Robust Control of an Underactuated Surface Vessel with Thruster Dynamics K. Y. Pettersen and O. Egeland Department of Engineering Cybernetics Norwegian Uniersity of Science and Technology N- Trondheim, Norway Abstract In this paper a continuous periodic time-arying feedback law is proposed that exponentially stabilizes both the position and orientation of a surface essel haing only two actuators. To this end, a stability result for a class of homogeneous time-periodic systems is presented. The exponential stability proided by the proposed feedback law does not depend on exact knowledge of the model parameters, and is thus robust to model parameter uncertainty. The surface essel model considered is nonlinear and describes both the dynamics and the kinematics of the surface essel. Furthermore the actuator dynamics is included in the control design. Simulation results are presented.. Introduction Control of underactuated ehicles, i.e. ehicles with fewer independent control actuators than degrees of freedom to be controlled, is a eld of increasing interest. It is a continuation of the research on nonholonomic systems. The nonholonomic systems hae a constraint on the elocity, while the underactuation leads to a constraint on the acceleration. Underactuated ehicles hae been studied by e.g. Leonard [, ] who showed how open-loop periodic time-arying control can be used to control both underactuated spacecraft and underwater ehicles. Morin and Samson [] presented a continuous time-arying feedback law that exponentially stabilizes an underactuated spacecraft, using the homogeneity properties of the system. Position and orientation control of surface essels is required in many oshore oil eld operations such as drilling, pipe-laying and diing support. Critical for the success of a dynamically positioned surface essel is its capability for accurate and reliable control, subject to enironmental disturbances as well as to conguration related changes, such as a reduction in the number of actuators. Furthermore, robustness to actuator failures can be crucial for the safety e.g. during diing support and during oil tanking between two surface essels. Robustness to actuator failures may be achieed by equipping the ehicle with redundant actuators. A less costly option would be obtained by changing to a control law that controls the ehicle using only the remaining actuators, when an actuator failure is detected. This represents a software solution to fault handling in the eent of an actuator failure, and is a cost-reducing alternatie to fault handling by actuator duplication or triplication, i.e. by hardware redundancy. In feedback control, as opposed to open-loop control, the control input is a function of the state and thus proides some robustness to disturbances and model errors. An interesting control problem is thus to nd a feedback law that asymptotically stabilizes a desired equilibrium point of the system, using only the reduced number of aailable control inputs. In this paper we consider position and orientation control of a surface essel haing only two aailable actuators. The surface essel model considered describes both the dynamics and kinematics of a surface essel, including Coriolis and centripetal terms, hydrodynamic damping terms, hydrodynamic added mass, and force and torque control inputs. The model is thus nonlinear and it has a drift ector eld. When considering control of nonholonomic and underactuated systems, the actuator dynamics has generally been left out in preious works. In this paper we furthermore take the actuator dynamics into consideration in the control design. The surface essel with only two actuators belongs to a class of ehicles that cannot be asymptotically stabilized by either continuous pure-state feedback (e.g. linear feedback or discontinuous pure-state feedback [5]. In [8] a continuous pure-state feedback law was proposed that instead asymptotically stabilizes an equilibrium manifold containing the origin. Howeer, this feedback law depends on cancellation of dynamics, and the stability of the system can thus be sensitie to model parameter uncertainties. Furthermore, this solution does not achiee the main objectie of this paper which is to asymptotically stabilize the origin of the surface essel. In this paper a continuous periodic time-arying feedback law is proposed that exponentially stabilizes the origin using only the two