A DIRECT ADAPTIVE CONTROL SYSTEM FOR MASS-VARYING UNDERWATER VEHICLES WITH MANIPULATOR. Mario Alberto Jordán* and Jorge Luis Bustamante

Transcription

1 A DIRECT ADAPTIVE CONTROL SYSTEM FOR MASS-VARYING UNDERWATER VEHICLES WITH MANIPULATOR Mario Alberto Jordán* and Jorge Luis Bustamante Argentinean Institute of Oceanography (IADO-CONICET), Dep of Electrical Engineering and Computers (DIEC), Uniersidad Nacional del Sur (UNS), Florida 8, CRIBABB, B8FWB Bahía Blanca, ARGENTINA Abstract: In this work, the design of an adaptie control algorithm for a general class of underwater ehicles with manipulator in the context of a path-following problem is presented The algorithm is based on speed-gradient techniques and state/disturbance obserer A stepwise changing mass is assumed during the operations of the ehicle manipulator The influence of the unknown mass on the control performance is indirectly captured by means of adaptie tracking control A case study illustrates the features of our approach by means of numerical simulations Keywords: Adaptie nonlinear control, Tracking control, Autonomous ehicles, Time-ariyng systems, Nonlinear systems 1 INTRODUCTION One of the most critical tasks in the teleoperation of manipulator arms attached on subaquatic ehicles, is the stabilization of the ehicle on part of the operator during operations at underwater constructions or on the sea bottom Sometimes, the demanded precision is of an order of magnitude within a few centimeters Such scenarios are ery common in operations of welding, assembling, drilling, sampling, among others (see Fossen, 1995) The distribution of mass about the main ehicle axes plays an important role by control systems of ROVs (remotely operated ehicles) in both respects, on one side, achieing desired dynamic positioning, and on the other side, counteracting reactie forces through the teleoperation of the? Corresponding author: Mario A Jordán mjordan@cribaeduar Tel , Ext 169, Fax: /1112/1519 arm Thus, bounds for inertia must be taken into account in the controller design to confer the teleoperation wide margins of comfortability and ehicle maneuerability, but in extreme cases, also helping to presere stability against capsizing Inertia properties can be determined ad-hoc in a commissioning phase before the control system be designed This task is particularly important in oceanographic applications when the instrumentation needed for a research mission is basically dierse and constantly changeable in weight, shape and olume Similarly, in more technical applications in the off-shore industry, the subaquatic transportation of elements of dierse masses, or, in research applications, the sampling of elements with unknown mass from the sea bottom, entail basicallythesamedifficulty As most commercial underwater manipulators belong to master-slae type, they usually require the operator be well trained and fully skilled When

2 the preiously mentioned scenarios occur, the operator will instinctiely attempt to compensate performance drops by closing the control loop through the human interface during teleoperation, which can lead to undesired instability Therefore, it is desirable that a complementary automatic control loop can fit somehownewdy- namics arriing from particular arrangement of instruments, or that it can make self-adjustments in the control behaior against mass changes or force perturbations in order to presere stability and accomplish acceptable leels of performance In this respect robust and/or adaptation abilities in the control system could be ery useful to the operator, in order for him/her to only concentrate on reference path generation or arm manipulation Some techniques appear in the specific literature, ranging from sliding-mode-based control systems for seomechanisms (Liu and Feng, 25) and parameter identification for model-based control of ariable-configuration ehicles (Caccia et al, 2) Also, adaptie and learning methods were deeloped in (Yuh, 199; Fossen and Fjellstad, 1995; Ishii et al, 1995; Cristi et al, 199; Venugopal et al, 1992; Leonard, 1995; Smallwood and Whitcomb, 24), among others Although all of them are concerned with the problem of uncertainties, in general, only a few ones tackle the problem of time-arying parameters