e. by hadwae edundancy. The softwae solution is futhemoe weight economical compaed to the hadwae solution, and this can be an impotant issue in undewate applications. In this pape contol of the attitude of an AUV with only two contol toques is consideed. As open-loop contol does not compensate fo model eos and extenal distubances, we seek a feedback law. An exponentially stabilizing feedback law is deived following a pocedue simila to that of []. Howeve, using the esults of [] diectly, hydodynamic damping, gavitation and buoyancy would have to be cancelled. Cancellation of estoing foces and toques is not desiable, and it would futhemoe equie exact knowledge of the model paametes. The cancellation of the gavitation and buoyancy toques is needed to obtain a homogeneous system. Using the fact that the homogeneity popeties of a system ae coodinate dependent, we popose a change of coodinates. In [] the Rodigues paametes ae used to epesent the attitude. Fo the AUV, the gavitation and buoyancy vecto eld then is non-homogeneous. Making a change of coodinates using the Eule paametes instead of the Rodigues paametes, we obtain a homogeneous system without having to cancel the gavitation and buoyancy. Then, using the esults of [] diectly except fom this change of coodinates, the hydodynamic damping and
cetain tems of the gavitation and buoyancy would still have to be cancelled. As a futhe development of [] we pesent a esult on stability of a class of homogeneous time-peiodic systems. Using this new esult, cancellation of hydodynamic damping, gavitation and buoyancy is avoided. Futhemoe, using this esult the contolle can be extended to exponentially stabilize systems that include thuste dynamics. The esulting feedback law does not cancel any dynamics. The exponential stability esult does not depend on exact knowledge of the model paametes, and consequently the exponential stability is obust to model paamete uncetainty. The pape is oganized as follows: In Section the stability esult fo a class of homogeneous time-peiodic systems is pesented. In Section the AUV attitude model is pesented, and a continuous peiodic time-vaying feedback law is poposed. It is poved that the feedback law exponentially stabilizes the oigin of the AUV. This is illustated by simulations in Section. Stability of a class of homogeneous time-peiodic systems Fo the design of stabilizing contolles fo undeactuated systems, one appoach may be to st design a stabilizing contol law fo a educed system whee velocities ae consideed as inputs, and aftewads show stability of the oiginal system. This appoach was poposed by []. [] pesented a stability esult fo a class of asymptotically stable homogeneous time-peiodic systems to which an integato has been added at the input level. To obtain a system with a pue integato at the input, it may howeve be necessay to cancel some of the dynamics. In this section we povide an extension of the esult in [], and conclude stability fo a class of asymptotically stable homogeneous time-peiodic system to which an integato and dynamics with cetain homogeneity popeties has been added at the input level. By using this esult, in Section we can design an exponentially stabilizing contol law that does not cancel any dynamics. Fo the denitions of dilations, homogeneity, homogeneous noms and exponential stability with espect to a dilation, the eade is efeed to [9]. Denition A T-peiodic function is a time-peiodic function with peiod T. In the following poposition we conside a system that is homogeneous of degee with espect to a dilation (x; t). We denote this dilation by (x; t) = ( x ; : : : ; n x n ; t) () We dene the dilation e (x; y; t) by e (x; y; t) = ( x ; : : : ; n x n ; q y ; : : : ; qm y m ; t) () whee the q i 's ae dened in the poposition. Poposition Conside the system x = f(x; v(x; t); t) () Let f(x; y; t) : R n R m R! R n be a continuous T - peiodic function and assume that the system () is homogeneous of degee with espect to a dilation (x; t). Let v(x; t) : R n R! R m be a vecto function whose components v (x; t); : : : ; v m (x; t) ae continuous T -peiodic functions, dieentiable with espect to t, of class C on (R n? fg) R and homogeneous espectively of degee q ; : : : ; q m > with espect to the dilation (x; t). Let g i (x; y ; : : : ; y i ; t) : R n R i R! R be continuous, T - peiodic functions that