Modeling and Control of Underwater Robotic Systems
Dr.ing. thesis
Ingrid Schj0lberg
Department of Engineering Cybernetics Norwegian University of Science and Technology
March 1996
Report 96-21-W Department of Engineering Cybernetics Norwegian University of Science and Technology N-7034 Trondheim, Norway
OBmetmoN of im doombst b mwia DISCLAI R
Portions of tins document may be Illegible in electronic image products. Images are produced from the best available original document Preface
This thesis is submitted for the Doktor ingenipr degree at the Norwegian University of Science and Technology (NTNU). The research was carried out at the Norwegian Institute of Technology (NTH), Department of Engi neering Cybernetics, in the period from January 1991 to March 1996. The work was funded by The Research Council of Norway (NFR) through the MOBATEL project. Professor Dr.ing. Olav Egeland was my supervisor. During the academic year 1992-1993 I worked at the European Organization for Nuclear Research (CERN). This stay was funded by TOTAL Norge. The work performed during this stay has not been included in this thesis. I am indebted to Professor Egeland for giving me the opportunity to take this study. I am also grateful for advice and comments during the writing of scientific articles and for the introduction to the fields of force control and vibration damping. I am thankful to the administrator of the MOBATEL project, Professor Jens G. Balchen, for letting me take part in this project. I am grateful to Professor Dr.ing. Thor I. Fossen for useful discussions on hydrodynamics. I want to express my gratitude to Dr.ing. Rakel K. Kanestrpm and Dr.ing. Gunleiv Skofteland, for valuable comments and remarks to the draft of this thesis. I am grateful to Prof. Egeland for proofreading the final manuscript.
I gratefully acknowledge
• NFR for my three year scholarship
• TOTAL Norge for one year scholarship in France The MOBATEL board for financial support to attend the 3rd Inter national Conference on Manoeuvring and Control of Marine Craft.
Iffifjd iclyJjxa Ingrid Sctijolberg
Trondheim, Norway Summary
This doctoral thesis describes modeling and control of underwater vehicle- manipulator systems. The thesis also presents a model and a control scheme for a system consisting of a surface vessel connected to an underwater robotic system by means of a slender marine structure. The recursive Newton-Euler scheme for computing the dynamics of robot- manipulators is extended to include the added mass forces, vortex-induced forces, buoyant forces, rotational damping moments and current loads act ing on an underwater manipulator. The dynamics of the underwater ma nipulator is written in a matrix-vector form. The equations of motion of an underwater vehicle are re rieved taking into account the forces acting from the manipulator on the vehicle. The equation of motion of the total system is written in a matrix vector form and structural properties like symmetry, skew-symmetry and positiveness are established. This simplifies the control design and facilitates the use of Lyapunov theory in the stability analysis. The feedback linearization technique is evaluated through a simulation study. Two decoupling schemes are considered, 1) decoupling of the end- effector velocity from the vehicle velocity and 2) decoupling of the end- effector velocity from the system momentum. Passivity-based controllers for set-point control of the vehicle and joint- space control of the manipulator are proposed. The scheme makes it possi ble to achieve manipulator tracking without perfect knowledge of the vehicle dynamics. The controllers can be implemented with different bandwidths, and the vehicle can be used for slow-gross positioning while the manipula tor performs fast joint-space tasks. This provides a large workspace for the system. A control scheme for coordinated motion control of the vehicle and ma nipulator in world coordinates is proposed. In both the proposed control schemes velocities are not measured and 1st order observers are used to reconstruct the velocity signals. IV
Equations of motion for a cable/riser system connected to a surface vessel at the top end and to a robotic system at the bottom end are presented. A control system is proposed for position and velocity control of the robotic system. A control scheme for coordinated position control of the surface vessel and the robotic system is proposed. The total system is shown to be passive. The concept of connecting a thruster unit/robotic system to the bottom end of a slender marine structure can be applied in the operation of connecting a production riser to an underwater well-head. Contents
Preface i
Summary iii
List of Tables ix
List of Figures xii
Nomenclature xiii
1 Introduction 1 1.1 Previous work...... 2 1.1.1 Underwater vehicle-manipulator systems ...... 2 1.1.2 Underwater robotic system connected to a cable/riser system...... 4 1.2 Contributions of the thesis ...... 5 1.3 Outline of the thesis ...... 7
2 Underwater vehicle and manipulator dynamics 9 2.1 Coordinate frames...... 10 2.2 System dynamics ...... 11 2.2.1 Fluid forces on a manipulator link ...... 11 2.2.2 Equations of motion for the manipulator ...... 15 2.2.3 Equations of motion for the combined system .... 17 VI CONTENTS
2.2.4 Equations of motion in the inertial frame...... 19
2.3 System properties ...... 21
2.4 Conclusions ...... 23
3 Control of underwater vehicle and manipulator 25
3.1 Feedback linearization ...... 26
3.1.1 Kinematic equations ...... 27
3.1.2 System momentum ...... 27
3.1.3 Control laws ...... 28
3.1.4 Simulation study...... 30
3.1.5 Discussion ...... 34
3.1.6 Conclusions ...... 34
3.2 Regulation and tracking control ...... 35
3.2.1 Control strategy...... 35
3.2.2 Stability analysis ...... 37
3.2.3 Simulations ...... 40
3.2.4 Discussion ...... 42
3.2.5 Conclusions ...... 45
3.3 Coordinated motion control ...... 46
3.3.1 Control law...... 47
3.3.2 Stability analysis ...... 47
3.3.3 Simulations ...... 49
3.3.4 Discussion ...... 50
3.3.5 Conclusions ...... 51
3.4 Conclusions ...... 51 CONTENTS vii
4 Lateral motion control of an underwater thruster module connected to a slender marine structure 53 4.1 Dynamic models ...... 54 4.1.1 Equation of motion of the thruster system...... 55 4.1.2 Equations of motion of the cable/riser system .... 56 4.1.3 Equations of motion of the total system on matrix- vector form ...... 61 4:2 Control strategy ...... 62 4.2.1 Stability analysis ...... 62 4.2.2 Robustness ...... 65 4.3 Simulations ...... 65 4.3.1 Frequency response ...... 68 4.3.2 Constrained modes...... 69 4.4 Discussion ...... 71 4.5 Conclusions ...... 72
5 Control of an underwater robotic system connected to a slender marine structure in 3 DOF 73 5.1 Dynamic models ...... 74 5.1.1 Coordinate frames...... 74 5.1.2 Dynamics of an underwater robotic system...... 75 5.1.3 Dynamics of the cable/riser system...... 75 5.2 Control strategy...... 80 5.2.1 Stability analysis ...... 80 5.2.2 Discussion ...... 81 5.3 Coordinated position control of the robotic system and sur face vessel...... 81 5.3.1 Stability analysis of the total system with PD-control 83 5.3.2 Discussion ...... 84 5.4 Conclusions ...... 85 viii CONTENTS
6 Conclusions and recommendations 87 6.1 Conclusions ...... 87 6.2 Recommendations for further work ...... 88
References 91
A Vehicle-manipulator simulation model 95
B Numerical data of cable/riser system 97 List of Tables
3.1 Design parameters for controllers (3.14) and (3.18) ...... 31 3.2 Performance and torque indices assuming completely known model...... 32 3.3 Performance and torque indices assuming 20% uncertainty in the model ...... 33 3.4 Performance and torque indices assuming partly unknown model ...... 33 3.5 Performance and torque indices with drift in the vehicle po sition ...... 34
4.1 Performance indices with varying thrust weight...... 68 4.2 Constrained modes for the cable/riser simulation model. . . 71
List of Figures
1.1 Duplus II, vehicle-manipulator system...... 3 1.2 A slender marine structure connecting an underwater thruster module to a surface vessel...... 6
2.1 Coordinate frames attached to vehicle-manipulator system. . 10 2.2 The flow velocity Vj, drag force D, lift force L, angle of attack a and side-slip 0...... 12 2.3 Denavit-Hartenberg frame assignment. C denotes the center of gravity, B denotes the center of buoyancy and P denotes the center of pressure...... 15
3.1 Simulated end-effector position/orientation error vector for controller (3.14)...... 31 3.2 Simulated vehicle position/attitude error vector for controller (3.14)...... 32 3.3 Simulated end-effector position/orientation error vector for controller (3.18) ...... 32 3.4 Simulated vehicle position/attitude error vector for controller (3.18) ...... 33 3.5 Simulated vehicle position error for controllers (3.26) and (3.29) ...... 41 3.6 Simulated joint position error for controllers (3.26) and (3.29). 41 3.7 Simulation results with PD-controller ...... 42 3.8 Vehicle control vector using controllers (3.26) and (3.29). . 42 3.9 Manipulator control vector using (3.26) and (3.29) ...... 43 XII LIST OF FIGURES
3.10 Norm of the compensation vector fm...... 43 3.11 Vehicle position error vector without compensation of ma nipulator motion ...... 44 3.12 Joint position error vector with unknown payload mass and without compensation for hydrodynamic damping ...... 44 3.13 Simulation results with noisy position measurements with S/N=l%...... 45 3.14 Simulation results with noisy position measurements with S/N=10%...... 46 3.15 Vehicle reference and position error vector for controller (3.57). 50 3.16 Manipulator reference and position error vector for controller (3.57)...... 50
4.1 An underwater thrust unit suspended to a surface vessel by a cable/riser ...... 54 4.2 Block diagram of the closed loop system...... 63 4.3 Position and velocity errors with PID-controller and fi = 0. 66 4.4 Control input UfX with fi = 0...... 67 4.5 Cable/riser vibration with fi ^ 0...... 67 4.6 Cable/riser force fCx acting on the thruster system...... 67 4.7 Simulation results for system without a thrust unit connected to the cable/riser bottom end-point ...... 68 4.8 Simulation results under noisy position and velocity measure ments, S/N=l%...... 69 4.9 Simulation results under noisy position and velocity measure ments, S/N=10%...... 70 4.10 Transfer function from desired reference to the measured end point position for the linearized model (4-85) ...... 70
5.1 Coordinate frame assignment to the vessel, robotic system and cable/riser system...... 74 Nomenclature
All symbols used in specific contexts are explained when first introduced. Bold types are used exclusively to denote vectors and matrices. Bold up percase denotes matrices and lowercase denotes vectors.
Vehicle-manipulator system a angle of attack (3 angle of side-slip £ system velocity vector 4 system position and orientation vector 77 vehicle position and orientation vector r/0 estimated vehicle velocity vector 77 j desired vehicle position and orientation vector p fluid density v linear and angular velocity of the vehicle in frame B r system control vector re system control vector re(q, 77) in frame I rm manipulator control vector Tr vehicle control vector Aij added mass coefficient b buoyant force Cd drag coefficient Cl lift coefficient C system Coriolis and centrifugal matrix C(q,C) Ce system Coriolis and centrifugal matrix Ce(q, 77, £) in frame I Cm manipulator Coriolis and centrifugal matrix Cm(q, q) Cr vehicle Coriolis and centrifugal matrix Cr(v) D value of the drag force D system damping matrix D(q, C) xiv N omenclature
De system damping matrix De(q, tj, in frame I Dm manipulator damping matrix Dm(q,q) Dt vehicle damping matrix Dr{v) fm forces and moments on the vehicle due to manipulator motion
Fd drag and lift force in link frame Fs linear damping force g acceleration of gravity g system gravity and buoyant vector g(q,r}) g e system gravity and buoyant vector g e(q,rj) in frame I g m manipulator gravity and buoyant vector g m{q) g r vehicle gravity and buoyant vector g T(tj) h angular momentum J system transformation matrix J(q,Tj) Jv vehicle transformation matrix from B-frame to I-frame Jv(v) L value of the lift force M system mass matrix M(q) Me system mass matrix Me(q, tj) in frame I Mm manipulator mass matrix Mm(q) Mr vehicle mass matrix p linear momentum Pqi momentum Jacobian P02 momentum Jacobian q estimated joint velocity vector q generalized manipulator joint coordinates qd desired joint coordinate vector -Ry transformation matrix from link frame k to flow frame f Trd rotational damping moments Ta current loads Vf flow velocity vr one-dimensional relative velocity vr relative velocity °®e end-effector position and orientation vector in frame 0 xxe end-effector position and orientation vector in frame I xq estimated system position and orientation vector
Newton-Euler scheme notation kc angular acceleration of link k Tfc torque on link k XV
Wfc angular velocity of link k ka k linear acceleration of link k bk buoyant force the vector from joint A; + 1 to joint k k+1d,k/kc the vector from joint k to the center of mass of link k k+1d •k/kpt the vector from frame k to the center of pressure of link k k+1d,k/kfy the vector from frame k to the center of buoyancy of link k fk force interaction between link k — 1 and link k Fk forces acting at the center of mass of link k 9k gravity vector Ik total inertia of link k Im inertia of link k mk dry mass of link k Mk total mass of link k rotation matrix from frame k+1 to frame k tk moment interaction between link k — 1 and link k Tk moments acting at the center of mass of link k kvk linear velocity of link k kVkc linear velocity of the center of mass of link k
Cable/riser notation ipi(z) longitudinal normal function 4>i{t) longitudinal vibration mode out-of-plane normal function 0i(t) out-of-plane vibration mode u)i lateral vibration frequency uiXi, uiyi, wZivibration frequencies in x, y and z-plane c structural damping El stiffness EI(z) EA axial stiffness fxs lateral fluid forces fx, fy, fz fluid forces in x, y, z direction g x{z) static deflection in x direction at top end-point of cable/riser hx{z) static deflection in x direction at bottom end-point of cable/riser mc mass per unit length Pi{z) lateral normal function XVI Nomenclature
%(<) lateral vibration mode Te effective axial tension Te(z) T mean value tension U (z, t) longitudinal displacement v(z,t) in-plane displacement w (z,t) out-of-plane displacement z displacement along the cable/riser axis
Robotic system attached to the cable/riser
Vx lateral thruster system position Vxd desired lateral position Vi position of robotic system in x, y and z-plane Tv vector of control forces Cv 3x3 matrix of Coriolis and centrifugal terms Cv(rj,rj) Dv 3x3 matrix of damping terms Dv(r},r]) .fCx forces acting from the cable/riser on the thruster system fc forces acting from cable/riser on the robotic system 9v gravity and buoyancy vector g v(r)) mt mass of thruster system Mv 3x3 mass matrix Mv(rj) u tx thruster force in lateral direction
Vessel connected to the cable/riser rs vector of control forces u>n most dominating wave frequency Cs 3x3 matrix of Coriolis and centrifugal terms Cs(r, r) Ds 3x3 matrix of damping terms Ds(r, r) fi forces due to vessel motion, wave- and current loads Ms 3x3 mass matrix Ms(r) rx lateral vessel motion r vessel position vector in frame I Uc lateral current velocity Ucj lateral current velocity at depth z = / w(z, t) lateral wave velocity Chapter 1
Introduction
The main subject of this thesis is modeling and automatic control of under water vehicle and manipulator systems. In the offshore industry the use of remotely operated and semi-autonomous/autonomous underwater robotic systems is essential in installation, inspection and maintenance work. The term remotely operated is used on tethered systems, semi-autonomous is used for system which have some communication with a human operator, while the term autonomous is used for systems which have no communi cation with a human operator. Tethered systems have the disadvantage that the cable may get tangled with the sea floor or an underwater con struction. Also, fluid forces induce oscillations and vibrations in the cable which result in wear in the system. However, tethered systems possess high information rate and unlimited power supply. Tether-less systems op erated through an acoustic link (semi-autonomous) are restricted due to limited bandwidth in the communication link, but employ high maneuver ability. A dynamic model of the system and its environment is useful due to the narrow communication link. In contrast to tethered systems, au tonomous systems can operate at larger depths and the operation time is only restricted by the capacity of the local power supply system. However, autonomous systems lack the human supervisory control which may be nec essary in complex underwater operations. The use of remotely operated or semi-autonomous /autonomous robotic systems depends on the task, and on task requirements like maneuverability, working capacity, information rate and operation security. The work presented in this thesis is part of the MOBATEL project. The project is a research program at The Norwegian Institute of Technology (NTH) sponsored by The Research Council of Norway (NFR) and several 2 Introduction
Norwegian companies. The project is administrated by Professor Balchen at Department of Engineering Cybernetics, NTNU. MOBATEL stands for
• MOdel BAsed TELeoperation of an underwater vehicle over a narrow communication link.