aailable actuators. The feedback law is deried following a procedure similar to that of []. As a further deelopment of [] we present a result on stability of a class of homogeneous time-periodic systems. Using this new result, cancellation of hydrodynamic damping is aoided, and furthermore actuator dynamics need not be cancelled. The resulting feedback law does not cancel any dynamics. The exponential stability result does not depend on exact knowledge of the model parameters, and consequently the exponential stability is robust to model parameter uncertainty. To apply the stability result for the class of homogeneous time-periodic systems, and certain other analysis tools, the model must hae the appropriate homogeneity properties. Howeer, the surface essel model includes trigonometric terms, and is thus not homogeneous with respect to any dilation. To oercome this problem, we
propose a global coordinate transformation that renders the system homogeneous. The paper is organized as follows: In Section. a stability result for a class of homogeneous time-periodic systems is presented. In Section. the surface essel model is presented. In Section. a coordinate transformation is proposed that renders the surface essel model homogeneous, and a continuous periodic time-arying feedback law is proposed. It is proed that the feedback law exponentially stabilizes the origin of the surface essel. This is illustrated by simulations in Section 5.. Stability of a class of homogeneous time-periodic systems For the design of stabilizing controllers for underactuated systems, one approach may be to rst design a stabilizing control law for a reduced system where elocities are considered as inputs. Then an inner control loop can be used to make the actual elocities track the desired elocity inputs. Finally, stability of the complete system must be established. This approach was proposed by []. [] presented a stability result for a class of asymptotically stable homogeneous time-periodic systems to which an integrator has been added at the input leel. To obtain a system with a pure integrator at the input, it may howeer be necessary to cancel some of the dynamics. For the surface essel in this paper this would amount to cancellation of hydrodynamic damping and actuator dynamics. In this section we proide an extension of the result in [], and conclude stability for a class of asymptotically stable homogeneous time-periodic system to which an integrator and dynamics with certain homogeneity properties has been added at the input leel. Using this result, in Section. we can deelop an exponentially stabilizing feedback law that does not cancel any dynamics. For the denitions of dilations, homogeneity, homogeneous norms and exponential stability with respect to a dilation, the reader is referred to []. Denition A T-periodic function is a time-periodic function with period T. In the following proposition we consider a system that is homogeneous of degree with respect to a dilation (x; t. We denote this dilation by (x; t ( r x ; : : : ; rn x n ; t ( We dene the dilation e (x; y; t by e (x; y; t ( r x ; : : : ; rn x n ; q y ; : : : ; qm y m ; t ( where the q i 's are dened in the proposition. Proposition Consider the system x f(x; (x; t; t ( Let f(x; y; t : R n R m R! R n be a continuous T - periodic function and assume that the system ( is homogeneous of degree with respect to a dilation (x; t. Let (x; t : R n R! R m be a ector function whose components (x; t; : : : ; m (x; t are continuous T -periodic functions, dierentiable with respect to t, of class C on (R n? fg R and homogeneous respectiely of degree q ; : : : ; q m > with respect to the dilation (x; t. Let g i (x; y ; : : : ; y i ; t : R n R i R! R be continuous, T - periodic functions that are homogeneous of degree q i, for i [; m], with respect to the dilation e (x; y; t. Assume that the origin x is an asymptotically stable equilibrium point of the system (. Then, for positie and suciently large alues of k ; : : : ; k m, the origin (x; y (; is an asymptotically stable equilibrium point of the system x f(x; y ; : : : ; y m ; t y g (x; y ; t? k (y? (x; t y g (x; y ; y ; t? k (y? (x; t (. y m g m (x; y ; : : : ; y m ; t? k m (y m? m (x; t For the proof of Proposition the reader is referred to [5]. The following corollary is easily deduced from Proposition and [, Prop. ]. Corollary Under the conditions gien in Proposition, for