specifically Fig 1 - Port/starboard symmetric ROV with manipulator (case study) In this paper, we present a noel approach to direct adaptie control for path following of underwater ehicles in 6 degrees of freedom with a manipulator arm A similar adaptie method was presented in (Jordán and Bustamante, 26), howeer it was required therein the a-priori knowledge of the mass-center coordinates and its constancy in time 2 VEHICLE DYNAMICS Let a body-fixed coordinate system be that illustrated in Fig 1 for an origin O coincident with the naigation sensor system So, the center of graity G has coordinates gien by r G = [x G,y G,z G ] T with respect to O The system states are the ehicle position ector, referred to as η, with respect to a ground-fixed coordinate system O, anhe speed ector with respect to the ehicle-fixed coordinate system O In details η=[x, y, z, φ, θ, ψ] T includes translations along the main axes (ie, surge, sway and heae) and tilts about them (ie, pitch, roll and yaw), and =[u,, w, p, q, r] T includes the respectie linear speeds and rotation rates in the ehicle-fixed system The general 6-DOF rigid-body equations are (cf Fossen, 1995) M = C() D( ) + F b (η)+f c + F t (1) η = J(η), (2) where M is a non-diagonal inertia matrix and C a Coriolis matrix, both conceied as a sum of a rigid-body and an added mass components, as respectiely indicated in M(t)=M b (t)+m a (t) (3) C()=C b ()+C a (), (4) D is a damping matrix with two components accounting for linear and quadratic skin friction due to laminar and turbulent boundary layers, respectiely, it is D() =D l +D q ( ), (5) F b is a restoring generalized force, F c is a reactie force of the umbilical cable and finally F t is the generalized thrust force Finally, J is the wellknown 6 6 dimensional rotation matrix (see Fossen, 1995) The notation is applied for a ector with elements of in absolute alues More specifically, the system matrices take the general form m mz G -my G m -mz G mx G M b = m my G -mx G -mz G my G I x -I xy -I xz mz G -mx G -I yx I y -I yz -my G mx G -I zx -I zy I z (6) M a = m aij,fori, j =1,, 6 (7) C b ()= -m (y G q+z G r) m (y G p+w) m (z G p-) m (x G q-w) -m (z G r+x G p) m (z G q+u) m (x G r+) m (y G r-u) -m (x G p+y G q) m (y G q+z G r) -m (x G q-w) -m (x G r+) -m (y G p+w) m (z G r+x G p) -m (y G r-u) -m (z G p-) -m (z G q+u) m (x G p+y G q) -I yz q-i xz p + I z ri yz r+i xz p-i y q I yz q+i xz p-i z r -I xz r-i xy q+i x p -I yz r-i xy p+i y qi xz r+i xy q-i x p (8)

3 c a3 c a2 c a3 c a1 C a () = c a2 c a1 c a3 c a2 c a6 c a5 c a3 c a1 c a6 c a4 c a2 c a1 c a5 c a4 (9) D l =(d qij ),fori, j =1,, 6 (1) " T # D q1 D q ( ) =, (11) T D q6 where m is the total ehicle mass, m aij the additie mass elements in each cross modes ij, I ij are elements in M b representing inertia moments with respect to axes ij, (x G,y G,z G ) are the coordinates of the mass center, the functions c ai in C a are c ai = P 6 j=1 m a ij j with j the element j of,and finally the D qi s in D q are 6 6 constant matrices The metacenter and mass center lie both in the plane x -z Thus, the buoyancy ector is expressed as F b (η) = (12) (W -W w ) s(θ) - (W -W w ) c(θ) s(φ) - (W -W w )cos(θ)cos(φ) - (Wy G -W w y B ) c(θ) c(φ) + (Wz G -W w z B ) c(θ) s(φ) (Wx G -W w x B ) c(θ) c(φ) + (Wz G -W w z B ) s(θ) - (Wy G -W w y B ) s(θ) - (Wx G -W w x B ) c(θ) s(φ) where (x B,y B,z B ) are the coordinates of the metacenter, W is the ehicle total weight mg, W w is the buoyancy, and s() and c()arethesineand cosine functions, respectiely The generalized thrust force F t is computed by the controller for the system O Thisforce can be decomposed into the thrust ector f = T f1,f 2,,f nf with nf the number of thrusters Both ectors are related by f = B T BB T 1 Ft, (13) with the matrix B containing constructie ehicle constants The thruster dynamics is n = C 1 (f) =C 1 B T (BB T ) 1 F t (14) µ n r = 1 N(s) n r + D(s) n (15) b b f r = C (n r ), (16) where n and n r are ectors with the true thruster rpm s and its reference, respectiely, f r is the reference for the thrust ector f, C is a static, nonlinear, continuous and inertible propeller characteristic in ector form, and finally, D(s) and N(s) are Hurwitz polynomials in the Laplace ariable s with b = N() (Jordán et al, 25) 3 THE ADAPTIVE