• Teleoperated and sensor controlled assembly and machining with a manipulator mounted on a controlled underwater vehicle.
The system studied in this project consist of a remotely operated under water vehicle with manipulator arms communicating with an operator sit uated far away, over a narrow ultrasonic communication link. Data from sensors and cameras located on the vehicle are transfered to the operator. An ultrasonic communication link has very limited information rate, there fore the operator controls the vehicle and manipulator using mathematical models of the system and the environment. Sensor information and camera images are used to update the mathematical models. Detailed information on the project can be found in Balchen & Yin (1994). In this thesis some important control aspects related to vehicle-manipulator motion control are investigated.
1.1 Previous work
Dynamics models of underwater robotic systems are required in advanced control design. Models of underwater vehicles have been presented by sev eral authors, see for instance Fossen (1994) and Yuh (1990). Triantafyl- lou & Hover (1989) describes the equation of motion of a tethered un derwater vehicle. Also dynamic models of robot manipulators are well- established, (Spong k Vidyasagar 1989). Dynamic models of underwater vehicle-manipulator systems are extensions of these models. Underwater robotic systems are highly nonlinear and are affected by fluid forces. This results in parametric and structural uncertainties in the dynamic model. The use of robust control design techniques is therefore advantageous.
1.1.1 Underwater vehicle-manipulator systems
Possible working modes for autonomous underwater vehicle-manipulator systems where advanced control would be beneficial are: 1.1. Previous work 3
Figure 1.1: Duplus II, vehicle-manipulator system.
• The vehicle is docked on an ocean pipeline and the manipulator per forms maintenance work. This mode requires tracking control of the manipulator while the local vehicle control system can be turned off. • Keeping the vehicle at a fixed position while the manipulator performs a service task. This mode calls for set-point control of the vehicle and tracking control of the manipulator.
• The vehicle follows an ocean pipeline during which the manipulator performs inspection around the pipe. This mode requires coordinated motion control of the vehicle and manipulator in world coordinates.
A manipulator submerged in fluid is affected by fluid forces. Janocha & Papadimitriou (1991) analyzed the motion of an underwater robot arm by simulation and compared the results with an arm driven above water. The conclusions were that the dynamic behavior of the robot in water is quite different from that in atmosphere. A higher torque is required under water and it was also found that under water the coupling between the axes is higher. Several authors have studied the dynamics of underwater manipula tor systems. Ioi & Itoh (1990) formulated a recursive Newton-Euler scheme for an underwater manipulator with a fixed base, including vortex shedding forces. Levesque & Richard (1994) presented an approximate method for computing the buoyancy and the drag force of the links of an underwater 4 Introduction
manipulator. The drag force was computed by numerical integration of the local drag. McMillan, Orin & McGhee (1994) developed a simulation algo rithm for an underwater vehicle-manipulator system, including fluid forces and the effect of a moving base. Mahesh, Yuh & Lakshmi (1991) derived the equations of motion for an underwater vehicle-manipulator system us ing the vector dyadic form (Frisch 1974). The matrix-vector formulation presented in this thesis, facilitates the study of structural properties and simplifies the use MIMO nonlinear controllers and Lyapunov theory in the stability analysis. Underwater vehicle-manipulator system dynamics are strongly nonlinear due to fluid forces acting on the system. Automatic controllers can be de signed for nonlinear mechanical systems using the feedback linearization technique, or by utilizing nonlinear control techniques. Sensor systems are an important part of the control system. Vehicle navigation sensors are transponder systems, sonar systems, camera images and laser systems (Yuh 1995). Sensors attached to the manipulator end-effector can be visual sensors, force sensors or tactile sensors. At depths greater than 400-500 me ters wave loads on the system may be neglected. However current loads are significant, and must be compensated for by the controller. It is reason able to assume that the sea current varies slowly with time and is therefore a static disturbance which can be compensated for by including integral action in the controller. Mahesh et al. (1991) suggests an adaptive control-strategy for coordinated control of the vehicle and manipulator, based on discrete-time approxi mation of the dynamic model. Fjellstad (1994) and Paulsen, Egeland & Fossen (1994) presented observer based controllers for underwater vehicles. Lizarralde, Wen & Hsu (1995) presented a control scheme for coordinated control of a sub sea vehicle-manipulator system using quaternion represen tation and a velocity observer. Yoerger, H.Schempt & D.M.DiPietro (1991) presents sea-test of a tethered vehicle/manipulator system with PD-control of the manipulator end-effector. Control systems designed for underwater vehicle control and robot manipulator control can be extended and applied to underwater vehicle-manipulator systems, employing the matrix-vector formulation of the system dynamics.
1.1.2 Underwater robotic system connected to a cable/riser system
The thesis includes a study of a system consisting of a cable, production riser, conductor or an other similar slender marine structure connected to 1.2. Contributions of the thesis 5
a surface vessel at the top end and to a mass module or to a robotic system at the bottom end. The robot system may be a tool system equipped with thrusters or a simple thruster system. The function of the robot system may be several, one is to perform repair on underwater installations. In this case the marine structure is a cable providing power and control signals for the tool system. Another function may be to connect the riser to a well head, in this case the robot system will consist of a thruster unit. Slender marine structures undergo deformations induced by the motion of the surface vessel, wave and current forces. These deformations lead to reduced performance of the robot system and this results in the need of a robust and high performance controller for the robot system. The usual way of connecting a riser (attached to a floating production vessel) to a well-head is by controlling the surface vessel. However, wave and current loads induce vibrations and oscillations in the riser and this makes the connection operation a time demanding one. In this thesis a solution to this problem is suggested: The proposed solution is to attach position and velocity controlled thrusters to the bottom end of the riser, this results in a more time- and cost efficient operation. To develop a controller and prove stability a mathematical model is needed. Mathematical models for underwater cables and risers have been presented by several authors, see for instance Triantafyllou (1990), Chakrabarti k Frampton (1982), Blevins (1990), Sarpkaya k Isaacson (1981) and Kam- man k Huston (1985). Triantafyllou (1990) describes the equations for dynamic tension and dynamic deflection angle. Chakrabarti & Frampton (1982) gives a review of the research on slender vertical members undergo ing deformations. In Chakrabarti k Frampton (1982), Etok k Kirk (1981) and Daring k Huang (1979) the model of the marine riser is a vertical beam in tension pinned at the bottom and top end. Modal analysis is used for analyzing the riser response. Weaver, Timoshenko k Young (1990) presents beams subjected to support motion and beams on elastic foundations. The cable/riser model needs to be combined with the equations of motion of an underwater thruster unit. This has not previously been presented in the literature.
1.2 Contributions of the thesis
The main contributions of the thesis are: • Recursive Newton-Euler scheme for an underwater manip ulator 6 Introduction
Figure 1.2: A slender marine structure connecting an underwater thruster module to a surface vessel.
A recursive Newton-Euler scheme is derived for underwater manipu lators. The standard Newton-Euler scheme for manipulators is mod ified to include added mass, buoyant forces, drag forces, vortex in duced forces, rotational damping moments and current loads. This scheme is presented in Section 2.2.2 and is based on Schjplberg & Fossen (1994) and Schjplberg & Egeland (1996c)
• Matrix-vector formulation of underwater vehicle and manip ulator dynamics The equations of motion of underwater vehicle-manipulator systems are formulated in a matrix-vector form. Structural properties of the equations of motion are investigated and properties like symmetry, skew-symmetry and positiveness are established. This formulation is described in Section 2.2.3 and is based on Schjplberg & Fossen (1994) and Schjqlberg & Egeland (1996c). 1.3. Outline of the thesis 7
• Controllers for regulation and tracking control of the vehicle and manipulator Controllers for set-point regulation of the vehicle and motion control of the manipulator are proposed. The vehicle and manipulator joint velocities are not measured and two observers are used to reconstruct these signals. Moreover, a tracking controller for coordinated control of the vehicle and manipulator is proposed. These controllers are presented in Section 3.2 and in Section 3.3. The presentation is partly based on Schjplberg & Egeland (1996c). • Models of a cable/riser system with a robotic system at tached to the bottom end Equations of motion for a cable/riser system connected to a surface vessel at the top end and to a thruster unit at the bottom end are presented. In Section 4.1 the equations of motion in the lateral di rection are described and the total system dynamics is written in a matrix-vector form. In Section 5.1 the equations of motion in 3 de grees of freedom are described. The presentation is partly based on Schjplberg k Egeland (1996a) and Schjplberg k Egeland (1996b). • Stability analysis of the total system consisting of cable/riser system connected to a surface vessel and a robotic system PID-controllers are proposed used for position and velocity control of the bottom end-point of the cable/riser system. Lyapunov stability for the total system is proved and good performance and robustness properties are established. A control scheme for coordinated position control of the vessel and robotic system is proposed. The results are presented in Section 4.2 and in Section 5.2. The presentation is partly based on Schjplberg k Egeland (1996a) and Schjplberg k Egeland (1996b).
1.3 Outline of the thesis
The chapters of the thesis are organized as follows: • Chapter 2: The equations of motion of the underwater vehicle and manipulator are described. The system kinematics and system prop erties are presented. • Chapter 3: The feedback linearization technique is applied to the underwater vehicle-manipulator system and evaluated through a sim ulation study. Passivity-based controllers for vehicle and manipulator 8 Introduction
control are presented. Stability of the closed loop system is proved and simulation results are presented.
• Chapter 4: The equation of motion for lateral motion of a cable/riser system connected to a surface vessel at the top end and to a thruster unit at the bottom end is described. Stability analysis and simulations are rresented.
• Chapter 5: The equations of motion in 3 degrees of freedom of the cable/riser, surface vessel and robotic system are given. Stability analysis of the total system with PD-controllers is presented.
• Chapter 6: Conclusion and recommendations for further work are given in this chapter. Chapter 2
Underwater vehicle and manipulator dynamics
For the purpose of designing nonlinear controllers for underwater vehicle- manipulator systems a dynamic model of the system is derived. Sarpkaya & Isaacson (1981), Olson (1980) and Faltinsen (1990) identified important fluid forces acting on rigid bodies at great depth, that is, at a depth where free surface effects are negligible. In this chapter these forces are presented and included in the recursive Newton-Euler algorithm to de rive the equations of motion of an underwater manipulator with cylindrical links. The hydrodynamic forces considered are added inertia forces, buoy ant forces, friction forces, rotational damping moments and current loads. The equations of motion of an underwater vehicle written in matrix vector form was presented by Fossen (1991). In this chapter the equations of motion for the underwater manipulator are combined with the equations describing the motion of an 6 degrees of freedom (DOF) underwater vehicle. The dynamic model of the total system is written in matrix-vector form. Emphasis is put on representing the nonlinear model such that mechanical system properties like symmetry, skew-symmetry and positiveness can be used in the control design. The matrix-vector representation facilitates the study of structural properties and simplifies the control design for MIMO nonlinear systems. Moreover, this gives a good starting point for using Lyapunov theory in the stability analysis. The chapter is organized as follows: In Section 2.1 a short description of the coordinate frame assignment is given. The system dynamics is derived 10 Underwater vehicle and manipulator dynamics
in Section 2.2 and the system properties are presented in Section 2.3. Sec tion 2.4 holds the concluding remarks.
The presentation is based on Schjolberg & Fossen (1994) and Schjolberg & Egeland (1996c).
2.1 Coordinate frames
The position and attitude of the vehicle system is described in 1R3 by 6 independent coordinates, 3 for the position and 3 Euler angles for repre sentation of the attitude. Alternatively, the attitude can be represented by quaternions (Fjellstad 1994). A coordinated frame is fixed in the vehicle body, and is denoted as frame B (body-fixed frame). A similar frame is fixed in space, and is denoted as frame I (inertial-fixed frame). The manip ulator kinematics are described by N generalized coordinates, where N is the number of manipulator joints. The manipulator end-effector position and orientation vector can be found by applying the manipulator forward kinematics. A frame is attached to each manipulator link according to the Denavit-Hartenberg convention. The base frame is denoted frame 0 and the frame fixed on link N is denoted frame N.
Link N I; Frame N
Link 1 Manipulator
Frame 0 Frame B
A Frame I
Vehicle
Figure 2.1: Coordinate frames attached to vehicle-manipulator system. 2.2. System dynamics 11
2.2 System dynamics
A formulation of the equations of motion of an underwater vehicle car rying a manipulator is presented in this section. The dynamic model of the manipulator can be computed recursively by computing the forces and moments acting at the center of gravity of each link, taking into account external fluid forces, the interaction forces between the links and the veloc ity of the vehicle. The reaction forces from the manipulator on the vehicle are computed from the recursive Newton-Euler algorithm. The equations of motion of the total system are derived by combining the equations of motion of the vehicle and manipulator.
2.2.1 Fluid forces on a manipulator link
A forced harmonic motion of a rigid body in fluid gives rise to pressure- induced forces and moments proportional to the rigid-body acceleration. These forces are denoted added mass and added moment of inertia. Fluid friction forces acting on homogeneous elements at depth are forces acting at the center of pressure of the manipulator link. These forces are the drag forces and act in the direction of the flow velocity Vf. Vortex shedding results in oscillatory forces in the drag direction and in the direction or thogonal to the drag direction (lift forces). A flow-axis system (Xf,yf,Zf) is defined such that the flow velocity is along the x-axis (x/) of the frame. This is illustrated in Figure 2.2. In the flow-axis system the drag force —D is along the x-axis and the lift force —L is along the y-axis. The z-axis (zf) is placed such that a right-hand system is formed. The transforma tion matrix between the flow-axis system and the link frame is found by introducing two orientation angles, the angle of attach a and the angle of side-slip /3. The angles are defined by performing a plane rotation a about the y^-axis and a rotation /3 along the new z^-axis. This results in a transformation matrix Rf between link frame k and the flow-axis frame f
R) = Rz(-(3)Ry(a) (2.1) cos(/3)cos(a) sin(/3) cos(/3)sin(a) -sin(/?)cos(a) cos(/3) -sin(/3)sin(a) (2.2) —sin(a) 0 The drag and lift force can thereby be expressed in the link frame ' -D' (2.3) 12 Underwater vehicle and manipulator dynamics
Link k
Figure 2.2: The flow velocity Vf, drag force D, lift force L, angle of attack a and side-slip (3.
Other fluid effects are the rotational damping, the buoyant forces and cur rent loads. The fluid damping moments denoted Tra are functions of the angular velocities of the manipulator link, of the angle of attack and of the angle of side-slip. It is reasonable to approximate that the damping moments as linear in the angular velocities of the manipulator link. The buoyant force & is a restoring force due to displaced water and attacks in the gravity center of the displaced water, the center of buoyancy. Current loads denoted Td exerted on the manipulator link are functions of the relative velocity of the link.
Manipulators are often built up of rectangular or cylindrical links. In the following the fluid forces on manipulators with cylindrical links are dis cussed, it is assumed that the manipulator is operating in ideal fluid.
