Cascaded Adaptive Control of Ocean Vehicles With Significant Actuator Dynamics

This paper presents a cascaded adaptive control scheme for marine vehicles where the nonlinear equations of motion include a model of the actuator dynamics. The adaptive controller does not require the parameters of the vehicle dynamics and the actuator time constants known a priori. Both the velocity and position tracking errors are shown to converge to zero by applying Barbb alat's lemma. Furthermore, all parameter estimates are shown to be bounded. Computer simulations of a remotely operated vehicle (ROV) is used to illustrate the design methodology.


INTRODUCTION
Adaptive control theory have been successfully applied to ocean vehicles like underwater vehicles, submersibles and surface ships.However, most of these methods are based on the assumption that the actuator dynamics can be neglected in the control design.Including a model of the actuator can improve the robustness and performance of the control system.Fjellstad, Fossen and Egeland (1992) propose to use the reference model decomposition (RMD) technique of Butler, Honderd and Van Amerongen (1991) to compensate for the unmodelled underwater vehicle thruster dynamics.However, global stability has not been proven for this method.This paper presents a new globally stable adaptive controller for marine vehicles where a model of the actuator dynamics is included.Consider the 6 degrees of freedom (DOF) nonlinear marine vehicle equations of motion in abbreviated form, Fossen (1994): (3) where (1) is the vehicle dynamics, (2) is the kinematics and (3) is the actuator dynamics.= (u; v; w; p; q; r) T is a vector of body-xed linear and angular velocity components and = (x; y; z; ; ; ) T is a vector of positions (x; y; z) and Euler angles ( ; ; ).The components of and corresponds to the 6 DOF motion variables in surge, sway, heave, roll, pitch and yaw.u 2 IR p (p 6) is a vector of actual control inputs, and u c 2 IR p is a vector of commanded actuator inputs.Furthermore, g( ) is an unknown vector of restoring forces and moments while B( ) is a known 6 p control matrix.J( ) is a 6 6 known block diagonal transformation matrix relating the body-xed reference frame to the inertial reference frame (usually the earth).It should be noted that J( ) only depends on the Euler angles ( ; ; ), while g( ) depends on and .T = diagft i g is a p p diagonal matrix of positive unknown actuator time constants (t i > 0).
The unknown matrices M, C and D have the following properties: Property 1 (M) Newman (1977) has shown that for rigid-body at rest under the assumption of an ideal uid, the inertia matrix (including hydrodynamic inertia) is symmetrical and positive de nite, that is: In a real uid the 36 elements of M may all be distinct.However, experience has shown that the numerical values of the added mass derivatives in a real uid are usually in good agreement with those obtained from ideal theory (see e.g.Wendel (1956)).For ocean vehicles, M will depend on the wave encounter frequency and thus the speed of the vehicle.This relationship has not been established for a general vehicle in 6 DOF.For simplicity, we will restrict our treatment to marine systems described by _ M = 0.
Property 2 (C) Sagatun and Fossen (1991) have shown by applying Kirchho 's equations of motion (Kirchho ( 1869)) that for a rigid-body moving through an ideal uid the Coriolis and centripetal forces C( ) (included hydrodynamic added mass) can always be parameterized such that C( ) is skewsymmetrical, that is: This implies that: x T C( )x = 0 8 ; x 2 IR 6 Property 3 (D) Since all ocean vehicles are dissipative, the hydrodynamic damping matrix D( ) will be positive, that is: D( ) > 0 8 2 IR 6 ; 6 = 0 For details on the modeling aspects, the interested reader is recommended to consult Fossen (1994).

CONTROL SYSTEMS DESIGN
Let us initially assume that M, C( ), D( ), g( ), T and B( ) are known.Consider the Lyapunov function candidate: where ~ = ?d and ũ = u ?u d are the tracking errors and d 2 C 2 and u d 2 C 1 are the desired velocity and actuator state, respectively.
Di erentiating V with respect to time and then applying Property 1 and 2, yields: (5) Hence, substituting (1) and (3) into the expression for _ V , yields: The control objective can be de ned as: which simply states that the velocity tracking error ~ and the actuator state tracking error ũ should converge to zero as t ! 1.This objective can be satis ed by applying the following lemma.
Lemma 1 (Globally Stable Velocity Control) Consider the system (1), ( 2) and ( 3) and the following control law: 11)   where K u > 0 and u d is computed from: Here K 0 and B + = B T (BB T ) ?1 is the right Moore-Penrose pseudo-inverse of B which exists if the matrix BB T is non-singular.Then the signals ũ and ~ converge asymptotically to zero as t ! 1.
In the implementation of the control law ( 11) and ( 12), _ u d must be computed.This implies that _ , and must be measured.In fact, only the components and of are needed.It seems reasonable to chose the maximum singular values of the gain matrices K u and K according to: where ( ) is the maximum singular value, to ensure that the bandwidth of the inner servo loop (actuator dynamics) will be higher than the bandwidth of the outer loop (vehicle dynamics).

Remark 1 (Negligible Actuator Dynamics)
If the actuator is fast compared to the vehicle dynamics, we have that T = 0 and consequently that u = u c .This implies that (4) reduces to: Di erentiating this expression with respect to time suggests that the control law should be chosen as: to yield: _ V 0.

