MEROS Project. Technical Advances in Modeling and Control. Dr. Lotfi CHIKH February (33)

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1 MEROS Project Technical Advances in Modeling and Control Dr. Lotfi CHIKH February (33)

2 Contents 1 Introduction 3 2 Dynamic model of the ROV: the mathematical background 4 3 Matlab/Simulink ROVs Simulator Example of a simple Cubic configuration MEROS ROV Modeling L 1 Adaptive control Control objective Formulation of the L 1 Adaptive controller Some comments Formulation of the ROV model for L 1 adaptive control theory Conclusions et perspectives 14 Appendices 16 A Jacobian transformation matrix J(η) 16 B Calculation of the inertia M, damping D and restoring forces g 16 B.1 Inertia matrix M B.2 Restoring forces g

3 1 Introduction This reports summarizes the advances in modeling and control of the MEROS project. The aim of the project is the design, modeling and control of a 6 DOF compact and low cost Remotely Operated Vehicle (ROV) for inspection tasks for depths less than 100m. The user implication in the ROV control has to be as minimal as possible, in order to limitate the infrastructures and the people concerned by the inspection task. The ROV must be able to handle three main tasks: Station keeping en one point of the workspace. Moving along a desired trajectory given by the operator. Displacement along a virtual plane surface. However, two of the six degrees of freedom (DoF) cannot be measured experimentally. Therefore, only four DoF are controlled. The two remaining ones (translational x and y) will be controlled in open-loop by an operator. Therefore, we will speak about co-control of the ROV as we have a combination of DoFs controlled by and operator and DoFs controlled by the navigation system. Two kinds of constraints have been identified: 1) constraints related to the design and 2) the constraints related to the modeling and control of the ROV. For the first family of constraints (concerning design), many efficient an reliable technologies in case of large/middle size ROVs cannot be integrated in a small, compact and low cost ROV. This implies an intelligent design strategy that optimizes the size and cost of the component whilst as long as possible it does not limitate the performances. The second family of constraints (linked to control of the ROV) includes: Imprecisions and uncertainties due to bad parameters estimation of the dynamic model (buoyancy, damping,... ) or to parameter changes during ROV functioning. Some variables cannot be measured and even their estimation is not accurate enough to be used in the control strategy. The good functioning and performances have to be guaranteed in presence of disturbances that can affect the ROV (waves, currents, seaweeds). The umbilical will have an important destabilizing effect that can lead to a big energy consumption from the actuators. This can deteriorate the performances and even lead to the unstability if there is a saturation of the motors. After the presentation of the ROV dynamic model equations, different ROV configurations are simulated in order to study the ROV behavior, after that the basis of the L 1 adaptive control theory are introduced and its application to our ROV will be discussed. 3

4 2 Dynamic model of the ROV: the mathematical background The dynamic model of a ROV relies the forces applied by the thrusters to the positions, velocities and accelerations of the vehicle. Applying Newton s second law, it is possible to obtain the classical model used in naval architecture. This model is complicated to use due to the large number of hydrodynamic coefficients that are involved. That is why the vectorial model representation introduced for the first time in [5] will be used. It is inspired from Craig s robot model [4] which is commonly used in robot control theory. The difference in case of ROVs comparing to robot manipulators is that it takes into account additional physical phenomena typical to underwater environment. A general dynamic model of a ROV that will be used for the controller synthesis was proposed by Fossen [6], and is based on SNAME notations. This model takes into account two types of physical contributions: the rigid body dynamics corresponding to the Newton-Euler formulation for rigid bodies (this formulation is classically used for robot manipulators); the hydrodynamic forces and moments that combine environmental disturbances (wind, waves, currents) and the so-called radiation induced forces [6] which are: i) the added mass due to the inertia of the surrounding fluid, ii) the radiation induced potential damping and iii) the restoring forces due to Archimedes (weight and buoyancy). The general formulation of the ROV model takes the following matrix form: η = J(η)ν M ν + C(ν)ν + D(ν)ν + g(η) = τ + w d (1) where ν = [u, v, w, p, q, r] T, η = [x, y, z, φ, ϑ, ψ] T represent respectively the vector of velocities (in the body fixed frame) and the position/euler angles (in the earth fixed frame). J(η) R 6 6 is the jacobian transformation matrix from the body fixed frame to the earth fixed frame. Matrices M, C and D represent respectively the inertia, Coriolis and Centripetal forces, and damping. g is the vector of restoring forces due to Archimedes (weight and buoyancy). τ denotes the control inputs while w d is the vector of external disturbances. In this study, one common assumption will be taken. It is supposed that the Coriolis terms can be removed (C(ν) 0) as the ROV will move at relatively low velocities. The expression of J is given in appendix A while appendix B tackles the issue of calculation of M, D and g. 4