• Added mass and added moment of inertia: The added mass loads can be expressed by Aa where A is the coefficient matrix and a is acceleration of the rigid body. The cross terms in the coefficient matrix are small due to symmetry of the cylinder and are therefore neglected. The coefficients can be derived for an arbitrary link ge ometry using strip theory. For a cylindrical link the coefficients are 2.2. System dynamics 13
found to be An = 0.1 m A22 = 7rpr2l A33 = wpr2l A44 = 0 A55 = jjTrpr 2/3 Aee = ^7rpr 2/3 where p is the fluid density, r is the radius of a link, d the diameter, l the length, and m is the dry mass of a link. The added mass can be included in total mass matrix Mk of the link
Mfc = diag(Aii,A22,A33) + ml
where I is the identity matrix. The added moment of inertia result in an increase in the total moment of inertia Ik of the link
Ik = diag(A44, A55, Aee) + Im
where Im is the inertia of the link. • Other forces and moments: The lift and drag forces are approx imated as pressure and viscous shear forces and also forces due to vortex shedding. The drag force can be approximated by a nonlinear expansion
F d = Dsvr + Br|vr|vr + Dg vl + 0{vl) (2.4)
where Ds, Dr and Dq are friction coefficients and vr is a one di mensional relative velocity between the fluid velocity and rigid body velocity. This formulation covers all linear and nonlinear fluid Motion forces, however the 3rd term may be neglected due to symmetry of the link whereas higher order terms are small compared to the 2nd order term in (2.4). The linear term includes linear skin friction forces whereas the 2nd order term includes drag and lift forces. In the three dimensional system the linear term in (2.4) can be written in a link-fixed frame
Fs = Dsvr (2.5)
where Ds is a 3 x 3 diagonal matrix of linear friction coefficients. The relative velocity between the three dimensional rigid body velocity and the one dimensional flow velocity is given by v r = v - VfR{xf = [vrz,v ry,v rz]T (2.6) 14 Underwater vehicle and manipulator dynamics
where v is the link translational velocity, Vf is the flow velocity and x/ is the unit vector along the z/-axis. The quadratic term in (2.4) has been approximated as the average value of the drag (Faltinsen 1990)
D = ipCx>( Rn, KC, a)A(a)|vr|v r
where Cx>(Rn, a) is the drag coefficient and A(a ) is the projected frontal area of the rigid body, a is the angle of attack, Rn is the Reynold number and KC is the Keulegan-Carpenter number. In the control model it is reasonable to assume KC independence. Vortex shedding results in forces in the drag direction, however these forces are small compared to the term Dr|vr|vr and are therefore neglected (Faltinsen 1990). The magnitude of the lift force L has been approximated as (Faltinsen 1990) L = ^pCi,(Rn,KC,a)A(o;)cos(27r/ T,t + 7)
where Cx(Rn,KC, a) is the lift coefficient and fv is the vortex shed ding frequency given by fv = where St is the Strouhal number. The phase angle 7 varies strongly along the longitudinal cylinder-axis. The drag and lift forces Fa = [F^, F Fdy = ^pAv(CDy + CLj/cos(2nf v + 7))|v rJvry (2.8) Fd, = |pAz{CD, + CL;COs(2wfv + 7))lvr« |vr, (2.9) where the subscripts z, y and z indicates x-, y-, and z-component in the link frame. The angular velocity of the manipulator changes the flow field around it and this results in an increase in the manipulator moments. This ef fect is modeled as linear rotational damping Trd = [Trd x, Trd y, Trdz]T Trdx ~ ^/?llv r||^xC'zp(Qf, /?, Rn)a?3; (2.10) Frdy = J/?|| v r ^dyCmqiot") Rn)(Vy (2.11) Trd z ~ ^/>l!v rl!dzC7nr(0!?/3, Rn)u?z (2.12) 2.2. System dynamics 15 where Qp, Cmq and Cnr are dimension-less rotation derivatives, and d x, d y and d z are characteristic lengths. The vector w = [a)r, tvz]T is the angular velocity of the manipulator link. • Current loads: The current loads are given as (Faltinsen 1990) Td = S(vr)[diag(An ,A22,A33)vr] (2.13) where S is a skew-symmetric matrix operator defined such that a x b = S(a)b. This effect is only relevant for rigid bodies with non circular cross sections. Important fluid friction forces and moments acting on the manipula tor link are in the following denoted nf = Fd + Fs (2.14) n t = Trd + Tcl - (2.15) • Buoyancy: The buoyant force is given by b = pgV where V is the displacement volume and g is the acceleration of gravity. 2.2.2 Equations of motion for the manipulator The manipulator considered is an N-link manipulator with N revolute joints. A coordinate frame is attached to each joint according to the Denavit-Hartenberg convention, see Figure 2.3. The linear velocity and Joint k Joint k -f 1 Figure 2.3: Denavit-Hartenberg frame assignment. C denotes the center of gravity, B denotes the center of buoyancy and P denotes the center of pressure. angular velocity of a link are calculated recursively as in the recursive Newton-Euler scheme (Spong & Vidyasagar 1989). 16 Underwater vehicle and manipulator dynamics Outward iterations: k: 0 —»JV : k+lUk+l = Rk+i(ku k +zqk+1) (2.16) k+1oc k+1 = Rkk+1(ka k + ku k X zqk+i + zqk+1) (2.17) k+1vk+i = iZ|+1 kvk + k+1u>k+ 1 x k+1d k+i/k (2.18) k+lvk+lc = Rkk+1 kvk + fc+1(v* 4.1 x k+1d k/kc (2.19) k+1a k+1 = Bj+1 ka k + k+1a k+1 x k+1d k+1/k + fc+1a?fc+i x (k+1a>fc+i x k+1d k+i/k) (2.20) where koc k angular acceleration of link k kUk angular velocity of link k ka k linear acceleration of link k k+1d k+i/fc the vector from joint k + 1 to joint k the vector from joint k to the center of mass of link k 9k gravity vector mk mass of link k 9k generalized joint coordinate Rkk+1 is the transformation matrix from frame k to frame k+1 fcVfc linear velocity of link k kVkc linear velocity of the center of mass of link k z unit vector along the z-axis Including the fluid forces (2.14) and (2.15) in the recursive Newton-Euler algorithm, gives the following equations for the force fk and moment tk interaction between link k — 1 and link k written in frame k: fk = J2fc+1 k+1fk+i + Rk~ mkg k + bk + n fk (2.21) tk = Rk+lk+1tk+i + d k/k+l x (Rkk+lk+1fk+1) +d k/kc x Fk+Tk + d k/kc x (-mkg k) + d k/kp x n/ t +d k/kb x bk + n tk (2.22) where the vector rifk is the linear and quadratic hydrodynamic damping forces. The vector d k/kp is the vector from frame k to the center of pressure of joint k, d kjkb is the vector from frame k to the center of buoyancy of link k, bk is a vector of buoyant forces and n Pk is the vector of hydrodynamic damping moments. The vector Fk are the forces acting at the center of mass Fk = Mk(ka k + ka k x kd k/kc + ku k x (kvk x kd k/kc)) (2.23) 2.2. System dynamics 17 where Mk is the total mass of link k. The vector T& are the moments acting at the center of mass of link k Tk = Ik kcck + ku, k x (Jfc ku k) (2.24) where Ifc is the total inertia of link k. The joint torques for the manipulator are given by Tfc = zTtk + u k (2.25) where z is the unit vector along the %-axis and u k is the torque increase due to the flow velocity. The recursive equations are linear in the parameters and linear in the generalized joint coordinates, therefore the equations of motion for the underwater manipulator can be written in the form Mm{q)q + Cm(q, q)q + Dm(q, q)q + g m(q) = rm (2.26) where q € R^ is the vector of generalized coordinates and rm € R^ is the control input vector and the scalar N is equal to the number of manipulator joints. The matrix Mm 6 JRNxN is the matrix of inertia and added inertia, Cm € TRNxN is the matrix of Coriolis and centrifugal terms, Dm 6 HNxN is the matrix of hydrodynamic damping terms and g m € is vector of gravity and buoyant forces. 2.2.3 Equations of motion for the combined system The Newton equations of motion for a rigid body can be written Mrbu + Crb{v )v = trb (2.27) According to Fossen (1994), the added inertia forces, the added Coriolis and centrifugal forces, the damping forces and buoyant and gravity forces on a rigid body in fluid can be expressed —M\i> - Ca {v )v - D{v)v - g(r)) = th (2.28) Hence the equations of motion of an underwater vehicle carrying a manip ulator can be written in the abbreviated form: Mru + Cr(v)u + Dr(v)v + g r(ij) + fm = rr (2.29) where Mr = Mrb + Ma , Cr = Crb + Ca , -Dr = D, g T = g and v € Rn and rj € Rm. The vector rr = trb — tr € Rn is the vector of control 18 Underwater vehicle and manipulator dynamics forces and moments. The scalars n and m indicates DOF in the body and inertial-fixed frame. According to the SNAME notation v = [vj, I/£]T = [-It, v, w, p, q, r]T (2.30) is the vector of linear and angular velocity of the vehicle in the body-fixed frame, and v = fai, f?i]T = k y, z, <£, o, V']T (2.31) is the vector of position and attitude of the vehicle in the inertial frame. The attitude vector will be represented in Euler angles in the following. The vector fm € !Rn are forces and moments on the vehicle due to the manipulator motion. The matrix Mr € Rnxn is the inertia and added inertia matrix, Cr-(v) € lRnxn is the matrix of Coriolis and centrifugal terms, Dr(u) € Rnxn is the hydrodynamic lift and damping matrix and g r(rj) e Rn is the vector of gravity and buoyant forces. The kinematic transformation between frame I and frame B is given by Fossen (1994) J v\ {Vi) o v — J v iyi)v — (2.32) 0 Jv2 (Vt) where ciftcff —sipc The forces fi and moments ti applied by the manipulator on its base are given by (2.21) and (2.22). Locating the body-fixed reference frame of the vehicle in the manipulator base, the force and torque from the manipulator on the vehicle are given by /m(9> 9,9, v) = [(ilo/i)T, (Ro*i)T1T (2-35) The vector fm can be expressed in the form fm = H(q)v + Ci (q,q, u)v + Di (q, q, v)v + Mc{q)q + C2{q, q)q + D2(q, q, v)q + g E{q) (2.36) 2.2. System dynamics 19 The matrix Mc — contains the terms due to the reaction forces be tween the vehicle and the manipulator, H is the increased inertia in the vehicle inertia due to the manipulator. The matrix H satisfies the property H = £fT > 0 due to positive kinetic energy. The matrices C\ and Cg are the changes in vehicle Coriolis and centrifugal terms due to the manipu lator movement. The matrices D\ and £>2 are the increased damping in the vehicle dynamics due to the manipulator motion. The vector g E is the increased vehicle gravity due to the manipulator. By inserting expression (2.36) for fm in (2.29) and setting the initial lin ear and angular velocity of the manipulator equal to the vehicle velocity [t>o tv Mr + H(q) Mc(q) M(q) (2.38) Mj(q) Mm(q) Cr(v) + Ci(q,q,v) C2(q,q) (2.39) C(q, C) Cz{q,q,v) Cm(q,q) Dr(v) + Di(q,q,v) D2(q,q, v) (2.40) D(q, 0 D3(q,q,v>) D±{q,q,v) 9 An) +9 e (q) g(q,v) (2.41) 9m(n) T [T^,TT]T (2.42) The matrix Mm is the added inertia and inertia matrix and Cm are the Coriolis and centrifugal terms of the manipulator. The matrix C3 describes the effect of the vehicle motion on the manipulator Coriolis and centrifugal terms. The matrices Dz and £>4 describe the damping on the manipulator due to joint motion and vehicle movement. The total system has n + N degrees of freedom (DOF). 2.2.4 Equations of motion in the inertial frame The configuration of vehicle and manipulator can be specified in frame I by the n + 6 dimensional vector £ = in T, 'a#? (2.43) 20 Underwater vehicle and manipulator dynamics where Ixe is the manipulator position/orientation vector in frame I. The equations of motion for the total system (2.37) can be written in frame I by applying the velocity transformation t = JC (2.44) where the transformation matrix J is to be specified below. To find the expression for J it is noted that t} = Jv (t])u (2.45) lxe = T^Xe (2.46) where °xe is the manipulator end-effector position/orientation vector in the base-frame 0 and Tj is given by To r°i o T/ r 0 Rj (2.47) where Rj is the rotation matrix from frame 0 to frame I. Moreover, the linear and angular velocity of manipulator end-effector in the base-frame can be written o °VN Xe °U)JV = JiV + J2q (2.48) where Isx3 Ti Ti/^i ... r^/3jv J 2 = (2.49) o /sx3 ... and r, = -jrsatii^.d (2.50) 3=i A = ar's (2.51) The matrix S is a skew-symmetric matrix operator. The vector dj/j_i is the length of manipulator arm j and z is the unit vector along the z-axis. Combining (2.43-2.51) the transformation matrix J is found to be Jv(v) 0 (2.52) J(q,il) = T°jJi(v) Tpiiq) 2.3. System properties 21 Consequently the equations of motion for the total system in frame I can be written in the form Me{q, ff)l + Ce(q, v,0k + De(q, V, <)£ + 9 e{q, v) = Te(9 , v) (2.53) where £ = [vT, 7a:^]r and Me(q,rj) = J~T(q, rj)M(q)J~l(q, q) (2.54) Ce(q,vX) = J~T {q>ri)C{q,C)J~l{q,v) + J~TM(q)j 1(q,n) (2.55) De(q,V,() = J~T(q,v)D(q,C)J~1(q,v) (2.56) 9e(g,?7) = J~T(q,v)g{q,v) (2.57) re{q,v) = J~T(q,v)T (2.58) 2.3 System properties The added mass matrix M^oi the vehicle is positive definite under the assumption that the added mass coefficients are frequency independent and that the system is moving in an ideal fluid (Fossen 1994). Thereby it is reasonable to assume that Mr — Mj > 0 for underwater vehicles operating outside the wave-affected zone. The inertia matrix for the total system is positive definite M(q) = MT(q) > 0 due to positive kinetic energy, whereas symmetry is guaranteed by applying Newtons ’ third law (action-reaction principle). For rigid bodies moving through an ideal fluid the centrifugal matrix CT of the vehicle can be parameterized such that it satisfies the skew-symmetric property (Fossen 1994). The change in the system energy can be expressed in terms of the change in linear and angular momentum p and h (Hughes 1986) N T Pk = CT (t - &(q, C)C - g(q, v)) (2.59) E hk where = [rj, a>J]T denotes the linear and angular velocity of body k, k = 0 indicates the vehicle rigid body. The change in the linear and angular momentum may be written (Newton-Euler formulation) (Hughes 1986) Pk = Mfcifc + Wfc(C)®fc (2.60) hk 22 Underwater vehicle and manipulator dynamics where Wk is skew-symmetric (Fossen 1994). The vector xk can be expressed xk = Jk(q) C (2.61) where Jo = [Iex6 Onxn) and j = [r0 rx rlZ ... rNz (2.62) k [ 0 7o 7i - In where for 0 < i < k To = R°k (2.63) 7< = K1* (2.64) (2.65) and for i > k, 7j = 0 and T, = 0. By substituting (2.60) and (2.61) into (2.59) it can be seen that the system mass matrix can be written N M{q) = y£JkMkJk (2.66) fc=0 and the system Coriolis and centrifugal matrix can be expressed N C(q, C) = £(jfcMfcJfc - JlWkJk) (2.67) k=0 The C matrix has the property that M(9)-2C(q,C) (2.68) is skew-symmetric. The dynamic model can be written so that the damping matrix D(qr, () consists of terms due to dissipative forces. The matrix will then satisfy D > 0. The matrix Me in (2.53) is symmetric since M = 1WT. The matrix Me is positive definite due to positive kinetic energy. The matrix Me{q,rj)-2Ce{q,V,0 (2.69) 2.4. Conclusions 23 satisfies the skew-symmetric property (Me - 2Ce)T = J~TM j 1 - jTMJ~l - 2C)T J-1 (2.70) -(Me - 2Ce) = - j_T M j-1 + J~tMJ - 2C)J~1 (2.71) Hence (Me - 2Ce)T = -(Me - 2Ce). 2.4 Conclusions A dynamic model of an underwater vehicle-manipulator system has been formulated in matrix-vector form. Mechanical properties like symmetry, skew-symmetry and positiveness have been established. The matrix-vector form simplifies the use of non-linear control theory in the control design and facilitates the use of Lyapunov theory in the stability analysis. The formulation is an extension of similar formulations of robot-manipulator dy namics and underwater vehicle dynamics. This means that well-established motion controllers derived for robot-manipulators and underwater vehicles can be extended and applied to underwater vehicle-manipulator systems and stability can be proved. Also, passivity-based and observer-based con trollers developed for robot-manipulators and underwater vehicles can be applied to underwater vehicle-manipulator systems and state convergence can be proved. 