ADAPTIVE CONTROL SYSTEMS DESIGN FOR MARINE VEHICLES
In this section, we will rst derive a globally stable adaptive velocity controller and then extend these results to a locally stable adaptive position/attitude scheme.This will be done under the assumption that M, C( ), D( ), g( ) and T are unknown and that the control matrix B( ) is known.

Adaptive Velocity Control
Consider the modi ed Lyapunov function candidate: where ?= ?T > 0 and i > 0 (i = 1..6).The parameter errors are de ned as: Theorem 1 (Globally Stable Adaptive Velocity Control) Consider the system (1), ( 2) and (3) with the following control law: The desired control input u d is computed from: with parameter adaption laws: Here K u > 0, K 0 and ?= ?T > 0. Then the signals ~ and T remain bounded and ũ and ~ converge asymptotically to zero as t ! 1.
Proof: See Appendix A.
Remark 2 (Negligible Actuator Dynamics) If the actuator is fast compared to the vehicle dynamics, we have that u = u c in view of T = 0. Hence, the Lyapunov function candidate (17) reduces to: which implies that the adaptive control law of Theorem 1 reduces to: where the parameter update law ( 21) is unchanged.Notice that for B( ) = I, this result is equivalent to the adaptive control law of Slotine and Li (1987) for robot manipulators.
Notice that, C = ?C T and _ M = 0 while C 6 = ?C T and _ M 6 = 0. Consider the modied Lyapunov function candidate: Here s can be interpreted as a measure of tracking de ned as, Slotine and Benedetto (1990): where > 0 may be interpreted as the control bandwidth and ~ = ?d where d 2 C 3 is the desired position and orientation of the vehicle.It is convenient to rewrite (28) as: (29) Assume that: M, C( ), D( ), g( ) are linear in their parameters.In the original work of Slotine and Benedetto (1990) the kinematics (2) enters the regressor ( ).However, this problem can be avoided by applying the parameterization of Fossen (1993) instead.Moreover: M _ r + C( ) r + D( ) r + g( ) = ( _ r ; r ; ; ) (30) Here ( ) is a known matrix and is the unknown parameter vector.This parameterization requires that the variables: r and _ r are computed as: r = J ?1 ( ) _ r (31) _ r = J ?1 ( ) r ?_ J( )J ?1 ( ) _ r ] (32) Hence, the following theorem yields a locally stable adaptive control law.

Theorem 2 (Locally Stable Adaptive
Position/Attitude Control) Consider the system (1), ( 2) and (3) with: Here u d is computed from: The parameter estimates ^ and T are updated through the di erential equations: _ ^ = ?? T ( _ r ; r ; ; ) J ?1 ( ) s (35) _ t = ?i ũi _ u di ; i > 0 (36) Here K 0, K u > 0 and ?= ?T > 0. Then the signals ~ and T remain bounded and ũ and s converge asymptotically to zero as t ! 1.In view of (28) this implies that ~ converge to zero as t ! 1.Hence, global convergence is proven for all points except the singular point = 90 o .
Proof: See Appendix B.

Remark 3 (Negligible Actuator Dynamics)
If the actuator is fast compared to the vehicle dynamics, we have that u = u c in view of T = 0. Hence, the Lyapunov function candidate (27) reduces to: V = 1 2 s T M s + ~ T ? ? ~ ] (37) which suggests that the adaptive control law should be chosen as: with parameter update (35) to yield _ V 0. This result is equivalent to the adaptive control scheme of Fossen (1991) for underwater vehicles.Healey and Marco (1992) propose to write the ROV speed equation according to:

SIMULATION STUDY
(m 1 ?X _u ) _ u 1 = X ujuj u 1 ju 1 j + X njnj njnj (39) where u 1 is the surge velocity and n is the propeller revolution.This system can be rewritten according to: We see from the simulation results that: ~ !0, ũ !0 and that m, do and T converge to their true values in less than 50 seconds.

CONCLUSIONS
A globally stable adaptive velocity controller and a locally stable adaptive controller for position/attitude control of marine vehicles in presence of actuator dynamics have been derived.The control laws exploit the physics and a priori information of the nonlinear vehicle dynamics and kinematics in terms of well known mechanical systems properties.In most control applications the dynamics of the actuator is neglected.Hence, parameter drift and robustness of the adaptive controller can be a severe problem.The new algorithm presented in this paper allows the designer to compensate for the vehicle and actuator dynamics in a systematic manner.For simplicity, the stability analyses is based on an actuator model described by a set of decoupled rst-order systems with unknown time constants.It should be noted, that it is straightforward to extend the adaptive control scheme to a more general nonlinear actuator model: ) where f(u) and G(u) are two unknown functions.If we in addition require that both func-tions are linear in their parameters, convergence of the tracking errors can be proven by small modi cations of the proofs in Appendix A and B. Finally, it is obvious that the results presented also have validation for other types of mechanical systems like spacecrafts, aircrafts and robot manipulators.

Figure 1 :Figure 2 :
Figure 1: Upper plot shows the surge velocity (t) and actuator state u(t) together with their desired values d (t) and u d (t) while the lower plot shows their corresponding tracking errors ~ (t) and ũ(t) as a function of time.
J ?T ( )B( )ũ = ũT B T ( )J ?1 ( )s, we can subtract B( )ũ = B( )u ?B( )u d from the rst bracket and add B T ( )J ?1 ( )s to the second bracket.Furthermore, we can use the parameterization