Adaptive Setpoint Control for Autonomous Underwater Vehicles

Transcription

1 Proceedings olthe 42nd IEEE Conference OD Decision and Control Maui, Hawaii USA, December TuP08-4 Adaptive Setpoint Control for Autonomous Underwater Vehicles Y.C. Sun and C.C. Cheah School of Electrical and Electronic Engineering Nanyang Technological University Block SI, Nanyang Avenue, Singapore Abstract-In this paper, we propose an adaptive saturated proportionalderivative (SP-D) setpoint controller for autonomous underwater vehicles. The proposed controller dws not require any knowledge of the inertia matrix, Coriolis and centripetal force, hydrodynamic damping, and parameters of the gravity and buoyancy forces. The structure of this setpoint controller is based on the SP-D feedback, plus an adaptive update law for grarity and buoyancy forces. By using Lyapunov s direct method and LaSalle s invariance principle, we provide simple explicit conditions on the regulator gains to ensure global asymptotic stability. Simulation results are presented to demonstrate the effectiveness of the proposed controller. Keywords-Underwater vehicles, adaptive control, setpoint control, gravity and buoyancy regressor, stability. I. INTRODUCTION In recent years, many research efforts have been deyoted to the development of autonomous underwater vehicles (AUVs) [11,[2],[3]. The potential applications of AWs vary from scientific research of ocean, surveillance and inspection of commercial undersea facilities and installations, to various military operations. The presence of hydrodynamic effects makes the control problem of the AW a very challenging task. Major factors that make it difficult to control the AUVs include: first, the vehicle has highly nonlinear, time-varying dynamic behavior. Second, hydrodynamics of the vehicle are poorly known and may vary with relative vehicle velocity to fluid motion. Third, a variety of unmeasurable disturbances by ocean currents. Setpoint control of underwater vehicles, also called station keeping or regulation of underwater vehicles, may be recognized as one of the most important aim in autonomous underwater vehicles control. The goal of global setpoint control is to move the underwater vehicle from any initial state to a desired destination point. The proportional-derivative (PD) control plus gravity and buoyancy compensation [I], [4] is the simplest global setpoint control for the underwater vehicles. The main advantage of this controller is that the tuning procedure to achieve global asymptotic stability reduces to selecting the proportional and derivative gains in a straightforward manner. However, a drawback of this control strategy is that the exact knowledge of the gravitational and buoyancy force of the underwater vehicle, which depends on some parameters like water density, is required. To overcome parametric uncertainties on the underwater vehicles dynamics, adaptive controllers have been introduced in [11,[51,[61,~71,~81,~91,[101 and [Ill. However, these controllers require a regressor of the dynamic model which include the inertia matrix, Coriolis and centripetal force, hydrodynamic damping, gravity and buoyancy force. Hence, the number of dynamic parameters to be updated by the adaptive law is very large. In addition, the inverse Jacobian matrix is required in these adaptive control laws. On the other hand, the common practice of using the proportional-integral-derivative (PID) control [I], 1121 in the control of underwater vehicles, which does not required any information of the underwater vehicles dynamics, is only valid in a local sense. In this paper, we propose an adaptive setpoint controller for underwater vehicles which does not require any howledge of the [I31 inertia matrix, Coriolis and centripetal