24 Underwater vehicle and manipulator dynamics Chapter 3 Control of underwater vehicle and manipulator To be able to perform operations like inspection, maintenance, repair and service work on underwater installations, accurate control of both vehicle and manipulator is essential. In this chapter control strategies for under water vehicle-manipulator systems are presented. The feedback linearization technique is commonly used in motion control of robot-manipulators and space vehicles carrying manipulators, see Spong & Vidyasagar (1989), Egeland & Sagli (1993), Xu & Shum (1994), Dubowsky & Papadopoulos (1993) and references therein. In Section 3.1 this tech nique will be applied to underwater vehicle-manipulator systems. The feedback linearization concept is studied through a simulation study and performance, required torque and robustness are evaluated. In the design of feedback linearization controllers it is not required that the dynamic model is written in the matrix-vector form presented in Chapter 2. Having represented the dynamic model in a matrix-vector form and es tablished mechanical properties, nonlinear controllers designed for robot- manipulators and for underwater vehicles can be extended and applied to the combined underwater vehicle-manipulator system. Here, attention will be given in particular to passivity-based controllers developed for robot manipulators (Takegaki & Arimoto 1981), (Paden & Panja 1988). In these control schemes the availability of velocity measurements were assumed. Recently, several authors have presented control schemes for robot manip ulators were velocity estimates derived from position signals are used. Kelly (1993) used a first order filter to reconstruct the velocity signal in set-point 26 Control of underwater vehicle and manipulator regulation of robot manipulators. Nonlinear observers for trajectory con trol of a manipulator are described by Nicosia & Tomei (1990). Berghuis (1993) presented passivity based control structures for robot manipulators using only position measurements, see also references therein. This chapter is organized as follows: In Section 3.1 the feedback lineariza tion technique is examined through a simulation study, and performance, required torque and robustness are evaluated. Discussion and conclusions are presented in Sections 3.1.5 and 3.1.6. In Sections 3.2 and 3.3 passivity- based controllers without the need of velocity measurements are proposed for motion control of the vehicle and manipulator. Computer simulations are presented in Sections 3.2.3 and 3.3.3 to illustrate performance and ro bustness properties of the proposed controllers. Discussion and conclusions are presented in Sections 3.2.4, 3.2.5, 3.3.4 and 3.3.4. Section 3.4 holds the concluding remarks. The presentation is based on (Schjplberg & Egeland 1995) and (Schjplberg & Egeland 1996c). 3.1 Feedback linearization The feedback linearization technique (Spong & Vidyasagar 1989) is a com mon approach to control design of robotic systems. The concept is to choose an appropriate feedback loop so that the nonlinearities in the sys tem are cancelled. This results in a linear system for which many control techniques are available. The main problem with the feedback linearization method is that robustness is difficult to establish for realistic model uncer tainties. Nevertheless, the simplicity of the technique and the possibility of using linear control techniques makes the feedback linearization method attractive. Dubowsky & Papadopoulos (1993) decoupled the spacecraft position and attitude vector from the manipulator position and orientation vector, and Egeland & Sagli (1993) suggested to decouple the manipulator motion and the system linear and angular momentum. This resulted in a more energy- efficient control scheme. Since fuel is limited also for underwater robotic systems, it is relevant to see if the same results are achieved for underwater vehicle-manipulator systems. Underwater vehicle-manipulator systems are affected by hydrodynamic forces and possess uncertainties in the hydrodynamic coefficients and damp ing terms. It is therefore of interest to study the performance and robust ness of the control schemes based on feedback linearization. In this section 3.1. Feedback linearization 27 decoupling schemes suggested by Egeland & Sagli (1993) for spacecraft- manipulator systems are applied to an underwater vehicle-manipulator sys tem and the schemes are evaluated by simulations, considering robustness, performance, torque and implementation simplicity. 3.1.1 Kinematic equations Typically, it will be required that the underwater control system allows for the control of the manipulator end-effector position and orientation vector /x e in the 1-frame, and in addition, the vehicle position and attitude vector rj. As shown in the previous chapter, the end-effector velocity vector °x e can be expressed in terms of the vehicle velocity v and the joint velocity q °x e = o N =J\V + Jiq (3.1) U)n In the frame I the end-effector velocity vector is expressed (3.2) 3.1.2 System momentum The system motion can be described in terms of the system momentum. The linear and angular momentum of a system of rigid-bodies are defined (Hughes 1986) (3.3) where mi is the mass of body i, u, is the linear velocity of body i, Ij is the moment of inertia, w* is the angular velocity, is the linear momentum of body i and r, is the position of the end-point of body i to the center of mass of body i. The system momentum can be written as a function of the system velocities (Sagli 1991) Ojj =PoiI' + Po29 (3.4) where Pqi and P02 are momentum Jacobians. The change in the system linear and angular momentum can be expressed = t - n(q,u) (3.5) 28 Control of underwater vehicle and manipulator where r is the control force acting on the system. The vector n(q,u), where u — [qrT, vT]T, contains the hydrodynamic forces, the Coriolis and centrifugal forces and the gravity and buoyant forces. 3.1.3 Control laws In the following section feedback linearization is applied to an underwater vehicle and manipulator system and two control laws are given for control of the vehicle and the manipulator end-effector. Feedback linearization of the system dynamics is achieved by defining a controller in the form t = M(q)a u + C(q, u)u + D(q, u)u +g(q, v) (3.6) This concept assumes that the vehicle velocity and the joint velocity are measured as well as the vehicle position and attitude and the manipulator joint angles. Substituting (3.6) into the equations of motion (2.37) for the total system, yields the linearized system u = a u (3.7) The augmented task velocity vector y is chosen according to the require ments of the task and can be written y — P qu, where Pq(q, rj) is a matrix defined according to the task. The time derivative y is y = P qu + Po« (3.8) This gives u = PoHy - P qu) = a u (3.9) y — Po&U -f- Po u — dy (3.10) and the control vector a u — Pq ( This yields the closed loop error dynamics of the linearized system y — a y = 0 (3.12) Two choices for the task velocity vector y are: 3.1. Feedback linearization 29 1. Decoupling of the manipulator end-effector velocity from the vehicle velocity is achieved by defining (3.13) This scheme is used in the control of spacecraft-manipulator systems Egeland & Sagli (1993). Tracking convergence of the system posi tion/attitude and velocity states is achieved using the control law ®ed -Kpll®e -®^dll®e (3.14) Vd ~ Kpl2W ~ Kdl2V where fj is the vehicle position /attitude error vector f) = rj — rjd . The subscript d indicates the desired value. The vector xe is the manipulator position/orientation error vector xe — xe l x&d (3.15) The end-effector velocity error vector is defined (3.16) Remark 1. The manipulator controller is able to compensate for unpredicted vehicle motion due to the control of the end-effector in the I-frame. The vehicle thrusters are used to compensate for the reaction forces due to the manipulator motion. 2. For a spacecraft-manipulator system, Egeland & Sagli (1993) sug gested to decouple the end-effector motion from the total system momentum to reduce energy consumption. Decoupling of the ma nipulator end-effector velocity from the total system momentum is achieved by defining the task velocity vector (3.17) The system motion can be controlled using the control law ®ed -Kp21®e -®^d21®e (3.18) P01 (Vd ~ Kp22V ~ Kd22W) Remark 2. From (3.5) it can be seen that the control vector r has to counteract the hydrodynamic forces to be able to change the system momentum. 30 Control of underwater vehicle and manipulator Integral action can be obtained by including the term /J zdr, where z is the position/attitude error vector, in the controllers. Scheme (3.14) is computationally more efficient than scheme (3.18) since the last scheme demand the computation of the Jacobian ’s P qi and Pq2- 3.1.4 Simulation study The objective of the simulation study was to evaluate the performance, torque consume and robustness of the closed loop system utilizing the con trollers (3.14) and (3.18). A simulation model of an underwater vehicle carrying a planar two-arm ma nipulator system was implemented in Matlab, see Appendix A for further details. A sinusoidal signal r was chosen for the end-effector reference for the x- and z-direction. r = 1 + sin(0.57rt)(m) (3.19) A constant was used as a reference for the y-component of the translation. The end-effector orientation reference was chosen to be constant. The re dundancy resolution produced the vehicle reference rjld in the x-, y- and z-direction. The vehicle attitude vector q2(j was chosen as V2d = [0.1,0.1,0.1]T (rad) (3.20) All state variables were initially zero. The vehicle and manipulator dynam ics were simulated using the 4th order Runge-Kutta method with sampling time equal to 0.1 s for the vehicle dynamics and sampling time equal to 0.01 s for the manipulator dynamics. The controllers (3.14) and (3.18) were tuned under the requirement of high tracking accuracy and speed, and assuming exact knowledge of the sys tem model, see Table 3.1. The tracking error of the end-effector posi tion/orientation vector and the vehicle position and attitude vector using the controllers (3.14) and (3.18) are shown in Figures 3.1 to 3.4. The performance was evaluated by computing A = ;Ei?ii (3.21) K i=l (3.22) 3.1. Feedback linearization 31 Table 3.1: Design parameters for controllers (3.14) and (3.18). Kpn ■^Pl2 Kp2i Kp22 1001 1001 I I Kd21 K-dii 201 20J 21 21 9 (*) 10 Figure 3.1: Simulated end-effector position/orientation error vector for con troller (3.14). where k was the number of samples. The amount of vehicle control torque required in the x, y and z direction was compared by calculating the indices j3=i:iTU (3.23) k J4 = 53lT2il (3.24) i—1 h = 53lT3 where ti,t2,T3 are vehicle control forces in the x, y and z direction re spectively. The performance and torque indices using the controllers (3.14) and (3.18) are shown in Table 3.2. The robustness of the manipulator and vehicle controllers to model uncer tainties was evaluated by including uncertainties in the manipulator dynam- 32 Control of underwater vehicle and manipulator »(•) 10 Figure 3.2: Simulated vehicle position/attitude error vector for controller Figure 3.3: Simulated end-effector position/orientation error vector for con troller (3.18). Table 3.2: Performance and torque indices assuming completely known model. ______Ji J2 Js J4 h a yi 0.44 0.41 156 6 185 a yi 0.44 0.30 190 76 176 ics and in the vehicle dynamics. Two cases of uncertainty were considered: • Case 1. A 20% uncertainty was added to the hydrodynamic coeffi cients. The vehicle damping was assumed to be linear and the ma- 3.1. Feedback linearization 33 Figure 3.4: Simulated vehicle position/attitude error vector for controller nipulator damping was assumed to be linear and quadratic, excluding the vortex shedding effect. The indices are shown in Table 3.3. • Case 2. The vehicle and manipulator damping matrices were assumed to be constant. The added mass and added moment of inertia coef ficients were assumed to be completely unknown and the reaction forces were neglected. The indices are shown in Table 3.4. Table 3.3: Performance and torque indices assuming 20% uncertainty in the model. ______■___ Case 1 Ji h h J4 h a yi 0.56 0.49 189 23 231 °2/2 0.7 0.46 209 91 226 Case 2 Ji h h <74 h a yi 2.6 1.2 188.9 180 297.7 a V2 1.45 0.9 158.6 93 329 Table 3.4: Performance and torque indices assuming partly unknown model. Robustness of the manipulator controller to unpredicted vehicle movement was investigated by adding a constant signal to the vehicle position/attitude 34 Control of underwater vehicle and manipulator vector, see Table 3.5. Table 3.5: Performance and torque indices with drift in the vehicle position. Jl h h J4 Js a y i 0.48 0.43 168 5.3 175 a V2 0.52 0.33 202 78 167.6 3.1.5 Discussion The results presented in Table 3.2 showed that the performance indices were almost the same for the two control schemes (3.14) and (3.18). However, larger torque was required using (3.18). This can be explained by (3.5), where the control forces have to counteract the hydrodynamic forces to be able to change the system momentum. This is in contrast to the control of spacecraft-manipulator systems (Egeland & Sagli 1993), where the control force directly changes the system momentum. The results presented in Ta ble 3.3 and 3.4 showed that for both controllers the performance decreased with increasing model uncertainty. The controller (3.14) was more robust to small uncertainty in the dynamic model while the controller (3.18) was more robust to large uncertainty in the dynamic model. The reason for this is that the control law (3.18) includes the dynamic term Pqi while control law (3.14) results in a purely kinematic decoupling. With high ac curacy in the dynamic model the torque was lower with the conventional scheme (3.14), but for systems with large model uncertainty (3.18) resulted in larger torque consume. Table 3.5 showed that unpredicted drift in the vehicle position resulted in almost the same performance for the two con trollers. 3.1.6 Conclusions In this section feedback linearization has been applied to an underwater vehicle-manipulator system. Two decoupling schemes have been studied, 1) decoupling of the manipulator end-effector velocity and the vehicle ve locity (conventional scheme), and 2) decoupling of the end-effector veloc ity and the total system momentum. The main aim of this work was to see if the same results, considering performance and energy consumption, 3.2. Regulation and tracking control 35 was achieved for underwater vehicle-manipulator systems as for spacecraft- manipulator systems using the two decoupling schemes. Fuel storage is lim ited for space and underwater systems and decoupling of the end-effector from the system momentum resulted in low reaction fuel consumption for spacecraft systems. This was not achieved for underwater systems since the vehicle thrusters must counteract the fluid forces to be able to change the system momentum. Both algorithms can be developed recursively and this makes the schemes easy to implement. However, the conventional scheme is computationally more efficient since it does not demand the computation of the momentum Jacobians. The main conclusion of this study is that decoupling of the end-effector velocity from the vehicle velocity is the best choice of control scheme. 3.2 Regulation and tracking control Standard working modes for underwater vehicle-manipulator systems de mand set-point control of the vehicle and motion control of the manipulator, assuming that the vehicle is stationed within the manipulators workspace. Utilizing the knowledge of the system properties and structure, a robust vehicle controller with reaction moment compensation for set-point regula tion of the vehicle position and attitude is proposed. A motion controller is proposed to control the manipulator. Thereby, the vehicle can be used for slow-gross positioning while the manipulator performs fast joint-space tasks and this provides a large workspace for the system. The proposed controllers are designed so that knowledge of the complete system dynam ics is not needed. This results in reduced computational requirements. Further, the controllers does not require that the system velocity vector is measured. This is convenient since velocity sensors will increase system volume, weight and cost. The choice of controller structures is inspired by the work of Berghuis (1993) and Paulsen et al. (1994). 