positie and suciently large alues of k ; : : : ; k m, the origin (x; y (; of the system ( is globally exponentially stable with respect to e (x; y; t, in the sense that there exists two strictly positie constants K; a such that along any solution (x(t; y(t of ( the following inequality is satised: e (x(t; y(t Ke?at e (x(; y( (5 where e (x; y denotes any homogeneous norm associated with e (x; y; t.. The surface essel model We consider a surface essel haing no side thruster, described by the following model: where M + C( + D( C( " ( J( (? T (? com (8? T (? com (9 M diagfm ; ; m g ( "? m u?m u D( diagfd + X i d u i j u j i? ; d + X i d i j j i? ; d + X i d r i j r j i? g J( " cos? sin sin cos ( ( (
The ector [u; ; r] T is the linear elocities in surge and sway, and the angular elocity in yaw, decomposed in the body-xed frame. The ector [x; y; ] T is the position and orientation decomposed in the earth- xed frame. The thruster force in surge is denoted by and the thruster yaw torque is denoted by. We dene [ ; ] T. Furthermore, i com is the commanded thrust and T i is the time constant for the thrust in surge (i and yaw (i. The matrix M is the inertia matrix, including added mass. C( is the Coriolis and centripetal matrix, also including added mass. D( is the damping matrix. Equation ( represents the kinematics. Both the inertia matrix M and the damping matrix D( are positie denite.. Exponential stabilization of the underactuated surface essel In this section a continuous periodic time-arying feedback law is proposed and proed to exponentially stabilize the origin of the surface essel using only the two aailable actuators. We consider the case where the surface essel has no side thruster, i.e.. Howeer, the control synthesis of this paper is easily extended to the cases where instead or are missing. The underactuated surface essel belongs to a class of ehicles that from [5] is known to not be asymptotically stabilizable by either continuous or discontinuous pure-state feedback. To eade this negatie stabilizability result we introduce explicit time dependence in the feedback law, an approach rst used for mobile robots by []. To use the stability result gien in Section. and certain other analysis tools, the model must hae the appropriate homogeneity properties. The drift ector eld of the surface essel in Section. howeer includes trigonometric terms, and is thus not homogeneous with respect to any dilation. Howeer, the homogeneity property is coordinate dependent. Motiated by this we seek a coordinate transformation that renders the system homogeneous. We propose the following coordinate transformation, which is a global dieomorphism: z cos( x + sin( y z? sin( x + cos( y ( z The state equations of the surface essel are then: z u + z r z? z r z r u m r? d m u + m (5? m ur? d r m? m u? d m r + m? T (? com? T (? com Proposition Consider the functions u d (z; ; t?k z + (z; sin(t" ( r d (z; ; t?k z + (z; (kz + d sin(t" ( d(z; ; t?k m (u? u d (z; ; t (8 d(z; ; t?k m (r? r d (z; ; t (9 where k ; k ; k ; k >, k >, d > and (z; is any homogeneous norm associated with the dilation (z; ; t (z ; z ; z ; ; t. Gien the following continuous periodic time-arying feedback law com (z; ; ; t?t k 5 (? d ( com (z; ; ; t?t k (? d ( Then there exists an " > such that for any " (; " and for positie and suciently large parameters k, k, k 5 and k, the feedback law ({ ( locally exponentially stabilizes the origin of the system (5 with respect to the dilation (z; ; ; t (z ; z ; z ; u; ; r; ; ; t. Proof. The system (5 with controls ({( may be written " z f(z; ; ; t + h(z; ; ; t ( where f(z; ; ; t u? z r r m (? d u? m ur? d m (? d r? T (? com? T (? com 5 ( and h(z; ; ; t consists of the remaining terms. Consider the dilation (z; ; ; t (z ; z ; z ; u; ; r; ; ; t ( As the functions u d and r d are homogeneous of degree with respect to the dilation, and continuous for (z; ;, they are also continuous at zero. The ector function f is thus continuous. It is furthermore T -periodic and f(; ; ; t. The ector function h is continuous. The system " z f(z; ; ; t (5 is homogeneous of degree with respect to the dilation, and the ector eld h is homogeneous of degree strictly positie with respect to. Thus the solution (z; ; (; ; of the system ( is locally exponentially stable with respect to if the equilibrium (z; ; (; ; is a locally asymptotically stable equilibrium of the system (5 [, Prop. ].