ae homogeneous of degee q i, fo i [; m], with espect to the dilation e (x; y; t). Assume that the oigin x = is an asymptotically stable equilibium point of the system (). Then, fo positive and suciently lage values of k ; : : : ; k m, the oigin (x; y) = (; ) is an asymptotically stable equilibium point of the system x = f(x; y ; : : : ; y m ; t) y = g (x; y ; t)? k (y? v (x; t)) y = g (x; y ; y ; t)? k (y? v (x; t)) (). y m = g m (x; y ; : : : ; y m ; t)? k m (y m? v m (x; t)) Fo the poof of Poposition the eade is efeed to [8]. Exponential attitude stabilization of the undeactuated AUV In this section a continuous peiodic time-vaying feedback law is poposed, and poved to exponentially stabilize the oigin of the AUV using only the two available contol toques. We conside the attitude dynamics of an autonomous undewate vehicle descibed by: M + C() + D() + g() = () = J() () It is hee assumed that the linea velocities ae zeo. The matix M is the inetia matix, including added mass. C() is the Coiolis and centipetal matix, also including added mass. D() is the damping matix and g() is the vecto of gavitational and buoyant toques. Equation () epesents the kinematics. The vecto = [p; q; ] T is the angula velocities in oll, pitch and yaw. The vecto = [ ; ; ] T is the vecto of contol toques. Fo paametization of SO(), both the Rodigues paametes and the Eule paametes give homogeneous kinematic equations. Howeve, to also obtain a homogeneous
gavitational and buoyancy vecto eld g() we have to use the Eule paametes [8]. To a otation about an axis of otation dened by the unit vecto the following fou paametes ae associated: = [ ; ; ] T = sin( ) = cos( ) () We assume in the following that jj <, i.e. >. Let =?. We dene = [ ; ; ; ] T. The kinematic equation () may then be witten = The matix M is The damping matix is D() = diagfd + X i +? +?? +??? p q (8) M = diagfm ; m ; m g (9) d p ijpj i? ; d + X i d q ijqj i? ; d + X i d ijj i? g () Both M and D ae positive denite matices. The matix C() is given by C() = m?m q?m m p m q?m p () The AUV is assumed to be neutally buoyant. The distance between the cente of gavity and the cente of buoyancy is dened by the vecto BG. It is assumed that BG = [; ; BG z ] T decomposed in the body-xed fame, and BG z >. Using theoy fom [] the gavitational and buoyancy vecto eld is found to be g() = BG zw (( + ) + ) BG z W (( + )? ) () We conside the case whee, due to an actuato failue =. The undeactuated AUV then belongs to the class of vehicles which it in [] is shown that cannot be asymptotically stabilized by eithe continuous state feedback (e.g. linea feedback) o by discontinuous state feedback. Instead we popose a continuous peiodic timevaying feedback law: Poposition Conside the functions v (; ; t) =?k? (; ) sin(t=") () v (; ; t) =?k + (; ) (k + d) sin(t=") whee d(m? m ) >, k(m? m ) >, k ; k > and (; ) is any homogeneous nom associated with the dilation (; ; t) = ( ; ; ; ; t). Given the following continuous peiodic time-vaying feedback law (; ; t) =?k m (p? v (; ; t)) (; ; t) =?k m (q? v (; ; t)) () Then thee exists an " > such that fo any " (; " ) and fo positive and suciently lage paametes k and k, the feedback law () locally exponentially stabilizes the oigin of the system (){(8), with espect to the dilation (; ; t) = ( ; ; ; ; p; q; ; t). Poof. The system (){(8) with contol () can be witten = l(; ; t) + h(; ; t) () whee l(; ; t) = p q + ( q? p)? ( p + q) m? d m p? BGzW m m? d m?m m m q? BGzW pq? d m m () and h(; ; t) consists of the emaining tems. The functions and ae homogeneous of degee with espect to and continuous fo (; ) = (; ). Thus they ae also continuous at (; ) = (; ). The vecto function l is thus continuous. It is futhemoe T -peiodic and l(; ; t) =. The vecto function h is continuous and is a sum of homogeneous vecto elds of degee stictly positive with espect to. The system = l(; ; t) () is homogeneous of degee with espect to. Thus by [, Pop. ] the solution (; ) = (; ) of the system () is locally exponentially stable with espect to if the equilibium (; ) = (; ) is a locally asymptotically stable equilibium of the system (). The vaiable is uniquely dened by since the Eule paametes satisfy the equation + + + = (8) and we have assumed that >. We educe the system () by emoving the equation fo. If the educed system is locally asymptotically stable, so is the system ().