5 Using the kinematic transformations [6], η = J(η)ν η = J(η) ν + J(η)ν M (η) = J (η)mj 1 (η) D (ν, η) = J (η)d(ν)j 1 (η) g (ν) = J (η)g(ν) τ = J (η)τ (2) w d = J (η)w d it is possible to express the ROV body-fixed frame dynamics into Earth fixed frame dynamics: M (η) η + D (v, η) η + g (η) = τ + w d (3) 3 Matlab/Simulink ROVs Simulator In the present section, some preliminary simulations are done in order to study the physical equations of the dynamic model and validate it. We mean by validation the observation in simulation of a good and logical behavior of the ROV whilst one or several thruster are actuated. The simulations are run within Matlab where the equations (1) are implemented. The developed simulator has to take into account the following points: good understanding of the physics phenomena. We will simulate some simple scenarios in order to observe and judge the ROV behavior. flexibility, as the thrusters configuration can be changed easily leading to different ROV configurations. visualization of the ROV movement. This is not the most crucial point but it can be useful to observe the movement of the ROV and validate the simulator. 3.1 Example of a simple Cubic configuration As a first step, a simple 6 DOFs ROV corresponding to a Cubic ROV [8] is simulated. It is represented in figure 1 The choice of Cubic ROV is motivated by its simplicity. Indeed, its exclusively either horizontal either vertical thruster configuration makes easy to understand which thrusters have to be activated in order to generate a movement in some direction of the workspace. Table 1 one summarizes this, It is then easy to deduce the expression of the configuration matrix T Cubic T Cubic = (4)

6 z (5) (4) (2) (3) y x (1) (6) Figure 1: Thrusters configuration positions of the CUBIC ROV example Table 1: Forces and corresponding thrusters in case of Cubic ROV Forces and moments Activated thrusters X: motions in the X direction (surge) 1 and 2 Y: motions in the Y direction (sway) 5 and 6 Z: motions in the Z direction (heave) 3 and 4 K: rotation about the x-axis (roll) 5 and 6 M: rotation about the y-axis (pitch) 3 and 4 N: rotation about the z-axis (yaw) 1 and 2 In figure 2, a graphical simulation of Cubic ROV is shown. As the dynamical parameters of the model (1) are not available, we have used some parameters from an existing ROV studied by our colleagues of Lirmm. The forces are displayed in red and the blue surface represents a virtual water surface. Different elementary tests were performed in order to assess the effectiveness of the model, among them: A simple test corresponding to a movement in the X direction (surge) is now performed. Figure 3 shows that when applying the same force with an inverse amplitude to thrusters 1 and 2, this leads to a movement in the positive X direction, which is the expected result. We also observe a change in the ROV depth which is also a natural result as the buoyancy was chosen larger than the weight. This is a common choice in ROVs and UAVs because in case of thrusters failure, the buoyancy of the ROV enables it to surface by his own action. By choosing a buoyancy value which is two times larger than the ROV weight, without any force on the thrusters, one can observe a more clear surfacing of the ROV (figure 4). By increasing the coefficient of the damping matrix D, for the same thruster forces, a slower ROV movement is observed. 6

7 Figure 2: Graphical Simulation of Cubic ROV: the forces are displayed in red and the blue surface represents a virtual surface water Figure 3: Applying forces on the thrusters 1 and 2 leads to a movement along the X axis Figure 4: Increasing the buoyancy without applying forces on the thrusters leads to a faster surfacing of the ROV (green line corresponds to the ROV trajectory) 7