d We reduce the system (5, by dening d and as control ariables: u? z r z r m ( d? d u (? m ur? d 5 m ( d? d r We furthermore reduce the system (, by dening u d u and r d r as control ariables: u d z? z r d r d 5 (? m u d r d? d With controls gien by ( and ( the \aeraged system", [], of ( is h z i?kz + kzz?kz 5 (8? m kkzz? m (kz + d? d As the linearization of (8 about (z; (; is asymptotically stable, the origin is a locally asymptotically stable equilibrium of (8. Furthermore the system ( is continuous, T -periodic and homogeneous of degree with respect to the dilation. Thus there exists an " > such that for any " (; " the origin of ( is locally asymptotically stable [, Th..]. The functions u d (z; ; t and r d (z; ; t are continuous, T -periodic, dierentiable with respect to t, of class C on (R R? f; g R, and homogeneous of degree with respect to the dilation. The equations for u and r in ( with controls (8{(9 are u? d m u? k (u? u d (z; ; t (9 r? d r? k (r? r d (z; ; t ( m Thus, by Proposition, for positie and suciently large alues of k and k the origin (z; (; of the system ( with controls (8{(9 is locally asymptotically stable. The functions d(z; ; t and d(z; ; t are continuous, T -periodic, dierentiable with respect to t, of class C on (R R? f; g R, and homogeneous of degree with respect to the dilation (z; ; t (z ; z ; z ; u; ; r; t. The equations for and in (5 are? T? k 5 (? d (? T? k (? d ( Thus, by Proposition, for positie and suciently large alues of k 5 and k the origin of the system (5 is locally asymptotically stable. Thus the origin is a locally exponentially stable equilibrium of the system ( with respect to. [, Prop. ]. 5. Simulations The action of the control law (-( has been simulated with control parameters k :5; k :5 k k ; k 5 k k : ; d : ; " The homogeneous norm used in the control law was q (z; z + jz j + z + jj ( and the initial alues were: [x(; y(; (; u(; (; r(; (; (] T [; ; ; ; ; ; ; ] T ( In Figure the trajectory of the surface essel in the xy plane is shown. Figure shows the time eolution of the position and orientation. Figure shows the time eolution of the elocities. In Figure and 5 the time eolution of the thruster surge force and yaw torque are shown. The function q (z; ; z + jz j + z + + jj + r u + + (5 is a homogeneous norm associated with. We see in Figure that the natural logarithm of (z; ; is upper bounded by a decreasing straight line. This illustrates the exponential conergence of the system to zero. y position.8... Position in the xy plane...... x position Figure : The trajectory in the xy-plane References [] N. E. Leonard. Control synthesis and adaptation for an underactuated autonomous underwater ehicle. IEEE J. of Oceanic Eng., (:{, July 995. [] N. E. Leonard. Periodic forcing, dynamics and control of underactuated spacecraft and underwater ehicles. In Proc. th IEEE Conf. on Decision and Control, pages 98{985, New Orleans, LA, Dec. 995.
5 8..5.8..............8. 5 8 Figure : The time eolution of the position and orientation x(?, y(?, (??. Figure 5: Yaw torque.... 5.. 5 8 Time [s]. 5 8 Figure : The time eolution of the elocities u(?, (?, r(??. [] R. T. M'Closkey and R. M. Murray. Nonholonomic systems and exponential conergence: some analysis tools. In Proc. nd IEEE Conf. on Decision and Control, pages 9{98, San Antonio, Texas, December 99. [] P. Morin and C. Samson. Time-arying exponential stabilization of the attitude of a rigid spacecraft with two controls. In Proc. th IEEE Conf. on Decision and Control, pages 988{99, New Orleans, LA, Dec. 995. [5] K.Y. Pettersen. Exponential Stabilization of Underactuated Vehicles. PhD thesis, Norwegian Uniersity of Science and Technology, 99. Figure : The natural logarithm of the homogeneous norm (z; ;. [] J. B. Pomet and C. Samson. Time-arying exponential stabilization of nonholonomic systems in power form. Technical Report, INRIA, Dec. 99. [] C. Samson. Velocity and torque feedback control of a nonholonomic cart. In C. Canudas de Wit, editor, Adanced Robot Control, Proc. of the Int. Workshop on Nonlinear and Adaptie Control: Issues in Robotics, Grenoble, France, No. 99, pages 5{5. Springer- Verlag, 99. [8] K. Y. Wichlund, O. J. Srdalen, and O. Egeland. Control of ehicles with second-order nonholonomic constraints: Underactuated ehicles. In Proc. Third Eur. Control Conf., pages 8{9, Rome, Italy, 995...8.......8 5 8 Figure : Surge force.