force, hydrodynamic damping, and parameters of the gravity and buoyancy forces in the control law. Motivated by the controllers introduced in [14],[15],[161 for robotic manipulator, we develop an adaptive setpoint control law for autonomous underwater vehicle which does not require the inverse of Jacobian matrix. The presence of hydrodynamics effects makes the control problem of the autonomous underwater vehicle a more challenging task. The contributions of this paper are: first, the result is valid globally in the presence of uncertainty in the gravity and buoyancy forces. As compare with the PID control law [I], the result is valid only in the local sense. Second, our approach requires only the gravity regressor as compare with the previous adaptive control laws ~11,~51,~61,~71,~81,[91,[101 which require the regressor matrix of inertia matrix, Coriolis and centripetal matrix, and hydrodynamic damping matrix. Since the dynamics of the underwater vehicles is very complex, the number of physical parameters to he updated by adaptive law is very large. Therefore, the proposed control law which used only the gravity regressor is very simple and hence easy to implement. Thud, the inverse Jacobian matrix is not required in the adaptive law as compared with the previous adaptive control laws in the literature. The organization of this paper is as follow. Section II summarizes the underwater vehicles kinematic and dynamics model and its main properties. Section 111 proposes an adaptive SP-D controller and provides conditions on the controller gains to ensure global asymptotic stability. Section IV shows the simulation result of the proposed control law. Finally, Section VI concludes this paper /$17.00 Q2003 IEEE 1262

2 11. PROBLEM FORMULATlON It is convenient to define the underwater vehicle state vectors according to the Society of Naval Architects and Marine Engineers (SNAMEj notation [l]. Two common vectors that being used in defining the undenvater vehicle state vector are q and U. The vector q is defined as: q = [qy $IT, where q1 = [x y is the vehicle position vector in the earth fixed frame and q2 = [4 6' $IT is the vehicle Euler angle in the earth fixed frame. And the vector U is defined as: U = [UT vflt, where q = [U U wit is the body fixed linear velocity vector and 212 = [p q vit is the body fixed angular velocity vector. In Fig. 1, the defined coordinate frames are illustrated. Beside the Euler angle representation, Euler parameters or unit quaternions [l], [17] can also be used. Fig. 1. Body fixed and eanh fixed reference frames The vehicle's motion path relative to the earth fixed frame coordinate system is given by the kinematics equation [I] as follow: where,j(q) is a 6 x 6 kinematics transformation matrix or Jacobian matrix. The dynamics behavior of an underwater vehicle is described through Newton's laws of linear and angular momentum. The equations of motion of underwater vehicles are highly nonlinear and coupled due to hydrodynamic added mass, lift and drag forces, which are acting on the vehicle. The equations of motion of the underwater vehicle is given as [l]: A.16 + C(v)v + D(v)v + g(q) = 7, (2) where U E R6 is the vector of linear and angular velocity of the AUV with respect to the vehicle's body fixed frame. The vector r E R6 is the vector of generalized forces on the AUV which is supplied by the thrusters. The matrix Af is the inertia matrix of the AUV which includes both the rigid body and the added mass term. The matrix C(u) denotes the matrix of Coriolis and centripetal forces which includes both the rigid body and the added mass terms. The vector D(u) represent the hydrodynamic damping and lift force and the vector g(q) denotes the gravitational and buoyancy force. The underwater vehicle dynamics