CONTROL APPROACH 31 Path-tracking problem We are inoled with the control objectie lim η(t) η r(t) = (17) t lim (t) r(t) =, (18) t for arbitrary bounded initial conditions η() and () and smooth positioning and cinematic references η r (t) and r (t), respectiely To attain (17)-(18), a control action law is designed to coneniently manipulate the thrusts f j in f The adaptie objectie supposes additionally that (6)- (12) are unknown To this end, let us define (cf Jordán and Bustamante, 26) η = η η r (19) = J 1 (η) η r + J 1 (η)k pη, (2) with a gain matrix K p = Kp T andanenergy cost functional Q( η, )= 1 η T 1 η + T M, (21) 2 2 which must be a radially unbounded and nonnegatie scalar function (Fradko et al, 1999) For an asymptotic stable controlled dynamics h i in the state space η T, T T, it is aimed that for eery initial mismatches η() and (), one accomplishes Q( η(t), (t)), fort (22) According to the speed-gradient approach, the manipulated ariable, in this case the generalized force F t, can be designed by considering certain properties of Q like smoothness, conexity and radially growth in some compact D in the state space So, taking the first deriatie of (21) with (19)-(2), one gets Q( η,, η,t) = η T T K pη + η J(η) (23) T C () T D l D q ( ) T F b (η)+ F c T M d J 1 (η) η r + µ + T dj 1 (η) M K p J 1 (η)kp 2 η+ + T MJ 1 (η)k p J(η) + T F t Thus, a suitable selection of F t for the aimed properties of Q is (cf Jordán and Bustamante, 26) 6X F t = U i C i ( i )+U 7 + (24) + 13X j=8 i=1 U j D j- 7 ( j-7 )+U 14 F b1 (η)+u 15 F b2 (η)+ + U 16 d ³ η, η,,t F c K J T (η) η, with ³ η, d, η, t = d J 1 (η) dj 1 (η) η r K pη + + J 1 (η)kp 2 η J 1 (η)k p J(η), (25)

4 where U i are controller matrices of unknown coefficients accounting for physical parameters that hae to be found adaptiely, C i and D i are elocity-dependent matrices explained later, F b1 and F b2 are tilt-dependent ectors also cleared next, and finally K is a matrix with K = K T Thenotations means element-by-element matrix product Moreoer, it was assumed the case that F c can be measured without error According to the general formulation of speedgradient algorithm any allowable candidate F t must be such a one that Q(U i ) result conex in eery element u kij of the U i s It straightforward to erify from (23) with (24) that this condition is also satisfied 32 Adaptie laws Now, one can calculate the functions U i s by means of adaptie control laws based on gradient functions as U Q i = Γ i U i The main idea consists in separating coefficients from ariables in the dynamics equations Towards this goal, consider (24) and let first the Coriolis matrix C in (4) be expressed as P 6 i=1 C i C i ( i ) with C i constant matrices So, it is alid for U i with i =1,, 6 the laws ũ i q ũ i r ṽ i p ṽ i r U i = Γ i w i p w i q p i p i w p i q p i r, q i u q i w q i p q i r r i u r i r i p r i q (26) where ũ,, r are elements of Next,forD l there corresponds the law U 7 = Γ 7 ũu ũ ũw ũp ũq ũr ṽu ṽ ṽw ṽp ṽq ṽr wu w ww wp wq wr pu p pw pp pq pr qu q qw qp qq qr ru r rw rp rq rr (27) Analogously, D q can be expressed as P 13 j=8 D q j D j 7 ( j 7 ) with D qj constant matrices, for which following laws there will be assigned U j = Γ j (28) ũ k u ũ k u ũ k w ũ k p ũ k q ũ k r ṽ k u ṽ k ṽ k w ṽ k p ṽ k q ṽ k r w k u w k w k w w k p w k q w k r p k u p k p k w p k p p k q p k r q k u q k q k w q k p q k q q k r r k u r k r k w r k p r k q r k r with k = j 7 Next, the buoyancy ector F b can be decomposed into two ector terms as B 1 F b1 + B 2 F b2, for which there corresponds each one the laws U 14 = Γ 14 diag (ũ sin θ,, w cos θ cos φ (29) p cos θ cos φ, q cos θ cos φ, r sin (θ)) U 15 = Γ 15 diag (, ṽ cos θ sin φ, (3), p cos θ sin φ, q sin θ, cos (θ)sin(φ)), respectiely Finally, for the inertia matrix M it is assigned the law ũd 1 ũd 2 ũd 3 ũd 4 ũd 5 ũd 6 ṽd 1 ṽd 2 ṽd 3 ṽd 4 ṽd 5 ṽd 6 U 16 = Γ 16 wd 1 wd 2 wd 3 wd 4 wd 5 wd 6 pd 1 pd 2 pd 3 pd 4 pd 5 pd 6, qd 1 qd 2 qd 3 qd 4 qd 5 qd 6 rd 1 rd 2 rd 3 rd 4 rd 5 rd 6 (31) where d i is the element i of the ector d in (25) Due to space constrain, the conergence of the adaptie laws are not shown here The reader can howeer refer to (Jordán and Bustamante, 26) for finding similar arguments for a formal proof 33 State/disturbance