3.2.1 Control strategy The vehicle position/attitude error vector is defined V — V ~ Vd and the joint error vector is defined q = q - qd . The subscript d indicates the desired state value. The desired joint trajectories qd and qd are bounded, llfldll ^ vm and \\qd \\ < a m. The proposed vehicle controller with reaction torque compensation is Tr = —Jv(ri)TKpi(tjq - 7jd) - Jv(.'n)TKdiVo + Mc{q)q d 36 Control of underwater vehicle and manipulator +C2{q, qd )qd + D2(q, qd , vd )qd +9 r(v)+9E(q ) (3-26) where u d is the vector of desired vehicle velocities, r]d is the vector of desired vehicle position and attitude, r)Q is the vector of estimated vehicle velocities and the design matrices Kp\ and Kd \ are symmetric and positive definite. Remark 1. The controller (3.26) consists of a linear PD controller and a compensatory term to counteract the forces acting from the manipulator on the vehicle. Measurement of the manipulator and vehicle velocity are not needed. Moreover, the reaction torque compensation makes the vehicle control system robust to changes in the manipulator dynamics. The equations for the vehicle velocity observer are (Paulsen et al. 1994) Vo = Kd i(~Kpo(Vo ~ V) + KpliVd ~ Vo )) (3.27) KpoiVo ~V) = Kpi(vd - Vo) + Kd\(Vd ~ Vo) (3-28) where the matrix Kpo is symmetric and positive definite and rj0 is an internal state of the observer. The proposed control law with a feedforward term for tracking control of the manipulator is Tm = Mm(q)q d + Cm(q, qd )qd + DA(q, qd , vd )qd + g m{q) -Kpq - Kd {k - qd ) (3.29) where the vector q contains the estimated joint velocities and Kp and Kd are symmetric and positive definite design matrices. Remark 2. The proposed controller (3.29) consists of a linear PD con troller and a feedforward of the manipulator dynamics. Measurement of the joint velocities and the vehicle velocity are not required. Remark 3. The main motivation for designing two control loops is that manipulator tracking can be achieved without perfect knowledge of the vehicle dynamics. Moreover, the controllers can be implemented with dif ferent bandwidths, which reduces the controller action of the vehicle ac tuators. The manipulator motion can be controlled accurately, using the vehicle for slow gross positioning which provides a large workspace for the system. Remark 5. The stability proof is simplified because the manipulator dy namics (2.37) can be written so that the manipulator Coriolis and centrifu gal terms C2 and Cm can be expressed independent of the vehicle velocity 3.2. Regulation and tracking control 37 v. The mass matrix Mm and the Coriolis and centrifugal matrix Cm of the manipulator utilized in the controller are well known. The equations for the proposed joint velocity observer are 9 = z + Li(q-q) (3.30) z = Qd~ -^2(9 - 9 3.2.2 Stability analysis Define the state vector By substituting the control laws (3.26-3.31) into the system model (2.37) it can be seen that e = 0 is a unique equilibrium point. Stability of the total system is proven by considering the following positive definite function where Mu = Mr + H and P\ and P2 are symmetric and positive definite matrices. The time derivative of V along the system trajectories becomes +(9 - 9 Substituting equations (3.26-3.31) into (3.33), utilizing the skew-symmetric property and choosing P2 = Kd L^[l and Pi = P2L2 gives a simplified expression for V V = -z/TDi(g,qr, v)v + vr {D2{q,q d ,vd )qd - D2(q,q,v)q) -qTD3(q, q, v)v + gT(£>4(g, qd , vd )qd - DA{q, q, v)q) -vTC2(q,q d )q - qTCm(q,q d )q - (f?0 - Vd^KdliVo ~ Vd) ~(q ~ Qd) TKd (q ~ Qd) (3.34) According to the Mean Value Theorem (Khalil 1992) D2(q, qd , vd )qd - D2(q, q, u)q = D2(q, qd , qd , vd ) - D2(q, qd , v) + D2(q, qd , v) - D2{q, q, q, v) dDii^q^x) dP 2(q,u,z) (ud - v) + (qd -q) (3.35) dx dz Z=(i2 where n x € [vd , u] and fi2 € [qd ,q\- Similarly Di(q, qd , u d )qd - D4(q, q, v)q dDi(q,q d ,x) (qd - q) dx *=/*! z=fi2 (3.36) For simplicity the following notation is used: dD? (3.37) A = dx ®=#*i dDi 02 = (3.38) dz Z=fjL2 dDi (3.39) 03 = dx 1 dDi ^4 = (3.40) dz z=H2 It is reasonable to assume that the vehicle velocity v and the manipulator joint velocity q are upper bounded. The hydrodynamic damping is modeled 3.2. Regulation and tracking control 39 as linear and quadratic damping terms (2.4) and this results in that is bounded. The Coriolis and centrifugal terms are linear in q and therefore C2(q, q) and Cm(q, q) satisfy the property C(q,x)y — C{q,y)x. The Coriolis and centrifugal forces are bounded (Berghuis 1993) (3.4i) Thereby V can be written dDo dD 2 V = —v tD\U — i/1 V q - qTD$v dx X=fil dz z=ii 2 :T 0D4 :T dD 4 ~q q dz »=**! -vTC2{q, qd )q - qTCm(q, qd )q ~(vo — Vd) TKd i(vo - vd ) — ( V is negative semi-definite when the matrix ||Di||+/?i ~\{02 + 03 + ll-Dsll + k2vm) (3.44) — \{02 + 03 + + k2vm) 04 + kmvm + Xm{Kd ) is positive definite, Xm(Kd ) denotes the upper bound on the eigenvalues of the matrix Kd . This requires 1 (02 +03+ 11^3 II + k2Vm)2 \n(*d) > {04 + kmVffi) (3.45) 4 WDiW+0! V < 0 implies that e is bounded and thereby tt and rm are bounded and the system is Lyapunov stable. Further this implies that is and q are bounded. Thereby V is bounded and V is uniformly continuous. Barbalat ’s lemma (Khalil 1992) states that for a positive semi-definite scalar function /, where / is negative semi-definite and uniformly continuous in time, / —*■ 0 as t —*■ oo. According to Barbalat ’s lemma, V —> 0 as t —» oo. This implies state convergence, that is e —*■ 0 as t —> oo. 40 Control of underwater vehicle and manipulator 3.2.3 Simulations The objective of the simulations was to show performance of the proposed controllers and to illustrate the significance of the fluid forces. The simu lation model is presented in Appendix A. The desired joint trajectory qd was generated using qdl = sin(0.l7rt)(rad) (3.46) qd2 = —sin(0.l7rt)(rad) (3.47) The desired vehicle position vector was chosen as Vdi = [1-0,1.0,1.0]T (m) (3.48) and desired vehicle attitude vector was chosen as fid* = [0.1,0.1,0.1]T(rad) (3.49) The vehicle state vector was initial set to zero and the manipulator state vector had initial value qd = [0 0jT (rad) and qd = [0.17T — 0.17r]T (rad/s). The controllers were tuned under the requirement of high tracking accuracy and speed. The vehicle control law parameters were chosen Kpl = 1500 diag(l, 1,1,1,1,1) (3.50) Kd i = 1500 diag(l, 1,1,1,1,1) (3.51) Kp0 = 1500 diag(l, 1,1,1,1,1) (3.52) and the manipulator control parameters Kp = 1000 diag(2,4) (3.53) Kd = 100diag(1.5,2.5) (3.54) Lx = 100diag(5,5) (3.55) L2 = 100diag(l,l) (3.56) The vehicle and manipulator dynamics were simulated using the 4th or der Runge-Kutta method and sampling time equal to 0.1 s for the vehicle dynamics and sampling time equal to 0.01 s for the manipulator dynamics. Tracking performance of the state vectors f) and q with the proposed con trollers (3.26) and (3.29) is shown in Figures 3.5 and 3.6. For compari son the system was also simulated using (3.26) without the reaction torque 3.2. Regulation and tracking control 41 Figure 3.5: Simulated vehicle position error for controllers (3.26) and Figure 3.6: Simulated joint position error for controllers (3.26) and (3.29). compensation for vehicle control and using (3.29) without the forward com pensation term for manipulator control. This corresponds to the classical PD-control structure. The simulation results are shown in Figure 3.7. The control vectors using the proposed controllers are shown in Figures 3.8 and 3.9. The vehicle system has been simulated without compensation for manipulator motion, see Figure 3.11, and the norm of the reaction torque compensatory vector is plotted in Figure 3.10. Figure (3.12) shows the system simulated with an unknown payload mass equal to 5 kg attached to the manipulator end-effector, and without compensation for the fluid damping acting on the manipulator. Position measurement noise v\ ~ N{0, Figure 3.7: Simulation results with PD-controller. the vehicle and joint position/attitude signals. Simulations were carried out with = 0.015 m and oi = 0.0014 rad such that the signal to noise ratio (S/N) was approximately 1%, and with Tr3(N) Tr6(Nm) Tr3(N) bme(e) Figure 3.8: Vehicle control vector using controllers (3.26) and (3.29). 3.2.4 Discussion The proposed controllers have a formulation which allows for a recursive implementation of the control law. The complexity of the algorithm is com parable to the standard recursive Newton-Euler algorithm used in robotics which involves 117n-24 multiplications and 103n-21 additions per sample. 3.2. Regulation and tracking control 43 Tml (Nm) rm2(Nm) Figure 3.9: Manipulator control vector using (3.26) and (3.29). \ :.... r ...... Figure 3.10: Norm of the compensation vector fm. The proposed algorithm will in addition require the computation of the hydrodynamic forces. Thus the total computational requirement will be in the order of a few thousand flops per sample. With present day processors this may be executed at 1kHz sampling rate, which is comparable with the sampling rate of commonly used velocity loops for electrical motors. As can be seen from Figures 3.5 and 3.6 compensation for the reaction torque in the vehicle controller (3.26) and the feedforward compensation in the joint controller (3.29) results in zero tracking convergence lim 7? = 0 t—* oo lim g = 0 t—*00 Figure 3.7 shows that the PD-controller results in convergence errors for the vehicle and manipulator state vectors. It can be seen from Figure 3.11 44 Control of underwater vehicle and manipulator Figure 3.11: Vehicle position error vector without compensation of manip ulator motion. Figure 3.12: Joint position error vector with unknown payload mass and without compensation for hydrodynamic damping. that controlling the vehicle without any consideration of the manipulator motion, fm = 0, results in convergence errors for the vehicle state vector. This means that the reaction torque compensation in the vehicle controller makes the vehicle system robust to changes in the manipulator dynam ics. Figure 3.12 shows that the proposed manipulator motion controller is robust to load disturbance, but removing the compensation for the hydro- dynamic damping results in a more oscillatory joint performance. With a signal to noise ratio of 1% the tracking errors show the same time-behavior as in the noise-less case, see Figure 3.13. However with a signal to noise ratio of 10% the state vectors are highly noise-corrupted, see Figure 3.14. The velocity estimates are sensitive to noise and therefore in a real system, 3.2. Regulation and tracking control 45 *sa»«s»5ssiss tirrife (s) 10 -0.04 -0.06 time (s) Figure 3.13: Simulation results with noisy position measurements with S/N=l%. the control parameters and Kd must be limited. In the tuning pro cess an effective trade-off between performance, bandwidth and robustness requirements has to be made. The performance of the vehicle control loop is less important, since the manipulator will compensate for vehicle con trol deviations. Therefore the vehicle controller should be tuned to satisfy robustness requirements. The manipulator control loop should be tuned under requirements for high performance and bandwidth. 3.2.5 Conclusions Station-keeping of the vehicle while the manipulator performs fast joint- space tasks is required in several working scenarios for vehicle-manipulator systems. A set-point controller with reaction torque compensation was proposed for set-point control of the vehicle and a feedforward tracking controller was proposed for motion control of the manipulator. In the pro posed vehicle controller knowledge of the vehicle model was not needed in the vehicle controller and this results in less computational requirements. 46 Control of underwater vehicle and manipulator 0 5 10 15 20 0 5 10 15 2 time (s) time (s) time (s) time (s) Figure 3.14: Simulation results with noisy position measurements with S/N=10%. The two control loops can easily be implemented to have different band- widths and this reduces the controller action of the vehicle actuators. 3.3 Coordinated motion control Some underwater inspection and service operations require that both the vehicle and manipulator follow desired trajectories. In this section a passivity- based controller is proposed for coordinated motion control of the under water vehicle and manipulator system. The controller consists of a PD controller with a feedforward compensatory. Controllers with this struc ture has successfully been employed to robot manipulators, see Kelly & Salgado (1994) and references therein. Complete knowledge of the system dynamics is required and the mechanical properties derived in Section 2 are needed in the stability analysis. The proposed controller requires only the measurements of the vehicle and manipulator position vectors. The system 3.3. Coordinated motion control 47 velocity vector is estimated from the position measurements. 3.3.1 Control law A model-based controller is proposed for tracking control of the vehicle and manipulator. Consider the PD-controller with feedforward compensatory given by r = Me{x)xd + Ce(x, Cd)vd + De(x, Cd)id + 9e(x ) —jKpx — fQ(xo — Xd) (3.57) where x = [t7T gT]T is the system position/attitude vector and xq is the estimated value of the velocity vector x. The vehicle position and orien tation error vector is defined V ~ V ~ Vd and the manipulator joint error vector is defined q = q — qd . The subscript d indicates the desired value. The matrices Kp and Kj are diagonal positive definite design matrices. The design matrices are selected such that the system error converges to zero lim x = 0 t—* oo The proposed observer equations are xo = z + Li(x - xo) (3.58) z = xd - Li(xo - xd ) (3.59) where Li and L% are diagonal positive definite matrices and are chosen such that lim (xq - Xd) = 0 t—»oo 3.3.2 Stability analysis The closed-loop system is obtained by substituting the control law (3.57) into the dynamic model of the underwater vehicle-manipulator system. Define the state vector (3.60) The point e = 0 is a unique equilibrium point for the closed-loop system. 48 Control of underwater vehicle and manipulator Properties utilized in the stability proof are: Property 1. The Coriolis and Centrifugal forces are continuous and dif ferentiable and satisfy the Lipschitz condition (Khalil 1992). l\Ce(x, C)*d — Ce(x, Cd)*d|| = \\f(x,xd ,v) - f(x,xd ,u)\\ = - u)\\ < ||^(x,Xd,ti7)||||(e -tt)|| 5 •C'llC — Cdll = L\\J-1(x-xd )\\ De(x, <)x - De(x, Cd)&d = fl(®, C, *) ~ ®d) dg(x,y,z) (sc - Xd) + (C — Cd) dz z € [®d, x] <9y z e [id,x] 2/ G [Cd) C] 2/e [G,C] < (A + /?y||«7-x ")i (3.62) where & = || §§ and (3y = || ||| 'z,y ’ "y\z,y Consider the positive definite function V — ix XMe® + ^xTKpx + i(x 0 - Xd)TPi(*o - Xd) +^(sco - Xd)TP2(®0 - *d) (3.63) where Pi = P2^2 and P2 = Kd L11 are symmetric and positive definite matrices. Differentiating V along the system trajectories yield V = ix TMex + xT(t - Ce(x, C)x - De(x, ()x - 0e(x) - Mexd ) +xrKpx + (x 0 - x d)TPi(x 0 - Xd) + (®o - Xd)TP2(*o - ®d) = -x T(Ce(x, C) - Ce(x, Cd))Xd - ®T(Be(x, C)x - Pe(x, Cd)®d) (®o - Xd)TJK’d(*o - *d) (3.64) 3.3. Coordinated motion control 49 The Lipschitz condition, the Mean Value Theorem and the skew-symmetric property imply V < Lam\\i\\2 + (Pz + PyVm)\\if ~ Am(Kd)||x 0 - xdf (3.65) < -(Am(^d) - La m - (pz + 0yo m))||i||2 - ||x 0 - x|i 2 (3.66) where Xm(Kd) denotes the largest eigenvalue of the matrix Kd and a m denotes the largest singular value of the matrix «7-1. The function V is negative semi-definite for Am(-Kd) > (L + Py)crm + Pz (3.67) and the state vector e is bounded. Thereby the closed loop system is Lyapunov stable. It follows that x is bounded and thereby V is bounded and V is uniformly continuous. From Barbalat ’s lemma (Khalil 1992) it follows that the system error x and the estimation error xq — xd converges to zero as t —*■ oo. 3.3.3 Simulations In this section simulation results are presented in order to illustrate the theoretical results. The simulation model is presented in Appendix A. The vehicle and manipulator dynamics were simulated using the 4th order Runge-Kutta method and sampling time equal to 0.01 s. The state vectors had initial value equal to zero. The goal of the control law was to track a time-varying trajectory generated by an appropriate reference model. The desired vehicle position trajectory was generated using the 2nd-order smooth function r)d + rjd + r)d = A2r)T, where rjT is the commanded position reference. The desired vehicle orien tation vector was chosen equal to zero. The desired joint position q\d was chosen equal to zero, and q2d was generated by the square function shown in Figure 3.16. The design parameters were chosen: Kp = 250diag(l, 1,1,1,1,1,1,1) Kd = 50diag(l, 1,1,1,1,1,1,1) Li = 50diag(l, 1,1,1,1,1,0.5,0.5) L2 = 50diag(l, 1,1,1,1,1,1,1) Figures 3.15 and 3.16 show the tracking errors of the closed loop system. 