described in (2) has the following properties [l]: Property 1. The inertia matrix, Af including the added mass is symmetric and positive definite such that Ai = AfT > 0. Property 2. The Coriolis and centripetal matrix, C(u) is skew-symmetric matrix such that C(v) = -CT(u)Vv E R6. Property 3. The hydrodynamic damping matrix, D(w) is strictly positive such that D(v) > 0 V U E R ADAPTIVE SATURATED PROPORTIONAL-DERIVATIVE (SP-D) WITH GRAVITY REGRESSOR In this paper, we proposed an Adaptive SP-D control law with uncertain gravitational and buoyancy force compensation using gravity regressor. Note that the gravity term can be completely characterized by a set of parameters 0 = (6'1, [I41 as, g(q) = zw, (3) where, Z(6) E Rnxm is the gravity regressor. The Adaptive SP-D control law is proposed as 7 = -KpJT(q)s(e)- K,U + z(q)b, 0 = -L- 'zt(q)(u + ajt(q)s(e)), where e = ~ l-qd = (el,..., e,)t E R" is the positional deviation from a desired position: s(e) = (si(el),...,~~(e,))~; si(.), i = 1,..., n, are saturated functions to be defined; Kp = k,i, K, = k,i are positive feedback gains for the position and velocity, respectively; L is a positive definite diagonal matrix; a is a positive constant. Note that the above controller does not require the knowledge of the inertia matrix, Coriolis and Centripetal matrix, hydrodynamic damping matrix and the inverse of the lacobian matrix, J(q). (4) (5) Let us define a scalar function Se(@) and its derivative si(0) as shown in Fig. 2 and with the following properties [14]: 1) &(e) > o for 6' # o and s,(o) = 0. 2) &(e) is twice continuously differentiable, and the derivative si(0) = is strictly increasing in 6' for 16'1 < 'yi with some 'yi and saturated for 16'1 2 'yi, i.e. si(6) = +si for 0 2 fyi, and 6' 5-7, respectively where si is a positive constant. 3) There is constant C, > 0 such that,.%(e) 2 4Si*(6'), (6) 1263

3 for B f 0 where 1 V = -vtalw 2 + as(e)tj(q)alv 1 +-(0 - B)TL(@- 8) 2 " + ak,)s+(ei), (12) +C(kp i=l J/--7 Ibl... -t Fig. 2. (a)quasi-natural potential: S(0) 7 (b)derivative of S(0): s(s) Some examples of the saturation function can be found in [14] and The purpose of introducing the saturation function, si(b) [I41 in this paper is to obtain global result for the proposed control law. Without the saturation function, the result is only valid in a local sense. Substituting (4) into (2) and using (3), we have the closedloop equation, Alir + C(u)u + D(u)v + KnJT(o)s(e) +K,v + Z(q)O- Z(q)e = 0. Let us define a vector y of the form y = v + ajt(q)s(e). (7) (8) To cany out the stability analysis for the closed-loop system with the proposed controller, we take the inner product of y with the closed-loop equation (7) to give, (v + ajt(q)s(e))t(afd +KpJT(q)s(e) + ~ " v which can be expressed as t i ~ + v ~ ~~(u)v i ~ + C(v)v + D(v)v + z(ds - z(o)b) = 0, + J D ( ~ ) U + utkpjt(v)s(e) +vtk,u + as(e)*j(q)afd + as(e)tj(q)c(v)w +as(ejtj(q)d(u)v + as(e)tj(0)kpjt(q)s(e) +as(e)'j(q)k,v + ~ Tz(~)(o +as(e)tj(a)z(q)(o Equation (10) can be written as d - v + I$' = 0, dt (9) - 8) - 8) = 0. (10) W = utk,v + vtdv + as(e)tj(q)kpjt(q)s(e) +a{s(e)tj(o)c(u)v + s(e)tj(q)d(u)v -S(e)TJ(q)Alv- s(e)tj(?,)alv}. (13) We will show that this will eventually lead to a Lyapunov function for the stability analysis of the underwater vehicle control problem. To show the positive definiteness of the Lyapunov function candidate V in (12), we note that btmv 4 + a~(e)~j(q)alv + C:=,(kP + ak,)si(ei) = f(u + ZaJT(q)s(e))TAl(w + ZaJT(q)s(e)) -a2s(e)tj(q)aljt(q)s(e) + xy=l(kp + ak,)si(ei) 2 C:=l(kp + ak,)zis:(ei) - azs(e)tj(q)aijt(q)s(e) 2 C:=l(kp+a(ku~; -aa,))s:(eij, (14) A where A, = A,,,[J(q)AIJT(q)]. Substituting this into equation (IZ), we have 1 1 v 2 -vtalw + -(B - O)TL(O - e) 4 2 n +c(kp + a(k& - aa,))s:(ei) > 0, (15) i=l where ku and a are chosen so that k& - aam > 0. (16) Therefore, the function V in (12) represents a Lyapunov function candidate for the global set-point control of the underwater vehicles. Next, we proceed to show that the time derivative of the Lyapunov function is negative semi-definite. As seen from ll), this is equivalent to showing that IY is positive semidefinite. From the last term on the right-hand side of (13), since.