obserer The control performance can be considerably enhanced if the thruster dynamics is inoled in the controller design The way to do this includes the estimation of the driers states and input so as to employ the inerse thruster dynamics in the controller design Therefore, we summarize the algorithm presented in (Jordán et al, 25) on the basis of (13)-(16) First, let us look for a reference thrust ector instead of (13), but calculated directly by ³ f r = C n r, (32) with n r being a reference rpm ector with elements ˆn r for each thruster As n r is unknown, it is estimated as perturbation To this end, let the state model for thrusters be x = Ax + b n r (33) n = c T x, (34) and an estimation of the state x = Ax + b ˆn r + k x (n ideal ˆn), (35) with f ideal = B T (BB T ) 1 F t (36) n ideal = C 1 (f ideal ) (37) So the perturbation results estimated as ñ r = k n c T + kṅc T A x, (38) with x = x x the state error, which in turn accomplishes x = A kx c T b k n c T + kṅc T A x, (39) in where its parameters are kṅ = 1 b n 1 an 1 1 kˆx = + k n,,,, b n 1 b n 1 (4) Hence, if A k x c T b k n c T + kṅc T A is Hurwitz, then both errors accomplish x, ñr for t (41)

5 In (Jordán et al, 25) there are gien examples for the application of the obserer and guidelines to tune the matrix A Finally, instead of (13), (32) is applied for the adaptie control This also contains the information of the control action F t through (36) 34 Reference flight path for sampling The reference trajectory for flying and picking up a targeted object from the sea bottom is particularly defined in such a way that the landing and takeoff of the ehicle on the sea bottom be done softly despite a mass change in this operation Other point to be considered is that usually the on-board camera to operate the manipulator is on the front and directed to the targeted object So the ehicle has to approach with a nonzero pitch angle, grab the object and retreat finally without collision A suitable flight path to this end could consist in a rectilinear approach to the target with asymptotic null elocity, then of a pause on a lower fixed point oer the bottom till the object is seized, next of a retreat backwards with increasing elocity, and finally a turn around itself or on a helicoid to recoer the initial rectilinear stretch of the trajectory This would ensure successfully and optimally the end of the mission On the other hand, the execution of such a path on part of the operator would demand his/her full skillful if the goal has to be performed rapidly and carefully due to the significant perturbation caused in pitch and heae modes mainly Our goal next is to performed this tracking automatically and adaptiely with the approach deeloped preiously an upper initial position in order to pick up a mass on the sea bed and afterwards return to the start position Different masses were tested, namely 1(Kg) and 2, 25(Kg), taking care therein that no thruster saturation be produced in the ehicle state when being at rest θ ( o ) z (m) q ( o /s) w (m/s) θ θ ref z ref z q w ref w q ref Δm Time t (s) Fig 3 - Eolution of pitch and heae modes for a mass change 1(Kg) at t = 71(s) Thrust f i (N) 2 THRUST THRUST THRUST saturation Δm THRUST Time t (s) Fig 4 - Vertical thrusts for a mass change of 1(Kg) at t = 71(s) 2 saturation Δm THRUST THRUST 6 Thrust f i (N) THRUST THRUST 8 Fig 2 - Adaptie control of an AUV along a reference fly path for sampling 4 NUMERICAL SIMULATIONS In this section, some numerical simulations are presented to illustrate the features of our adaptie approach A reference trajectory such as that described preiously is applied (see Fig 2) At the beginning, no information is aailable of the large amount of system parameters in (6)-(12) A 11-meters stretch is coered by the ehicle from saturation Time t (s) Fig 5 - Horizontal thrusts for a mass change 1(Kg) at t = 71(s) Fig 3 depicts the main modes of motion for a mass change of 1(Kg) One sees that after a short commissioning phase of the adaptie control and obserer, both the pitch and heae acquires the desired geometric and cinematic paths without appreciable error This is also noticed for the time point when a mass is picked up by a simulacrum of the manipulator operation On the other side, the 8 thrusts show signs of a greater sensibility than the states at this point This is illustrated in