50 Control of underwater vehicle and manipulator Q* 1 -I- ...... t------1— 0 5 10 15 20 25 30 35 0.6 »?(m,iad) o4 0 -0 2 -0.4 0 5 10 15 ^(,1° 25 30 35 Figure 3.15: Vehicle reference and •position error vector for controller -1 0 5 10 15 20 25 30 35 40 time (a) 0.1.... 9i(nd) 0 -0.1 0 5 10 IS 20 25 30 35 40 time (s) Figure 3.16: Manipulator reference and position error vector for controller C&J7). 3.3.4 Discussion Figures 3.15 and 3.16 shows that the vehicle and manipulator follows the desired trajectories. Lower bounds have been found for the design matrix K influence of measurement noise in the system. Controlling the manipulator end-effector may be necessary in some operations, and this can be done using the kinematics given in (2.44). 3.3.5 Conclusions A feedforward motion controller for the underwater vehicle-manipulator system has been presented. The closed loop system is robust to changes in the vehicle and manipulator buoyancy and weight. Tracking convergence and stability was proved, and lower bounds on the tuning of the control pa rameters were given. The controller is similar to tracking controllers derived for robot manipulators. However, the manipulators considered previously are manipulators with a fixed base and without the effect of fluid forces. The controller is simple to implement and can be computed recursively. 3.4 Conclusions In Section 3.1 feedback linearization was applied to an underwater vehicle- manipulator system. Simulations showed that the best performance was obtained using the conventional decoupling scheme. In Section 3.2 and 3.3, passivity-based controllers were proposed to control the underwater vehicle and manipulator. In Section 3.2 a set-point con troller with reaction torque compensatory was proposed for control of the vehicle and a feedforward tracking controller was proposed for joint-space motion control of the manipulator. In Section 3.3 a feedforward motion controller was presented for tracking control of the vehicle and manipulator in world coordinates. Stability properties were proved for the closed-loop systems, and performance and robustness were illustrated by simulation. Velocity estimates were derived from position signals, which are less sensi tive to noise than velocity signals. With highly noise-contaminated position measurements Kalman filter estimates are more robust than the 1st order observer estimates presented in this chapter. However, 1st order observers are simple to implement and are model-independent. In underwater robotic systems operating at depth it is advantageous to minim ize the amount of equipment and sensors. The main motivation for employing passivity-based control schemes is that they are more robust to parametric and structural uncertainties than feed back linearization controllers. 52 Control of underwater vehicle and manipulator Chapter 4 Lateral motion control of an underwater thruster module connected to a slender marine structure Underwater cables, risers and other similar slender marine structures con nected to a surface vessel, undergo deformations due to fluid forces and vessel motions, hence vibrations and swinging motions (oscillations) are in duced in the system. The main aim of the present work is to demonstrate that by appropriate control of the bottom-end point of the structure, vi brations and oscillations in the system can be suppressed. The system considered in this chapter consists of a cable/riser connected to a surface vessel at the top end and to a thruster unit at the bottom end. A control system is proposed for position and velocity control of the thruster unit. The thruster control systems receives position reference signals from underwater transponder systems. In this chapter only the motion in the lateral direction is considered. In chapter 5 the system in 3 DOF is presented and coordinated position control of the vessel and thruster system is studied. By connecting thrusters to the bottom end-point of risers, the task of connecting a riser to a well-head can be performed more time- and cost- efficiently. After the connection operation has been performed, the thrusters can be disconnected from the riser by releasing a locking mechanism or by Lateral motion control of an underwater thruster module 54 connected to a slender marine structure help of an underwater vehicle-manipulator system supervising the opera tion. The equation of motion for the lateral motion used for the cable/riser sys tem is based on beam theory (Blevins 1990), (Sarpkaya Sc Isaacson 1981) and (Chakrabarti Sc Frampton 1982). This is combined with the equations of motion of the underwater thruster unit, taking into account the reaction forces between the two systems. The model of the total system is written in a compact form which facilitates the control system design and stability analysis. Lyapunov theory is used to prove that the closed loop system is globally asymptotically stable and thereby robust to environmental distur bances. This chapter is organized as follows: In Section 4.1 the equations of motion for the system is developed. A stability analysis is presented in Section 4.2 and Section 4.3 contains simulation results. Section 4.4 and 4.5 holds the discussion and concluding remarks. This presentation is based on (Schjplberg Sc Egeland 1996a) and (Schjplberg Sc Egeland 1996b). 4.1 Dynamic models i z Figure 4.1: An underwater thrust unit suspended to a surface vessel by a cable/riser. Figure 4.1 illustrates the system to be investigated. It consists of a ca ble/riser system connecting a surface vessel and to an underwater module 4.1. Dynamic models 55 equipped with thrusters. The equation of motion of the cable/riser system are similar to those of a slender beam under influence of external forces. It is assumed that the cable/riser is connected to the vessel and to the thruster system by means of ball-joints and that this results in small derivatives of the angles of deflection and zero bending moment. Further, it is assumed that the mass of the vessel is much greater than the mass of the cable/riser system and therefore the vessel will not be influenced by the cable/riser motion. The cable/riser system is affected by sea waves and water current loads. The lateral wave velocity w(z,t) for regular waves is defined by (Faltinsen 1990) W(z, t) = (vn £ae-6zsin ( 4.1.1 Equation of motion of the thruster system The lateral equation of motion of the thruster system at depth z = l is given by mtvx(t) + di-q x{t) + 4.1.2 Equations of motion of the cable/riser system The horizontal equation of motion of an underwater cable/riser for small angles of deflection can be described by the differential equation (Chakrabarti & Frampton 1982), (Sarpkaya & Isaacson 1981): +cv(z, t) + mcv(z, t) = fxs(z, t) (4.5) where El is the cable/riser stiffness, Te is the cable/riser tension, c is the structural damping and mc is the total mass of the cable/riser per unit length. The lateral displacement v(z, t) is normal to the cable/riser axis and varies with the time t and the longitudinal displacement z, that is the displacement along the cable/riser axis. It is reasonable to assume that the cable/riser is designed so that the stiffness is constant, EI(z) = El. It is also assumed that the cable/riser has a moderate tension, and that the effective axial tension Te(z) is approximated by the mean value T. This gives a good approximation of the vibration frequencies. Equation (4.5) then reduces to ~ T o~2 + cv + mcv - fxs(z,t) (4.6) The fluid loading is according to Morison ’s equation given by (Blevins 1990) fxs(z,t) = c2(z)(w(z, t) - v(z,t) - Uc(z))\w(z,t) - v(z, t) - Uc{z)I +Ci(z)w(z,t) (4.7) where ci(z) = jpCm(z)DQ and c2(z) = ^pCd(z)Do, and where p is the mass density of water, Do is the diameter of the cable/riser, Cm and Ca are drag coefficients. Fluid forces due to out-of-plane motions have been neglected. It is common to assume that Cm and Cd does not vary along the cable/riser (Nordell & Meggitt 1981). Moreover it is common to assume that the wave and current velocities are much larger than the cable/riser velocity (Blevins 1990). The expression for the fluid loading (4.7) then becomes fxs(z, t) = Ciw{z, t) + c2(w(z, t) - Uc(z))\w(z, t) - C7c(z)| (4.8) 4.1. Dynamic models 57 Natural vibration modes The natural vibration modes of the cable/riser can be found by solving the homogeneous equation + cv + mcv = 0 (4.9) The method of separation of variables gives a solution for mode i in the form (Meirovitch 1986) v*(z,t) =Pi{z)qi(t) (4.10) Inserting (4.10) into (4.9) gives — ki (4.11) Pi(z) Pi(z) qi(t) qi(t) where k{ is a constant. This leads to two differential equations of motion, where one is a function of time mcqi(t) + cqi(t) + &2%(t) = 0 (4.12) The solution to (4.12) is an oscillator with resonant frequency ut = — (4.13) mc and relative damping Q = . The second differential equation is a function of the longitudinal displacement z El Pi W - Tpi (z) - kipi(z) = 0 (4.14) and has a solution in the form Pi (z) — sinhn n z + At-2 coshn^ z + Aj3 sinnj 2 z + A,4 cosm, 2 z (4.15) where «ii = l^(2T + 2VT*+AkiEI) (4.16) n i2 = l-^±i{-2T + 2jT* + ±kiEI) (4.17) The frequency modes n and rii2 are found by solving (4.14) in Maple. The coefficients of (4.15) and the constrained modes can be determined using the boundary conditions. Lateral motion control of an underwater thruster module 58 connected to a slender marine structure Boundary conditions Chakrabarti & Frampton (1982) presented a review of riser dynamics with different boundary conditions. Ball-joint connections between the vessel and riser was used in most studies. Other connectors are: linear and rota tional springs, dampers and stiff masses. Boundary conditions may also be given as specification of deflections and moments. The boundary conditions affect the frequency and mode shapes of the cable/riser. Ball-joint connections are assumed in the present work. The boundary conditions at the top end-point of the cable/riser (z = 0) are given by the lateral movement of the vessel rx(t)g x(z). The ball-joint connection results in zero bending moment at the top end-point of the cable/riser v(0, t) = rx(t)g x{ 0) (4.18) v(0,t) = rx(t)g x( 0) (4.19) v(0,t) = fx (t) =0 (4.21) OZ 2=0 where yx(z) is the static deflection with the vessel displaced by one unit. The boundary conditions at the bottom end-point of the cable/riser (z — l) are given by the lateral movement of the thruster system r}x(t)hx(z) and by the zero bending moment at the bottom end-point of the cable/riser v(f,f) = Vx(t)hx(l) (4.22) v(l,t) = r}x(t)hx(l) (4.23) v(l,t) - rjx{t)hx(l) (4.24) (4.25) where hx(z) is the static deflection with the thruster system displaced by one unit. Equation of motion in normal coordinates The general solution to (4.9) is determined by adding the vibration motion to the end-point motions (Daring & Huang 1979). The total response is given by OO v(z,f) = g x(z)rx(t) + hx(z)r]x(t) + 53pi(z)gi(<) (4.26) 4.1. Dynamic models 59 The deflection functions gx and hx are determined by static analysis and are for pinned cable/riser connections defined according to Weaver et al. (1990) 9x(z) = 1 ~ y (4-27) = y (4-28) Inserting (4.26) into the equations for the boundary conditions gives the conditions Pi(0) = Pi(l) = 0 (4.29) Pi'(0) =Pi(0 = 0 (4-30) The coefficients of (4.15) are found from the conditions (4.29) and (4.30) Ai2 + Ai4 =0 (4.31) Ai2nh ~ Aiin i2 = 0 (4-32) A^sinhn^Z + Aj3sinn, 2Z = 0 (4.33) A^nfjSinhnijl — Ai3n| 2sinni 2Z = 0 (4.34) From (4.31) and (4.32) it follows that Aj2 = A,4 = 0. Substituting Aj3sinni 2i =-AjjSinhnijl (4.35) into (4.34) gives the trivial solution sinhn^Z = 0. Substituting A^smhrtiJ = — Aj3sinn 32Z (4.36) into (4.34) gives the solution sinnj 2Z = 0. The constrained modes are given by rii2l = iir, hence ^ = (y)2(EJ(y)^+T) (4.37) The influence of fluid forces, vessel and thrust unit motion on the equation of motion for vibration mode i can be found utilizing the conditions for orthogonality and normalization (Weaver et al. 1990). The orthogonality properties of the eigenfunctions are found by considering mode i and mode j of the eigenvalue problem El Pi' (z) — T p ” (z) = ktpi(z) (4.38) Elp'j"(z) ~ Tp” (z) = kjpj(z) (4.39) Lateral motion control of an underwater thruster module 60 connected to a slender marine structure Multiplying (4.38) by Pj and (4.39) by pi and integrating over the length of the cable/riser yield El f p’i 'pjdz -T f p”pjdz = ki f piPjdz (4.40) Jo Jo Jo El p'j'pidz ~T fQ PjPidz = kj pjPidz (4.41) Integration of the left-hand sides of (4.40) and (4.41) give Ellp'-'pjfo - Ellp'-pjfo + El fo PiPjdz -T[p^>j]o +T [ p'iPjdz = ki f p^jdz (4.42)- J o Jo EI\p'j'pi]l0 - EI\p"jp'i\l0 + El fQ PjPidz -T\pjPi]lQ + T Jq PjPidz = kj PjPidz (4.43) Subtracting (4.43) from (4.42) and utilizing the boundary conditions (4.29- 4.30) gives (4.44) For i j and u>i ^ u)j the eigenfunctions are orthogonal (4.45) For i = j the integral in (4.44) is equal to a constant (4.46) For convenience the constant is often chosen as a* = 1. Substituting the general solution (4.26) into (4.6) result in ElPi'qi - Tp”qi + cpi Multiplying (4.47) by pj and integrating over the length of the cable/riser yield El f pi’Pjdzqi -T f p'iPjdzqi + c [ PiPjdzcp + mc f PiPjdzqi = Jo Jo Jo Jo 4.1. Dynamic models 61 [ Pjfxs(z,t)dz -c f pjg x(z)dzr x - mc [ pjg x(z)dzf x Jo Jo Jo -c / Pjhx(z)dzrjx -mc Pjhx{z)dzrjx Jo Jo (4.48) Utilizing the conditions for orthogonality (4.40) and normalization (4.46) the equation of motion (4.6) reduces to the form mcqi(t) + cqi(t) + &%(() = Gi (4.49) where i = 1,2,3,.., oo. The forces induced by fluid loads, vessel and thrust unit motion cure summarized in 1 ft Gi = — Pi{z)\fx${z,t) - cg x(z)rx(t) - chx{z)r}x(t) a,i Jo -mcgx{z)rx{t) - mchx(z)ijx{t)]dz (4.50) The cable/riser force on the thruster system is according to the reaction- action principle (Newton ’s third law) /cz(2)*) = (4.51) where 1 lx — — / Pi(z)hx(z)dz (4.52) Oi Jo 4.1.3 Equations of motion of the total system on matrix- vector form Defining the state vector x = [r}x,qi]T, the equations of motion (4.4) and (4.49) can be expressed in the form Mt x + Dt(x) + Ktx — u + f (4.53) where Tf%t /yi Mt = (4.54) TTT-x 0 0 (4.55) 0 ki Lateral motion control of an underwater thruster module 62 connected to a slender marine structure and diVx + fd 2(Vx) (4.56) lici]x + cqi where fd 2(Vx) = diirjx -Uq)\Vx- UCl\ (4.57) The vector u = [utx, 0]T is the system input. The forces induced by the sea wave and current loads and the vessel motion are summarized in the vector / = [0, /i]T where 4.2 Control strategy In the following, only the vibration mode i has been considered. However, the model utilized in the control design can be extended to include other vibration modes. A controller based on the dynamic models presented in the previous section is developed. The objective of the controller is to control the position and velocity of the thruster system such that the position error rjx = r)x — r]Xd and the velocity error fjx = r/x are equal to zero, 7fXd is the desired position. Moreover the controller is designed to attenuate the vibrations induced in the system by the sea loads and the vessel motion. Consider the PID-controller (4.59) where kp, kd and kj are appropriately selected constants. The integral ac tion handles the sea current influence. The closed loop system is illustrated in Figure 4.2. 