8 3.2 MEROS ROV Modeling A CAD representation of the omnidirectional MEROS ROV is shown in Figure 5a. It is composed of 6 thrusters that provide movements and forces in the 6 DoFs. Figure 5b shows the positioning of the thrusters that corresponds to a tetrapod configuration [8]. (5) z x (4) (1) (6) (2) (3) y (a) CAD representation of the Meros ROV (b) Thrusters configuration positions corresponding to Meros ROV Figure 5: Meros ROV representation The deduction of the configuration matrix of the MEROS ROV is more complicated than for the Cubic ROV. For Meros, the tetrapod disposal of thrusters involves more thrusters in each DoF. Table 2 one summarizes this, Table 2: Forces and corresponding thrusters in case of Meros ROV Forces and moments Activated thrusters X: motions in the X direction (surge) F3 x, F4 x, F5 x and F6 x Y: motions in the Y direction (sway) F y 1, F y 2, F y 5 and y 6 Z: motions in the Z direction (heave) F1 z, F2 z, F3 z and F4 z K: rotation about the x-axis (roll) F3 z, F4 z, F y 5 and F y 6 M: rotation about the y-axis (pitch) F1 z, F2 z, F5 x and F6 x N: rotation about the z-axis (yaw) F y 1, F y 2, F3 x and F4 x We recall that the force/torque vector applied on the ROV is given by the following expression: τ = K T Meros u + τ r (5) where T Meros represents the configuration matrix, u is the electrical control input, τ r is the reaction torque vector and K is the force coefficient. From table 2, it is now easy to deduce the expression of the configuration matrix T Meros : 8

9 T Meros = s s 0 0 s s 1 1 s s s s s s s s 0 0 s r s r 1 1 s r s r 0 0 s r s r s r s r s r s r s r s r 0 0 where s represents sin( π ) and r is the radius of the sphere where the thrusters 4 are placed. Additional terms due to the reaction torques generated by the thruster rotations have to be taken intro account. They are given by the following expressions: (6) τ r φ = k(ω 3 ω 3 ω 4 ω 4 ω 5 ω 5 + ω 6 ω 6 )s τ r θ = k( ω 1 ω 1 + ω 2 ω 2 + ω 5 ω 5 + ω 6 ω 6 )s τ r ψ = k(ω 1 ω 1 + ω 2 ω 2 ω 3 ω 3 ω 4 ω 4 )s (7) with τ r = [0, 0, 0, τ r φ, τ r θ, τ r ψ ]. The inertia matrix is given by: M = M RB + M A (8) where M RB is the rigid body inertia matrix which is unique and satisfies M RB = MRB T > 0 and ṀRB = It is given in case of MEROS by: M RB = m m m I x I y I z where m is the mass and I x = I y = I z represents the moments of inertia. At the other hand, M A is the added-mass inertia. It represents the inertia of the surrounding fluid. M A = M A = diag{x u, Y v, Zẇ, Kṗ, M q, Nṙ} (10) During simulations, the values of the added mass coefficients are taken equal to 40% of the values of the rigid body inertia terms. The terms of damping will be chosen arbitrarily, and readapted once they will be identified. The remaining term of dynamic model of equation 1 is g corresponding to the restoring forces due to Archimedes (weight and buoyancy). The complete expression is given in appendix B. In case of MEROS, the buoyancy will be chosen 2% greater than the weight. (9) 9

10 4 L 1 Adaptive control L 1 adaptive control refers to a very recent adaptive control technique created by Chengyu Cao and Naira Hovakimyan in 2006 [3] [1]. The journal paper appears two years later [2] and the book in 2010 [7]. It can be viewed as a modified Model Reference Adaptive Control (MRAC ) scheme where the basic architecture is based on an internal model principle. The motivation of this novel technique is that in conventional adaptive controllers (in opposition to L 1 ), fast adaptation which is necessary for improving performances, was leading to high frequencies in control signals and increased sensitivity to time delays [7] which will affect robustness. This motivates the development of an architecture that allows fast adaptation without losing robustness which is the main contribution of L 1 adaptive control theory which decouples adaptation from robustness. This is possible because of two main theoretical results: the error norms are (uniformly) inversely proportional to the square root of the adaptation gains. High values of the adaptation gains are thus advantageous. the control signal is filtered to avoid high frequencies. The filter is also used to shape the nominal response. The conditions are given in terms of L 1 -norms of certain transfer functions that involve the filter and the largest values of the unknown parameters. That is why it is called L 1 adaptive control theory. 4.1 Control objective Consider the following SISO system dynamics ẋ(t) = Ax(t) + b(u(t) θ x(t)) y(t) = c x(t) x(0) = x 0 (11) where x(t) is the system state vector (supposed to be measurable), u(t) R is the control signal, b, c R n are known constant vectors, A is a known n n matrix, (A,b) is controllable, the unknown parameter θ R n belongs to a given compact convex set θ Ω and y(t) R is the regulated output. The control objective is to design an adaptive controller to ensure that the system output y(t) follows a given reference signal r(t) with quantifiable transient and steady-state performance bounds. 4.2 Formulation of the L 1 Adaptive controller Consider the following structure u(t) = u 1 (t) + u 2 (t) u 1 (t) = K x(t) (12) where K renders A m = A bk Hurwitz 1, while u 2 (t) is generated by the adaptive controller. It leads to the following system: 1 In control theory, a matrix A m is called Hurwitz (or stable matrix) if every eigenvalue of A has a strictly negative real part 10