(e) is bounded, there exist constants CO > 0 and c1 > 0 so that [I41 a I s(e)tj(dc(a)u + s(e)tj(do(v)u -S(e)TJ(q)Alv- s(e)tj(q)alv 2 -acollvll* - acllls(e)l12. (17) Substituting inequality (17) into (13) yields 1Y 2 vt(k,l+ D(v) - acol)v +s(e)t(aj(q)kpjt(v) - acios(e), 2 (ku + Xmin[D(u)l- a%)llvl12 +a(kp.\m+n[j(7/)jt(1)l - a)lls(e)l12, (18) I 1264

4 where kp and k, can be chosen sufficiently large and a chosen sufficiently small so that ko + Xmm[D(u)l - QCO > 0, kpxm*"[j(li)jt(0)l - c1 > 0, (1% and hence W is positive definite in z' and s(e) The asymptotic stability of the equilibrium position (vd,o) with uncertain gravitational and buoyancy force compensation can be made explicit in the following theorem: Theorem: The equilibrium position (qd,o) of the closed loop system described by (7) and (5) is asymprotically srable if the feedback gains kp, k, and a are chosen to.saris& conditions (16) and (19). Proof: Since V is positive definite and 1V is positive semi-definite, from (1 I), we have d -v dt = -wso Therefore, LaSalle's invariance Theorem implies that Vd , 'U + 0 as t + 03 for any initial position q(0) and g(7) = - (mg- $rr3pg)sin(@ -(mg - ~nr3pg)cos(b)sin(o) -(w 3 - ~ ~~~P~)cos(s)cos(o) ZGm.gCOS(s)Sjn(@) zgmgsin(8) IV. SIMULATION RESULTS To demonstrate the performance of the proposed controller, simulation study is performed on an underwater vehicle, ODIN (Omni Directional Intelligent Navigator) [191. ODIN is a spherical A W designed in the University of Hawaii which has 4 horizontal thrusters and 4 vertical thrusters. The parameters in the dynamic model of ODIN are given as follow [20]: m 0 -mzg 0 0 i7rp mw 0 -mw 0 0 mu -mu -mu 0 0 mu mu 0 0 Note that in the proposed control law described by (4), only the structure of the gravity regressor, Z(q), is required. In addition, only two parameters need to he updated for the vehicle ODIN. Hence, it is simpler and easy to implement. The uncertain parameters in the gravitational and buoyancy force are assumed to be unknown. In this simulation, the position and velocity gain is chosen as Kp = 450J and K, = 3001 respectively, the gain in the adaptive update law is chosen as L = 100 and a = 0.1. The vehicle is required to move from the origin to a desired location. First, the vehicle is required to move to [-2,3,5,0.17, -0.09, -0.44IT. Fig. 3 shows that both the position and orientation errors converge to zero. Second, the destination point is changed to [l, -4, -3, -0.26,0.09,0.35IT. Fig. 4 shows that the position and orientation errors converge to zero. Simulation results show that all the position errors converge to zero with the same position and velocity gain in the control law. V. CONCLUSION In this paper, we have presented an adaptive SP-D feedback control law for the setpoint control of the underwater 1265

5 CRB(IJ) = mzcr mw -mu mw mzgt mu m(v - zcp) -m(u + zcq) 0 -mzct mw -m(v - zcp) 0 IT -Iq -mw -mzg m(u+zcq) -Ir 0 IP in U -mu 0 Iq -IP 0 Z I I 4 s P Tlm IbSq I 4 s 5 10 IS 20 *I a T m Wl I Fig. 3. Destination point = 1-2: 3,5,0.17, -0.09; -0.~1' Fig. 4. Destination point = [Il-4, -3, ;0.35]T 1266