def We futhe educe the system by dening v = p and = q as contol vaiables: v def = v v + (v? v ) v v? d m?m m m (9) The contols v and v ae chosen to be the functions v (; ; t) and v (; ; t) given in Equation (). The \aveaged system" of the closed loop system is then = m?m m? k? k + (k? k ) (k k? k? d)? d m () The lineaization of () about (; ) = (; ) is asymptotically stable as k ; k >, d(m? m ) > and k(m?m ) >. The system () is thus locally asymptotically stable. The system (9) is continuous, T -peiodic and homogeneous of degee with espect to the dilation (; ; t) = ( ; ; ; ; t). Thus by [, Th..] thee exists an " > such that fo any " (; " ) the oigin of (9) is locally asymptotically stable. The functions v (; ; t) and v (; ; t) ae continuous, T -peiodic, dieentiable with espect to t, of class C on (R R? f; g) R and homogeneous of degee with espect to the dilation (; ; t). The equations fo p and q in () ae p =? d p? BG zw? k (p? v (; ; t)) m m q =? d q? BG zw? k (q? v (; ; t)) m m () () The st two tems of these equations ae continuous and homogeneous of degee with espect to. Thus, by Poposition, as the oigin of (9) is locally asymptotically stable, fo positive and lage enough values of k and k the oigin (; ) = (; ) of the system () is locally asymptotically stable. Thus, by [, Pop. ] (; ) = (; ) is a locally exponentially stable equilibium of (), with espect to the dilation. Remak Using the new esult of Section it can be shown that the feedback law () can be extended to exponentially stabilize the attitude of an AUV with actuato dynamics descibed by i =? T i ( i? com i ) () without having to cancel any dynamics. Hee i com is the commanded thust and T i is the time constant fo the oll contol toque (i = ) and pitch contol toque (i = ). This is done by including and in the state vecto. Simulations The action of the contol law () has been simulated fo an AUV of mass m = 8 kg, given by the following paametes: M = diagf; 8; 8g () D() = diagf + jpj; + jqj; + jjg () The matix C() is given by () and (). The AUV is neutally buoyant, and the cente of buoyancy coincides with the cente of gavity. The contol paametes wee chosen to be k = k =?: k = d =?: k = " = k = () Note that since m < m the paametes k and d ae chosen negative to satisfy the conditions d(m?m ) > and k(m? m ) >. The homogeneous nom used in the feedback law was q (; ) = + + j j + jj () and the initial values of the attitude dynamics wee [ (); (); (); (); p(); q(); ()] T = [:; :; :;?:; ; ; ] T (8) The time evolution of the Eule paametes and the angula velocities ae shown in Figue and espectively. It is seen that all the state vaiables conveged to zeo. In Section it was shown that the oigin of the system (){ (8) with contol () is exponentially stable with espect to the dilation, fo small enough values of " and lage enough values of k and k. The function (; ) = q + + j j + j j + p + q + jj (9) is a homogeneous nom associated with. In Figue it is seen that the natual logaithm of was uppe bounded by a deceasing staight line. This illustates the esult of Section. The simulations futhemoe illustate that " does not have to be vey small and k and k need not be vey lage to have exponential stability of the closed loop system.
........ 8 8 Figue : The time evolution of the Eule paametes (?), (?), (??) and ( ). Figue : The natual logaithm of the homogeneous nom. Refeences......... 8 Figue : The time evolution of the angula velocities p(?), q(?) and (??) [ ]. s Conclusions Robustness to actuato failues may be obtained by, when detecting an actuato failue, changing to a contol law that contols the vehicle using only the emaining actuatos. Such a contol law has been poposed fo attitude stabilization of an AUV with only two contol toques. It is shown that it exponentially stabilizes the oigin of the AUV using only the two available contol toques. The contol is a continuous, peiodic time-vaying feedback law. It does not cancel any dynamics. The exponential stability does not depend on exact knowledge of the model paametes, and the exponential stability of the contolled system is thus obust to uncetainties in the model paametes. [] C.I. Bynes and A. Isidoi. On the attitude stabilization of igid spacecaft. Automatica, :8{9, 99. [] T. I. Fossen. Guidance and Contol of Ocean Vehicles. John Wiley & Sons Ltd., Chicheste, 99. [] N. E. Leonad. Contol synthesis and adaptation fo an undeactuated autonomous undewate vehicle. IEEE J. of Oceanic Eng., ():{, July 99. [] N. E. Leonad. Peiodic focing, dynamics and contol of undeactuated spacecaft and undewate vehicles. In Poc. th IEEE Conf. on Decision and Contol, pages 98{98, New Oleans, LA, Dec. 99. [] R. T. M'Closkey and R. M. Muay. Nonholonomic systems and exponential convegence: some analysis tools. In Poc. nd IEEE Conf. on Decision and Contol, pages 9{98, San Antonio, Texas, Decembe 99. [] P. Moin and C. Samson. Time-vaying exponential stabilization of the attitude of a igid spacecaft with two contols. In Poc. th IEEE Conf. on Decision and Contol, pages 988{99, New Oleans, LA, Dec. 99. [] K. Y. Pettesen and O. Egeland. Exponential stabilization of an undeactuated suface vessel. In Poc. th IEEE Conf. on Decision and Contol, Kobe, Japan, Dec. 99. [8] K.Y. Pettesen. Exponential Stabilization of Undeactuated Vehicles. PhD thesis, Nowegian Univesity of Science and Technology, 99. [9] J. B. Pomet and C. Samson. Time-vaying exponential stabilization of nonholonomic systems in powe fom. Technical Repot, INRIA, Dec. 99.