11 ẋ(t) = A m x(t) bθ x(t) + bu 2 (t) ŷ(t) = c ˆx(t) ˆx(0) = x 0 (13) For the linearly parameterized system in (13), we consider the state predictor ˆx(t) = A mˆx(t) bˆθ x(t) + bu 2 (t) y(t) = c x(t) x(0) = x 0 (14) along with the projection-type adaptive law ˆθ(t) ˆθ(t) = ΓP roj(ˆθ(t), x(t) x P b) ˆθ(0) = ˆθ0 x(t) = ˆx(t) x(t) (15) where Γ > 0 is the adaptation gain, and P = P > 0 solves A mp +P A m = Q for some Q > 0. Let u 2 (s) = C(s)( r(s)+k g r(s)) k g = 1/(c H o (0)) H o (s) = (si A m ) 1 b (16) where r = ˆθ (t)x(t), while C(s) is an asymptotically stable and strictly proper transfer function with dc gain C(0) = 1. s represents the adaptation gain. The L 1 adaptive controller consists of (12), (14), (15) and (16), with K and C(s) such that λ Ḡ(s) L 1 θ max < 1 Ḡ(s) = H o (s)(c(s) 1) θ max = max θ Ω n θ i (17) It was proven (Lemma 8 in [2]) that condition in (17) can be straightforwardly satisfied by increasing the bandwidth of C(s). In (17), L 1 refers to the L 1 norm which is defined for a SISO and MIMO system respectively by Definition 1 The L 1 norm of an asymptotically stable and proper SISO system is defined as: H(s) L1 = h(t) dt, where h(t) is the impulse response of 0 H(s). Definition 2 For an asymptotically stable and proper m inputs n outputs system H(s), the L 1 gain is defined as H(s) L1 = max i=1,...,n ( m j=1 H ij(s) L1 ), where H ij (s) is the ith row jth column entry of H(s). 4.3 Some comments The advantages of L 1 adaptive controller presented in the former section can be hightlighted by theorem 2 of [2]. i=1 11

12 Theorem2 of [2] The L 1 adaptive controller consisting in (12), (14), (15) and (16) guarantees lim x(t) x ref(t) = 0 t lim u(t) u ref(t) = 0 (18) t x(t) x ref (t) L γ 1 / Γ u u ref L γ 2 / Γ (19) where γ 1 = H 2 (s) L1 ) θmax /λ min (P ), H 2 (s) = I + (I Ḡ(s)θ ) 1 (Ḡ(s)θ + (C(s) 1)I), γ 2 = C(s)[1/c o H o (s)]c o L1 θmax /λ min (P ) + C(s)θ K L1. and, u ref (s) = C(s)(θ x ref (s) + k g r(s)) K x ref (s) (20) x ref (s) = H o (s)(k g C(s)r(s) + (C(s) 1)θ x ref (s)) + (si A m ) 1 x 0 y ref (s) = c x ref (s) (21) Relations (18), (19) of theorem 2 lead to the following remarks and conclusions: Remark 1 From ( 18), one concludes that lim t y(t) = r. Remark 2 The control objective is reduced to the selection of K and C(s) as large adaptation gain values for Γ becomes possible due to relation ( 19) in theorem 2. Remark 3 In case of MRAC, C(s) = 1 and C(s)[1/c o H o (s)]c o L1 cannot be finite since H o (s) is striclty proper. Therefore, it is not possible to conclude to a uniform performance bound from Formulation of the ROV model for L 1 adaptive control theory The system formulation in (11) corresponds to a SISO linear system and does not match with our highly nonlinear system specifications. That is why in [7], the authors present different other formulations corresponding to a more realistic representation of the system. It is given by: ẋ 1 (t) = x 2 (t) ẋ 2 (t) = A 2 x 2 (t) + f 2 (t, x(t)) + B 2 ωu (22) where x(t) = [x 1 (t), x 2 (t)] represent complete state vector of the system, A 2 is a known n n matrix, B R n m is a full rank matrix. u(t) R m is the control signal (m n) rank matrix, ω R m m is the uncertainty on the input gain, C R m n is a known full rank matrix, y R m is the measured output and f 2 is an unknown nonlinear function. The system takes the following matrix form: ( 0n n I where: A = 2 0 n n A 2 ẋ = Ax + f + Bωu (23) ) ( ) ( ) 0n 1 0n m, f = and B =. f 2 B 2 12