6 Figs 4 and 5 for ertical and horizontal thrusters, respectiely The tenor of this response in more intense for a larger mass change of about 225(Kg) (not shown in this paper) Finally, Fig 6 characterizes the eolution of one of the numerous controller parameters, just one physically inoled in the mass change Despite the abrupt change of mass, the adaptie law leads this parameter slowly to a steady-state alue (U 16 ) 1,1 (Kg) Δ m Time t (s ) Fig 6 - Eolution of the element (1, 1) of U 16 for a mass change of 225(Kg) at t = 31(s) 5 CONCLUSIONS A noel approach to direct adaptie control that combines a speed-gradient algorithm with a state/disturbance obserer is presented for stepwise mass-arying The control employs a coordinate system on a fixed known point instead of the mass center This enables to generate adaptie laws for any dynamics change related to inertia, damping and hydrodynamics ariations In order to achiee high performances of the transient control behaior, the thruster dynamics is incorporated using an inerse dynamics technique and obserer of the unknown states and input of the driers using the ehicle states and the control action A suitable flight path for sampling with manipulator is conceied for reference trajectory generation An illustration of the features of the approach in 6 degrees of freedom is accomplished by simulations of a case study Results depict slight perturbations of the states during mass changes, nothoweerthesameinthethrusts,whichshow significant staged ariations reacting sensitiely to these disturbances Also all changes in the multiple controller parameters are moderate The all-round transient behaior of the adaptie loop is extremely short, traelling to the steady state rapidly with practically null tracking errors Future work is directed to construct minimal-time controls on geometric flight paths using automatic elocity generation 6 ACKNOWLEDGEMENTS The authors will thank IADO-CONICET and DIEC-UNS for the support of this research and Prof Dr E Kreuzer and Dr V Schlegel of MUM- TUHH, Germany, for haing introduced us in this application field REFERENCES [1] Caccia, M Indieri, G and Veruggio G, "Modeling and Identification of Open-Frame Variable Configuration Unmanned Underwater Vehicles," IEEE J Oceanic Eng, 25(2), (2) [2] Cristi, R, Papoulias, FA and Healey, AJ, Adaptie Sliding Mode Control of Autonomous Underwater Vehicles in the Die Plane, IEEE J Oceanic Eng, 15(3), , (199) [3] Fossen, TI, "Guidance and Control of Ocean Vehicles," John Wiley&Sons, (1994) [4] Fossen TI and Fjellstad, IE, Robust Adaptie Control of Underwater Vehicles: A comparatie study, Proc 3rd IFAC Workshop on Control Applications in Marine Systems, Trondheim, Norway, (1995) [5] Fradko, AL, Miroshnik, IV and Nikiforo VO, Nonlinear and adaptie control of complex systems, Kluwer Acad Pub (1999) [6] Ishii, K, Fujii, T and Ura, T, "An On- Line Adaptation Method in a Neural Network Based Control System for AUVs, IEEE J Oceanic Eng, 2(3), , (1995) [7] Jordán, MA and Bustamante, JL, "A Speed- Gradient Adaptie Control with State/Disturbance Obserer for Autonomous Subaquatic Vehicles," accepted for the IEEE Control and Decision Conf, San Diego, CA, USA, Dec, (26) [8] Jordán, MA, Bustamante, JL and Pinna- Cortinas, JM, "Design of Adaptie Control Systems for ROVs Using Inerse Dynamics and State/Disturbance Obseration," in 6th Argentine Symposium on Computing Technology, Rosario, Set 1-3, (25) [9] Leonard, NE, "Control Synthesis and Adaptation for an Underactuated Autonomous Underwater Vehicle," IEEE J Oc Eng, 2(3), , (1995) [1] Liu, Q and Feng, P-E, "Robust Tracking Control of Mechanical Sero Systems with Inertial Parameters Varying in Large Range," Control Theory and App, 22(6), , (25) [11] Smallwood, DA and Whitcomb, LL, "Model Based Dynamic Positioning of Underwater Robotic Vehicles: Theory and Experiment," IEEE J Oceanic Eng, 29(1), , (24) [12] Venugopal, KP, Sudhakar, R and Pandya, AS, On-Line Learning Control Of Autonomous Underwater Vehicles Using Feedforward Neural Networks, IEEE J Oceanic Eng, 17(4), , (1992) [13] Yuh, J, Modelling and Control of Underwater Robotic Vehicles, IEEE Trans Syst, Man, Cybern, 2(6), , (199)