6 vehicles. We have shown that the global asymptotic stability can be guaranteed even when the gravity and buoyancy force is uncertain. In the proposed control law, only the gravity regressor is required and no inverse of the Jacobian matrix is required. A Lyapunov function is proposed for the stability analysis. Sufficient conditions for choosing the feedback gains to guarantee the asymptotic stability are presented. Simulation results illustrated the performance of the proposed controllers. VI. REFERENCES [l] T. I. Fossen, Guidance and Contml of Ocean Vehicles. John Wiley & Sons, [21 K. P. Valavanis, D. Gracanin, M. Matijasevic, R. Kol- IUN, and G. A. Demetriou, Control architectures for autonomous underwater vehicles, IEEE Control Systenis Magazine, vol. 17, pp , Issue 6. [31 L. L. Whitcomb, Underwater robotics: Out of the research laboratory and into the field, in Proc. of the 2000 IEEE International Conference on Robotics and Autonurtion, (San Francisco, CA), pp , [4] M. Takegaki and S. Arimoto, A new feedback method for dynamic control of manipulators, ASME Journal of Dynamic Systems, Measurenient and Control, vol. 102, pp , [5] S. Zhao, J. Yuh, and H. T. Choi, Adaptive dob control of underwater robotic vehicles, MTVIEEE Conference and Exhibition, vol. 1, pp , [61 J. Yuh, J. Nie, and C. S. G. Lee, Experimental study on adaptive control of underwater robots, in Pmc. of the 1999 IEEE International Conference on Robotics and Automation, (Detroit, Michigan), pp , [71 G. Antonelli, S. Chiaverini, N. Sarkar, and M. West, Adaptive control of an autonomous underwater vehicle: Experimental results on odin, IEEE Trans. on Control Systems Technology, vol. 9, no. 5, pp , T. I. Fossen and S. I. Sagatun, Adaptive control of nonlinear underwater robotic systems, in Pmc. of the 1991 IEEE Int. Conference on Robotics and Automatian, Sacramento, California), pp , [91 J. Yuh, Modeling and control of underwater robotic vehicles, IEEE Trans. on Systems, Man, and Cybernetics, vol. 20, no. 6, pp. 1, , Cl01 T. I. Fossen and S. 1. Sagatun, Adaptive control of nonlinear systems: A case study of underwater robotics systems, Journal of Robotic Systems, vol. 8, pp , [lll J.-J. E. Slotine and W. Li, Adaptive manipulator control A case study, IEEE Transaction on Automatic Control, vol. 33, no. 11, pp , [12] S. Arimoto and E Miyazaki, Stability and robustness of pid feedback control for robot manipulators of sensory capability, in Proc. of Int. Symposium <$ Robotics Research, (Couvieuv, France), pp. -, [131 M. Caccia, G. Bruzzone, and G. Veruggio, Guidance of unmanned underwater vehicles: Experimental results, in Pmc. of rke 2000 IEEE International Conference on Robotics &Automation, (San Francisco, CA), pp , [141 S. Arimoto, Control Theory of Non-linear Mechanical Systems. A Passivity-based and Circuit-theoretic Approach. Clarendon Press, [IS] H. Yazarel, C. C. Cheah, and H. C. Liaw, Adaptive sp-d control of a robotic manipulator in the presence of modeling error in a gravity regressor matrix: theory and experiment, in IEEE Transactions on Robotics and Automation, pp , Vol. 18, Issue 3. [16] C. C. Cheah, K. Li, S. Kawamura, and S. Arimoto, Approximate jacobian feedback control of Jobots with kinematic uncertainty and its application to visual servoing, in IEEE Inr. Con$ on Robotics and Automation, (Seoul, Korea), pp , [ E. Fjellstad and T. I. Fossen, Quaternion feedback regulation of underwater vehicles, in Pmc. of the Third IEEE Conference on Control Applications, pp , vol R. Kelly, Global positioning of robot manipulators via pd control plus a class of nonlinear integral actions, IEEE Trans. on Automatic Contml, vol. 43, no. 7, pp , [19] S. Choi, J. Yuh, and N. Keevil, Design of omnidirectional underwater robotic vehicle, in OCEANS 93. Engineering in Harmony with Ocean. Proceedings, pp , vol.1. [201 T. K. Podder and N. Sarkar, Fault tolerant decomposition of thruster forces of an autonomous underwater vehicle, in IEEE Int. Con$ on Robotics and Automation, (Detroit, Michigan), pp. 8489,