4.2.1 Stability analysis The environmental and the vessel motion disturbances are set equal to zero in the stability analysis. Consider the function v-5* Tp * (4.60) 4.2. Control strategy 63 V fi cable and Ixi Vx tool dynamics Figure 4.2: Block diagram of the closed loop system. where z = [rjx, fjx, f* fjxdr, g,]T and mt mt 0 7i-rac 0 mt kp + kd + di ki -yimc 0 P = 0 kj kj 0 0 (4.61) 7,-mc 7,mc 0 mc 0 0 0 0 0 ki Choosing the control parameters such that kp > ki and kd + d\ > mt the function V can be made positive definite. The time-derivative of V along the system trajectories of (4.53) is V = -fd 2rix - g (kd + di — - fd 2Vx - ^(kp - ki)f)l ~2ptp ~ ~ 2^d + " mt)vl -7i(c - mc)qifjx - cqf (4.62) The first four terms in (4.62) can be made negative semi-definite by selecting the control parameters kp and kd sufficiently large. This is seen from the following: Let the 1st two terms in (4.62) be denoted V\ and the 2nd two terms in (4.62) be denoted V2. Vi = -fd 2f)x ~kr)l (4.63) V2 = ~fd 2fjx ~ ^(*p - k^ffl (4.64) where k — A(fcd + d\ - mt). Consider the case (% — UCl) > 0 in (4.57). Then |%| = \Uci| + 6, 6 > 0 and V\ = ~kr)l - d262r)x (4.65) Lateral motion control of an underwater thruster module 64 connected to a slender marine structure < -k\Vxf + d 2S2\'nx\ (4.66) + (4,67) where |%| < o. Next consider (r)x - UCl) < 0 in which case |Z7Ci| = \r}x\ + 8, 8 > 0 and Vi - -kril + d 28 2fix (4.68) < —k\rix\2 + d 262\rix\ (4.69) < (-k + d 2^)rjl (4.70) Hence, by choosing kd such that d kd + d\ rrif > a V\ is negative semi-definite. Similarly, it is found that (4.71) 5 ~(g(kp ~ hi) - d2 —)fjl (4.72) where |%| < b. By choosing kv such that kP-kj>^f- V2 is negative semi-definite. The constants a and b are upper bounds on the velocity state and the position error state. (4.62) then becomes V < — g(&p - kj)rj2 - ~{kd + di — mt)\r)x\2 +\li{c - mc)\\qi\\r)x\ - c|&|2 < -g(tp - h)vl ~ g(&d + d 1-mt- - mc)2)|i7z|2 -c(|gi| - ^|7i(c - mc)||7?I|)2 (4.73) The function V is negative semi-definite for kp ^ kj 4.3. Simulations 65 and It follows from Lyapunov ’s stability theorem (Khalil 1992) that the unique equilibrium point z = [0,0,^UCl\UCl\,0,0]T (4.74) is stable and that z is bounded. Furthermore, it follows from the LaSalles’s theorem (Khalil 1992) that the equilibrium point is global asymptotically stable. 4.2.2 Robustness The environmental and vessel motion disturbances are assumed to be bounded by ||/i||2 < <$, where 6 is a positive constant. With ^ 0 the time-derivative of V becomes Vi = Qifi “ 5100 (4.75) where 5iW — —fdiVx ~ g(&d + d\ — mt)ril — fd 2Vx — g(&p — ;2 2 'X .2 'X (4.76) It was shown in the previous section that —V = g\ >0. With V\ lower bounded the system dissipates energy if and only if [%,%,<%] =£ 0. The energy input of the system is <&/; and 51 is the dissipated power. Since the system is globally uniformly asymptotically stable it is totally stable (Slotine & Li 1991), that is it has the ability to withstand small distur bances. 4.3 Simulations To illustrate the theoretical results and the properties of the closed loop system, simulation results are presented. The numerical data utilized in the simulations are given in Appendix B. The system has been simulated in Lateral motion control of an underwater thruster module 66 connected to a slender marine structure Figure 4.3: Position and velocity errors with PID-controller and fi = 0. Matlab using sampling interval of 0.01 s. The state variables were initially set to zero and the position reference was generated by the 2nd order system "b 2^Vxd d* A TjXd = A ho where A = 1 and the commanded input ho was chosen equal to 1. The disturbance function fi was selected as fi = Ci (sin(w n t) + cos(wn Z)) where uin is the most dominating wave frequency. The control parameters were chosen kp = 2000 (4.77) kd = 1000 (4.78) kt = 700 (4.79) Figure 4.3 shows the thruster system position error fjx and velocity rjx. As proved in the previous section the states fjx and yx are asymptotically stable and the desired position is achieved within 15 s. The cable/riser states % and qi are shown in Figures 4.3 and 4.5. It was shown in the previous section that qi and qi are asymptotically stable for fi = 0, simulations confirm the theoretical results. For /, =£ 0, the vibrations in the cable are damped and oscillate around zero. The control input shown in Figure 4.4 is stationary due to the constant current velocity. A small DC thruster will be capable of delivering a torque of 800 N. The cable/riser force acting on the thruster system is shown in Figure 4.6. Performance indices were 4.3. Simulations 67 «tx(N) Figure 4.4: Control input utx with fa = 0. Figure 4.5: Cable/riser vibration with fa ^ 0. Figure 4.6: Cable/riser force fCx acting on the thruster system. calculated for varying mass of the thruster module, see Table 4.1. The performance indices were Ji = rl^dt (4.80) K «=o J2 = r£9idt (4.81) K i=0 J3 = (4.82) K i-0 J4 = T^^dt (4.83) K i=0 where k was the number of samples. The system was simulated with a Lateral motion control of an underwater thruster module 68 connected to a slender marine structure Table 4.1: Performance indices with varying thrust weight. Mass Ji Ji Jz Ji 10 kg 0.0391 0.0187 0.0882 0.0013 80 kg 0.0147 0.0082 0.0880 0.0012 300 kg 0.0145 0.0082 0.0916 0.0011 Figure 4.7: Simulation results for system without a thrust unit connected to the cable/riser bottom end-point. mass module without thrusters connected to the bottom end-point of the cable/riser system. The results are shown in Figure 4.7. Position measure ment noise v\ ~ JV(0, oj) and velocity measurement noise V2 ~ 7V(0, a 2) were added to the position and velocity signals. Simulations were carried out with d 1 = 0.007 m and 02 = 0.01 m/s, that is a signal to noise ratio (S/N) of approximately 1%. The simulation results are shown in Figure 4.8. Simulation results with S/N approximately equal to 10% are given in Fig ure 4.9. 4.3.1 Frequency response The frequency response of the closed loop system was found by linearizing the drag function / /d, = ^ (4.84) dtyr f)x= 0 4.3. Simulations 69 -1000 time (s) time (s) -0.1 - time (s) Figure 4.8: Simulation results under noisy position and velocity measure ments, S/N=l%. The linearized model of the total system can be written MtX + Dlx + Ktx = u + f (4.85) where d\ + 2^2 C7Cj 0 Dl = (4.86) 1iC c The frequency response of the system is shown in Figure 4.10. The lin earized system is robust to disturbances with frequencies ujn < 1 rad/s. 4.3.2 Constrained modes The numerical values of the constrained modes of the system are given in Table 4.2. The cable/riser system will have resonances with periods equal to 4.24 s, 1.87 s, 1.07 s, and so on. The calculation of the resonances is based Lateral motion control of an underwater thruster module 70 connected to a slender marine structure 0 10 20 30 time(s) time (s) time (s) Figure 4.9: Simulation results under noisy position and velocity measure ments, S/N=10%. Frequency (mtkeee) Frequency (na/mc) Figure 4.10: Transfer function from desired reference to the measured end point position for the linearized model (4-85). 4.4. Discussion 71 Table 4.2: Constrained modes for the cable/riser simulation model. Ui (rad/s) II Modes hi S 1 33.44 1.49 4.21 2 169.8 3.36 1.87 3 517.4 5.87 1.07 4 1256 9.15 0.69 on the assumption that the effective axial tension can be approximated by an average effective tension. The first three modes are approximately equal to the natural frequencies of a straight tensioned string. The values give an indication of the resonance frequencies of the cable/riser simulation model. 4.4 Discussion Figure 4.3 illustrates the system performance using the PID-controller for the thruster unit control. The cable/riser vibrations are suppressed and the cable/riser bottom end-point stabilizes at the desired position. The cable/riser force fCx is most dominating on the thruster module in the transient phase, see Figure 4.6. The compensation of fCx is important to avoid large transient swinging motions in the system. Environmental disturbances result in increased vibrations and oscillations in the system, however the system is proved to be robust to limited disturbances, see Figure 4.5. Frequency response of the simulation model with the given design parameters, shows that disturbances with frequencies below 1 rad/s are damped. As shown in Figures (4.8) and (4.9), a similar time-behavior was achieved for noisy position measurements. With a signal to noise ratio of 10% the state vector x and control input r contain quite some noise. This illustrates that the system is robust to limited measurement noise. Table 4.1 shows that the system performance is improved by increasing the weight of the thruster module. A heavy module attached to the bottom end-point of the cable/riser will reduce vibrations in the system. Figure 4.7 shows that by removing the thruster forces, a constant vibration and increasing position deviation is induced in the system due to constant current flow. Lateral motion control of an underwater thruster module 72 connected to a slender marine structure 4.5 Conclusions This chapter describes modeling and control of a system consisting of a cable/riser connecting a surface vessel to an underwater thruster system. The main aim of this work was to show that by controlling the cable/riser bottom end-point, vibrations in the system are reduced and oscillations are removed. A stability analysis of the system has been presented and shows that a PID-controller for position control of the thruster system gives good system performance and that the system is robust to limited environmental disturbances. Simulations illustrates these results. The control system has desirable properties like simplicity, robustness and ease of implementation. The process of connecting risers to deep ocean well heads can be more time-efficient and robust by connecting thrusters to the bottom end-point of the riser. Chapter 5 Control of an underwater robotic system connected to a slender marine structure in 3 DOF The system presented in Chapter 4 is here extended to describe motion in 3 DOF. The system considered in this chapter consist of a slender marine structure connected to a surface vessel at the top end and to a robotic system at the bottom end. The structure is connected through ball-joints to the vessel and robotic system. Wave and current loads give rise to in plane and out-of-plane structure motions. The structure is also affected by the vessel motions. The equations of motion for the slender marine structure are developed from beam theory and are found in Sarpkaya & Isaacson (1981) and Weaver et al. (1990). The model presented for the underwater robotic system is found in Fossen (1994). Defining the wave and current loads and vessel motion as a system distur bance, a PD-controller with gravity compensation is used to control the position of the robotic system. The closed loop system is shown to be asymptotically stable. The models of the slender marine structure, robotic system and surface vessel are combined in the following, taking into account the reaction forces between the systems. A coordinated control of the vessel and robotic system is proposed and the total system is shown to be asymptotically stable. Control of an underwater robotic system connected to a slender 74 marine structure in 3 DOF Figure 5.1: Coordinate frame assignment to the vesselrobotic system and cable/riser system. This chapter is organized as follows: In Section 5.1 the equations of motion for the cable/riser system and robotic system is presented. A stability analysis is presented in Section 5.2. Coordinated position control of the vessel and robotic system is presented in Section 5.3. Section 5.4 hold the concluding remarks. 5.1 Dynamic models 5.1.1 Coordinate frames A coordinate frame is fixed on the surface vessel and is denoted S. The position of the surface vessel is described relative to an inertial coordinate frame I. A frame C is fixed some place along the cable/riser axis. The z- axis is vertical and passed through the connecting joint between vessel and cable, while the x- and y-axis are placed to coincide with the in-plane and out-of-plane motions, such that a right-hand system is formed. Similarly a frame is fixed on the robotic system and denoted B. The position of the robotic system is described relative to the inertial frame. The vector p0 is the vector from the cable/riser frame C to frame I. The frame assignment is illustrated in Figure 5.1. 5.1. Dynamic models 75 5.1.2 Dynamics of an underwater robotic system The equations of motion of a robotic system operating in the inertial frame can be written in the matrix-vector form Mv{r)1)ij1 + Cv(T}1,rj1)rj1 +Dv{r}l:rjl)ri1+g v = Tv +fc (5.1) where the vector % € 1R3 is the position vector with coordinates in the inertial frame I. The matrix Mv is the mass/added mass matrix, Cv is the matrix of Coriolis and centrifugal forces, Dv is the matrix of hydrody namic lift and damping terms and g v is the vector of gravity and buoyant forces. The vector r„ E R3 is the vector of thruster forces and fc are the cable/riser forces acting on the robotic system. 5.1.3 Dynamics of the cable/riser system The equations of motion of the in-plane, out-of-plane and longitudinal mo tion of the cable/riser system are expressed in the cable frame C. The drag forces act in the in-plane and induce in-line vibrations while the lift forces act in the out-of-plane and induce cross-flow vibrations. The longitudinal equation of motion of an underwater cable/riser can be described as an extension of the longitudinal motion of a beam (Weaver et al. 1990) -EA- - + cu(z, t) + mcii(z, t) = fz(x, y, z, t) (5.2) where EA is the axial stiffness, c is the structural damping and mc is the total mass of the cable/riser per unit length. The longitudinal displacement u(z,f) is along the cable/riser axis and varies with the time t and the longitudinal position z, where z is the equilibrium position of a point on the cable/riser axis. The function fz is the fluid loading acting along the cable/riser axis. The in-plane equation of motion of an underwater cable/riser can for small angles of deflection be described by the differential equation (Chakrabarti & Frampton 1982), (Sarpkaya & Isaacson 1981): d 2v{z,t) d dv(z,t) ) ~ Tz{%(z) ) + cv(z,t) dz 2 dz 2 dz +mcv(z,t) = fx(x,y,z,t) (5.3) Control of an underwater robotic system connected to a slender 76 marine structure in 3 DOF where El is the cable/riser stiffness, Te is the cable/riser tension, c is the structural damping and mc is the total mass of the cable/riser per unit length. The displacement v(z, t) is modeled to be normal to the cable/riser axis. It is reasonable to assume that the cable/riser is designed so that the stiffness is constant, EI(z) = EL The tension Te(z) is approximated by the mean value T. The function fx is the in-plane fluid loading. The out-of-plane motion is given by the equation +mcw(z, t) = fy(x, y, z, t) (5.4) where w(z, t) is the out-of-plane displacement and fy is the out-of-plane fluid loading. It is assumed that the coupling between the in-plane and out-of-plane motions are dominated by the fluid forces. The functions fx, fy and fz includes vortex-induced forces, wave and current forces. Vortex- induced forces lead to correlated in-plane and out-of-plane motions, and are complicated to derive. Also, models of three dimensional wave- and current loads are difficult to establish. Therefore fluid forces are usually approxi mated in terms of Morisons ’ equations. In the following it is assumed that the fluid loads give rise to motion in a fixed x-y plane. Natural vibration modes The natural vibration modes of the cable/riser in the x, y and z-plane can be found by solving the homogeneous equations (5.5) (5.6) (5.7) The method of separation of variables give solutions for mode i in the form (Meirovitch 1986) (5.8) (5.9) (5.10) 5.1. Dynamic models 77 Inserting (5.8) into (5.5) yields two differential equations mc4>i + c (5.16) and relative damping C,x, = . c The second differential equation is a 2 ^kx^Tflc function of the longitudinal displacement z El Pi (z) ~ T pz (z) - kXipi(z) = 0 (5.17) and has a solution in the form Pi{z) = AjjSinhnjjZ + Ai2 coshn 2l z + Aj3sinrij 2z + A,4 cosn; 2 z (5.18) where 1 iJ^j(2T + 2y/T^+4kXiEI) (5.19) 1 ]/-±I(-2T + 2yjTi + 4kXiEI) (5.20) The coefficients of (5.18) and the constrained modes can be determined using the boundary conditions. Similar results are achieved by inserting (5.10) into (5.7). Control of an underwater robotic system connected to a slender 78 marine structure in 3 DOF Boundary conditions The boundary conditions at the top end-point of the cable/riser, z — 0, are given by the vessel motions r(t)g(z ) in the I-frame v(0,t) w(0, t) = (p 0 4- R^r(t))g u(0, t) = Cr(t)g= C[rI,ry,rz]Tg (5.21) where Rq is the transformation matrix between the inertial frame and the cable/riser frame. The function g{z) = diag {g x{z),9 y{z),g z{z)) is the static deflection with the vessel displaced by one unit. The boundary conditions at the bottom end-point of the cable/riser, z = l, are given by the movement of the robotic system r^hiz) in the I-frame w (l, t) = (p 0 + RcVi (0)^ u(l, t) = CVi(t)h = C[Vix,Viy,Vu}Th (5-22) where the function h(z) = di&g(h x(z),hy(z),hz(z)) is the static deflection with robotic system displaced by one unit. Pinned cable/riser connections at the bottom end-point of the cable/riser give the boundary conditions (5.23) (5.24) Equations of motion of the cable/riser The general solution to (5.5-5.7) is determined by adding the vibration motion to the end-point motions (Daring & Huang 1979) OO v(z, t) = g x(z) Crx(i) + hx{z) C»?lz (t) 4-^]%(z)%(t) (5.25) 5.1. Dynamic models 79 00 W (z, t) = 9y (z) cry(i) + h y(z) c77ly (() + ]T #i(z)0i(t) (5.26) 1=1 OO u(^,*) =9z(z) Crz(i) + /iz(z)c77lz(t) + 53 Qi + 2(XiujXiqi + w^.