13 From 3, it is possible to formulate the system equations in the form of (22) by choosing x 1 = η 1 = [x, y, z, φ, θ, ψ] and x 2 = η 2 = [ẋ, ẏ, ż, φ, θ, ψ] ( η1 η 2 ) = [ 06 6 I D M ] [ η1 η 2 ] [ g M + w d M ] + [ M ] ωτ (24) 13

14 5 Conclusions et perspectives This report presents the advances in modeling and control of the MEROS ROV at Tecnalia. The dynamic model of the ROV is introduced and a simulator is done within Matlab/Simulink. An introduction to L 1 adaptive control was given and the formulation of the ROV model in the suitable form for adaptive control was given. The next steps will concern the simulation of the controller in Matlab/Simulink using the MEROS dynamic model. 14

15 References [1] C. Cao and N. Hovakimyan. Design and analysis of a novel l1 adaptive controller, part ii: Guaranteed transient performance,. In In Proceedings of the American Control Conference, Minneapolis, MN, pp , [2] C. Cao and N. Hovakimyan. Design and analysis of a novel l1 adaptive control architecture with guaranteed transient performance. IEEE Transactions on Automatic Control, 53: , [3] C. Cao and N. Hovakimyan. Design and analysis of a novel l1 adaptive control architecture, part1: Control signal and asymptotic stability. In In American Control Conference, Minneapolis, MN, pp , June [4] J. J. Craig. Introduction to Robotics. Addison-Wesley. Reading, Massachussets, [5] T. Fossen. Nonlinear Modeling and Control of Underwater Vehicles. PhD thesis, Department of Engineering Cybernetics. Norwegian University of Science and Technology. Trondheim, [6] T. Fossen. Marine Control Systems:Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles [7] N. Hovakimyan and C. Cao. L1 Adaptive Control Theory, Guaranteed Robustness with Fast Adaptation (ISBN: ). [8] F. Pierrot, M. Benoit, and P Dauchez. Optimal thruster configuration for omni-directionalunderwater vehicles. samos: a pythagorean solution. In OCEANS 98 Conference Proceedings,

16 Appendices A Jacobian transformation matrix J(η) [ ṗn Θ η = J(η)ν ] = [ R n b (Θ) T Θ (Θ) ] [ v b 0 ω b nb ] (25) where p n = [n, e, d] R 3 is the NED position, Θ = [φ, θ, ψ] is the attitude (Euler angles), v0 b represents the body-fixed linear velocity and ωnb b is the bodyfixed angular velocity. Therefore, the transformation matrix linking ṗ n to v0 b denoted Rb n (Θ) is given by the following expression Rb n (Θ) = R z,ψ R y,θ R x,φ cφcθ sψcφ + cψsθsφ sψsφ + cψcφsθ = sψcθ cψcφ + sφsθsψ cψsφ + sθsψcφ sθ cθsφ cθcφ (26) where c and s denote respectively sinus and cosinus notations. In the Marine Systems Simulation toolbox (MSS) developed Thor I. Fossen, this function is executed using R zyx. ωnb b = φ Rx,φ By inversing T 1 (Θ), we obtain: Θ Θ = T Θ (Θ)ω b nb (27) 0 θ 0 T Θ (Θ) = + Rx,φR y,θ 0 0 φ 1 sφtθ cφtθ 0 cφ sφ 0 sφ/cθ cφ cθ := T 1 Θ (Θ) Θ (28) (29) where t denotes the tangent function. The matrix J is calculated by eulerang function of the MSS Matlab toolbox. B Calculation of the inertia M, damping D and restoring forces g B.1 Inertia matrix M M = M RB + M A (30) M A = M A = diag{x u, Y v, Zẇ, Kṗ, M q, Nṙ} (31) 16

17 [ ] mi3 3 ms(r M RB = g) b ms(rg) b I o m mz g my g 0 m 0 mz g 0 mx g = 0 0 m my g mx g 0 0 mz g my g I x I xy I xz mz g 0 mx g I yx I y I yz my g mx g 0 I zx I zy I z B.2 Restoring forces g g(η) = (W B)sθ (W B)cθsφ (W B)cθcφ (y g W y b B)cθcφ + (z g W z b B)cθsφ (z g W z b B)sθ + (x g W x b B)cθcφ (x g W x b B)cθsφ (y g W y b B)sθ (32) (33) 17