% = —Gxi (5.28) "lc 6i + ^Cyi^Vi + Uy.Qi — —Gyi (5.29) 77lc GZi = tr of u. f W(z)[A(z,W,z,t) - C9z{zfr z{t) - chz(z)cr)lz(t) /0V?|(z)dz Jo -mcg z(z)CTz{t)-mchz{z)C;qlz{t))diZ (5.33) The cable/riser force on the robotic system is found applying the reaction- action principle (Newton ’s third law) 1 ____ 55 'W = ~mc'YiRtf Oi = —mc7i-R/ *i (5.34) A where x{ = and 7{ = diag[7Zj,7yi,7Zi] where 1 fl 7x * = 2/ \, / Pi(*)M*)d* (5.35) f0pf(z)dz Jo Control of an underwater robotic system connected to a slender 80 marine structure in 3 DOF (5.36) (5.37) The equations of motion of the cable/riser and robotic system can be ex pressed in the form ^v{rh)rii +Cv{'nl,ijl)ijl + Dv(rjl,Tj1)rjl + gv = t - % (5.38) mcXi + c£i + KiXi + crfi cijl + mc7j crj1 = /i (5.39) where = diag{kXi,kyi,kZi). The forces induced by the sea wave and current loads and.the vessel motion are summarized in the vector = Pi(z)[fx(x, V, z, t) - cg x(z)crx - mcg x(z)crx]dz ■di(z)[f y(x,y,z,t) - cg y{z)cry - mcg y{z)c ry)dz 1 - .MWdz/o"...... — - 5.2 Control strategy Let the vessel be kept at rest by a DP-system, and consider the PD- controller with gravity compensation for control of the robotic system = -Kvrjl - Kd rj1 + g v (5.40) where Vi — Vi ~ Thd denotes the position error and r\ld is the desired position. The design matrices Kp and Kd are positive definite matrices. 5.2.1 Stability analysis Stability of the system can be shown using Lyapunov theory . The distur bance vector fi is set equal to zero in the stability analysis. Consider the function 1 1 V = + -mcxjxi+mc7}1T'yiR + \vTKXi + ^Tf^lCpTh (5.41) 5.3. Coordinated position control of the robotic system and surface vessel 81 (5.41) is positive definite for sufficient large mass Mv and mc. Utilizing the skew-symmetric property yT(Mv - 2Cv)y = 0 established for underwater robotic systems (Fossen 1994), the time-derivative of V becomes V = -rjxr(Kd + Dv)rjx - cxjii - cxjjirjx < - (Am(Kd ) + ||Dy||) ||7h|| 2 - c||ii||2 + c||7i|III^IINill = - (W-K-j) + IID.II - jlhill2) Will2 - C (lliill - l|l7illlWill)J (5.42) where Xm(Kd ) is the largest eigenvalue of Kd . The function V is negative semi-definite for Am(lifd) + ||Dv||>^||7i||2 (5.43) It follows firom Lyapunov ’s stability theorem that the system is stable, and according to LaSalle’s theorem the system is globally asymptotically stable. The system is thereby robust to small environmental disturbances. 5.2.2 Discussion PD-control(5.40) of the robotic system yields a globally asymptotically sta ble system. The system is robust disturbance, /t- , consisting of wave and current loads and vessel motions. In a real system the design matrix Kd must be bounded above to limit the influence of measurement noise in the system. Integral action can be included in the controller and the stability proof will then proceed as in chapter 4. 5.3 Coordinated position control of the robotic system and surface vessel The problem of moving a surface vessel holding slender marine structures from one position to another is a challenging one. This due to vibrations and swinging motions (oscillations) which arise in the structure due to Control of an underwater robotic system connected to a slender 82 marine structure in 3 DOF current and wave loads. By attaching thruster modules to the bottom end point of the structures and by coordinating the control of the vessel and thrusters this operation can be performed securely and efficiently. The vessel motion in surge, sway and heave can be written in the form (Fossen 1994) Ms(r)r + Cs(r, r)r + Ds(r, r)r = rs (5.44) where r is the vessel position vector in the inertial frame, Ms is the added mass/mass matrix, Cs is the Coriolis and centrifugal matrix and Ds is the damping matrix. The vector rs is the vector of thruster forces. Wave filtering of the vessel states is assumed to be included in the model. The vessel is not affected by the cable/riser motion. The cable/riser system is assumed to be connected to the vessel such that it is not affected by the vessel motion in roll, pitch and yaw. The cable/riser motion given by (5.39), may be expressed mcXi + cxi + K&i + crfi crjl + crjx + c/3* cr + mc0j cr = fs. (5.45) where & = diag(/3Xi, 0yi, (3Zi) and ^ = j^-JlPi{z)9Az)iz (546) ^ = j^-JlUz)9 ^z)iz (5-47) 1 fl Pzi = ri / The sea wave and current loads are summarized in the vector /Sl = [fsxi, fsyt, fsziF where 1 [l faxi = fi 2. / Pi{z)fx(x,y,z,t)dz (5.49) IoPi(z)dz Jo 1 fm = ri a2/ / Mz)fy(x,y,z,t)dz (5.50) f0i9f( z)dz Jo 1 fs*i = rl 2/ / 5.3.1 Stability analysis of the total system with PD-control For the surface vessel, consider the PD-controller given by rs = —KPa r - Kda r (5.52) where f is the position error vector and KPa and Kda are positive definite design matrices. The vector fs is set equal to zero in the stability analysis. Consider the positive definite function V V = + ^mcxJxi + mcTj^jiRfxi + ixjKxi \rTMsr + mc^^Rfxi + \r^KPa r (5.53) The time-derivative of V along the system trajectories is V = »?iT(rv - 9V) ~ 'ni£Dvtil - cxjxi - cxJjiVx -rT(Kda + Ds)r - cxj far + mcrTfaRfxi (5.54) The mapping from tv to ij1 is passive if the controller r„ includes a gravity compensation term g v, and if the system possess sufficient damping Dv and Ds V = rj^Ty - rhT(Dv - |7?7i)»7i - |(*i - ~ -rT(Kda +DS- j(3jfa - mca xvs)r -|(*i - \p&)T{xi - i/3ff) (5.55) where ||sj|| < a x and ||f|| < vs. Asymptotic stability of the total system is ensured by including a damping term Kd rjl in the controller rv. For the robotic system, consider the PD-controller with gravity compensation tv = -Kvi\x - Kd r}1 + g v (5.56) Let V\ be given by Vl = V + \rjiTKprjl (5.57) Control of an underwater robotic system connected to a slender 84 marine structure in 3 DOF The time-derivative of V\ along the system trajectories is Vi = -rj^iKd + Dv)ijx - cxjxi - cxJ'TiVi + mcrrfaR$Xi -rT(Kds + Ds)r - cxj far < ~{\m{Kd ) + ||DV|| - ^c||7i||2)h- 1||2 - \c(\\xi\\ - bill Will)2 ~(Xm(Kda) + ||DS|| - -c||/3i||2 - mcazvs)||r||2 -^(Hxill - li^H ||r||)2 (5.58) where Am indicates the largest eigenvalue of the matrix. The function Vi is negative semi-definite for W*d)-H|D,||>^b,f (559) and Am(Kda ) + ||DS|| > |||ft||2 + meax vs (5.60) It follows from LaSalle’s theorem that the system is asymptotically stable. This implies that the system is robust to limited environmental distur bances. 5.3.2 Discussion With linear static deflection functions g x,9y,9z and hx,hy, hz the diagonal elements of the matrices 7j and fa consist of cosines and sinus functions which have maximum norms equal to 1. (5.59) then becomes AmC^d) + ||D„|| > | and (5.60) becomes Am(^ In a real system the design matrices Kd and Kds must be bounded above to limit the influence of measurement noise in the system. The PD-controllers applied in the present work requires the measurement of the position and the velocity of the robotic system. However, velocity signals can be gen erated from the position measurements by using observers. Integral action 5.4. Conclusions 85 can be included in the controller, to counteract the current forces fs.. In the proposed coordinated control scheme integral action can only be in cluded in one of the control loops (5.52) or (5.56). If included, the integral action should be included in the vessel controller since static position error of vessel will result in large position error for the total system. 5.4 Conclusions A PD-controller with gravity compensation has been applied to control a robotic system in 3 DOF, connected to a surface vessel by means of a cable/riser system. The closed loop system is shown to be asymptotically stable and thereby" robust to limited environmental disturbances. Coordinated control of the vessel and robotic system has been suggested and the total system is shown to be asymptotically stable. This scheme can be applied when transferring surface vessels holding marine structures from one position to another. Control of an underwater robotic system connected to a slender 86 marine structure in 3 DOF Chapter 6 Conclusions and recommendations 6.1 Conclusions The presented matrix-vector form of underwater vehicle-manipulator dy namics has made it feasible to establish mechanical properties like symme try, skew-symmetry and positiveness for the system. This makes it possi ble to use nonlinear control design and Lyapunov theory in the stability analysis. Well-established control algorithms developed for underwater ve hicles and robot manipulators can be extended and applied to underwater vehicle-manipulator systems. The feedback linearisation technique has been applied to the vehicle-manipulator system and performance and robustness properties evaluated by simulation. The evaluation showed that with small model uncertainties, decoupling of the manipulator end-effector velocity from the vehicle velocity was preferred compared to the scheme of decoupling the manipulator end-effector velocity from the total system momentum. This was in contrast to the results achieved for spacecraft-manipulator systems. Passivity-based controllers, where velocity estimates are derived from po sition measurements, have been proposed. Model-based controllers have the advantage that they are more robust to parameter and structural un certainties than controllers based on the feedback linearisation technique. In some underwater operations set-point control of the vehicle and mo tion control of the manipulator is necessary. A control scheme has been suggested for this purpose, and perfect manipulator tracking is achieved 88 Conclusions and recommendations without perfect knowledge of the vehicle dynamics. The vehicle controller and manipulator controller can be implemented with different bandwidths, using the vehicle for slow-gross positioning and the manipulator for fast joint-space tasks. This reduces the controller action of the vehicle actu ators and provides a large work space for the system. Performance and robustness properties have been illustrated by simulation. A controller for coordinated control of the vehicle and manipulator in world coordinates has been proposed. The controller has a similar structure as controllers developed for robot-manipulators. The controller consist of a feedforward term and a PD-term. Stability and tracking convergence has been proved for the closed-loop systems. Attaching a position and velocity controlled robotic system to the bottom end-point of a cable/riser connected to a surface vessel at the top end results in vibration and oscillation attenuation in the system. Good performance of the robotic system is achieved by using the classical PID-controller. A stability analysis of the system has been presented, showing global asymp totic stability of the system and robustness to small disturbances. The control system has desirable properties as simplicity, robustness and ease of implementation. This concept will make the operation of connecting a production riser to a well head more time- and cost-efficient. The thruster unit can be loosened from the riser after the connection has been made, by releasing a locking mechanism or by the help of an underwater vehicle- manipulator system. 6.2 Recommendations for further work Suggestions for further research are: • Verify the modeling of underwater vehicle-manipulator system dy namics by comparing simulation results with data obtained from an actual system. • Consider hydrodynamic forces which arise due to the vehicle motion, in the modeling of the manipulator. • Perform experiments to evaluate the most efficient control strategy for motion control of the vehicle and manipulator considering imple mentation, performance and robustness properties. 6.2. Recommendations for further work 89 • Connecting the proposed simulation model to a graphical represen tations of the vehicle-manipulator system in its environments, thus making it possible to plan, run and test underwater operations in advance. • Analyze the system consisting of a cable/riser connecting a surface vessel to an underwater robotic system, considering structural and parametric uncertainty in the cable/riser model. • Expand commercial simulation programs for cable/rise system to in clude the dynamics of an underwater robotic system. This to verify the oscillation and vibration damping obtained by controlling the bottom end-point of the cable/riser system. 90 Conclusions and recommendations References Balchen, J. G. & Yin, S. (1994). The MOBATEL Project, Technical report, The Norwegian Institute of Technology, Trondheim, Norway. Berghuis, H. (1993). Model-Based Robot Control: From, Theory to Practice, PhD thesis, University of Twente, Enschede, The Netherlands. Blevins, R. 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Appendix A Vehicle—manipulator simulation model The simulation model consists of a 6 DOF autonomous underwater vehicle holding a planar two-arm manipulator. The parameters in the vehicle model were based on the experimental values derived for the Norwegian Experimental Remotely Operated Vehicle (NEROV) (Fossen 1991). The planar two-arm manipulator was chosen to have cylindrical links and revolute joints. The mass of the manipulator link was set to 25 kg and the link had diameter d = 0.1 [m] and length l = 0.5 [mj. The manipulator dynamics were generated recursively in Maple, and linear and quadratic friction forces and current loads were included in the model. The hydrodynamical friction coefficients were functions of the angle of attack, angle of sideslip and the Reynold number. It was assumed that the fluid had Reynolds number Rn = 105 and that Cd = 0.02 + 0.25a2, Cl — 0.25a, Ds = 100, Cip = —0.1a, Cmq = —0.1a and Cnr = —2/3. The Strouhal number was chosen St = 0.2 and the phase angle 7 = 0. The flow velocity Vf [m/s] was generated by a 1st order Gauss-Markov Pro cess Vf(t) = w(t) — yrVf(t), were w(t) was a zero mean Gaussian white noise process and T was the sampling time. Further, Vf was limited by 0.1 < Vf < 0.6. 96 Vehicle-manipulator simulation model Appendix B Numerical data of cable/riser system Cable/riser data Structural damping c = 5 Ns/m2 Added mass coefficient Cm ~ 1 Drag coefficient Cd = 1.6 Outer diameter Do = 0.1 m Stiffness El = 4109 Nm2 Total mass mx = 15 kg/m Mean tension T = 1.11106 N Thruster system data Cross section area A — 1 m2 Friction coefficient di = 100 Mass/added mass mt = 80 kg Wave and current data Water density p — 1025 kg/m3 Wave amplitude Co = 1 m Wave frequency wn = 0.5 Hz Disturbance amplitude Ci = 40 Depth l = 600 m Current UCl — 1 m/s