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C O N T E N T VOL. / NO. - 4 Lagrangian Modelling of an Underwater Wave Glider By Andrea Caiti, Vincenzo Calabró, Seraglio Grammatico, Andrea Munafó & Mirko Stifani This paper introduces a control-oriented modelling approach for a hybrid autonomous underwater vehicle, the Underwater Wave Glider. The vehicle can accomplish both surface and underwater tasks by changing its shape: it can operate as a wave glider at the sea surface, exploiting wave and solar energy to recharge the onboard batteries, and switch to the typical torpedo-shaped configuration to operate as a self-propelled autonomous underwater vehicle. The vehicle dynamics is modelled using a novel Lagrangian approach. Simulation results are provided. 2
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Lagrangian Modelling of an Underwater Wave Glider By Andrea Caiti 1,*, Vincenzo Calabró 1, Seraglio Grammatico 1, Andrea Munafó 2 & Mirko Stifani 3 ABSTRACT This paper introduces a control-oriented modelling approach for a hybrid autonomous underwater vehicle, the Underwater Wave Glider. The vehicle can accomplish both surface and underwater tasks by changing its shape: it can operate as a wave glider at the sea surface, exploiting wave and solar energy to recharge the onboard batteries, and switch to the typical torpedo-shaped configuration to operate as a self-propelled autonomous underwater vehicle. The vehicle dynamics is modelled using a novel Lagrangian approach. Simulation results are provided. Key words: Propulsion, Robot, Wave-Body Interaction 1 Introduction Autonomous Underwater Vehicles (AUVs) have been widely used for oceanic research purposes. The requirements of underwater missions have become more and more challenging in terms of the required goals and mission endurance. These two objectives have always been kept separated in the vehicle design: traditional AUVs are in fact characterised by high maneuverability, allowing performing complex missions but also have a very limited operative range and endurance. On the other hand, gliders are designed to minimise the energy consumption to perform very long missions but because of this reason they are characterised by a very limited maneuverability. There has been a growing interest in the investigation of not only new ways for the optimisation of the energy consumption of underwater vehicles but also for the exploitation of renewable power sources to guarantee (at least in theory) endless energy while maintaining necessary level of maneuverability. Several clean-energy powered systems have been developed so far: the Pelamis Wave Power, Jones (2008), is an example of an active power plant system which converts wave energy into electric power using relative motion between its parts. The same idea was used in ships and autonomous marine vehicles. The Suntory Mermaid II, N.N. (2008), reached the Hawaiian coast starting from Japan, travelling for 6400 km in 108 days. This vessel is powered by solar panels providing energy for electronic and communication devices and uses a spring-fin system to convert the wave energy directly into propulsion. Another example is the Wave Glider (WG) autonomous 4 1 University of Pisa, Pisa, Italy 2 Integrated Systems for Marine Environment (ISME),???, Italy 3 Italian Navy * Corresponding author. Present address: a.caiti@dsea.unipi.it
vehicle, Willcox et al. (2009), Manley and Willcox (2010), which completed a travel of 4000 km from Hawaii to San Diego in only 82 days. Again, solar panels were used to provide power to the on-board electronics. Propulsion was achieved with a submerged multiple wings system connected with the surface floating part through a cable. The motivating idea of this work was to combine the above solutions into a single hybrid AUV, characterised by a long endurance of a wave glider together with typical underwater exploration and navigation capabilities of AUVs and gliders. This paper investigates the design of such an Underwater Wave Glider (UWG) vehicle capable to combine such properties into a single, compact and clean energy-powered system. Such UWG vehicle will be equipped with solar panels to recharge batteries when on surface and with variable buoyancy links and underwater wings to use wave motion for propulsion. The resulting vehicle will be able to change its geometry depending on the specific task and battery level. The main contribution of the paper is to present a novel modelling approach for the dynamics of the vehicle incorporating its geometric variability. The model can be directly used for the development of control laws for the navigation and guidance of the UWG. 2 Underwater Wave Glider (UWG) concept Existing hybrid AUV/glider vehicles, e.g. Alvarez et al. (2009), suffer from the usual energy density problem and cannot achieve complete autonomy and long endurance. Consider the UWG in Fig. 1 which can change its shape and hence its operating mode depending on the mission requirements. The analysis is focused on the control-oriented modelling of the wave-glider configuration, Fig. 1 (top), but the presented approach can be extended on the vehicle in AUV/glider shape, Fig. 1 (bottom). The dynamic model of torpedo-shaped underwater systems have been established in the literature, e.g. Fossen (2002), Antonelli (2006). Complex models have been proposed for a hybrid class of AUV/glider vehicles with variable mass and centre of gravity (CoG) by Caiti and Calabró (2010). The state-of-the-art techniques for modelling underwater variable geometry robots, Antonelli (2006), are restricted to AUVs equipped with serial manipulators. Since in the wave gliding mode the overall mass and the CoG position are constant, the model can be derived from the standard form proposed by Fossen (2002), with the exception of body-fluid interactions considered here. In order to derive a dynamical model for the longitudinal plane motion (surge, heave and pitch), a Lagrangian approach is used. The wave glider configuration is a conceptual innovation for AUVs. The main difference between the UWG vehicle and a wave glider, Manley and Willcox (2010), is that the latter uses a cable to link the floating part to the underwater wings and the UWG uses a rigid rod exploiting the vertical oscillating motion due to waves. 3 Direct kinematics Fig. 1: UWG operating modes. The traditional approach for modelling complex robotics systems is based on the use of several reference frames combined together to describe the position, orientation and speed of the different parts of a multi-body system. One of the most frequently used convention was proposed by Denavit Ship Technology Research Schiffstechnik VOL. / NO. - 20 5
and Hartenberg (1955). According to this convention it is necessary to describe successive rototranslations of reference frames using two pure rotations described by angles θ and α and two pure translations described by displacements d and a. Once the i-th part of the multi-body system is described in terms of the Denavit-Hartenberg (DH) parameters ξ i = [d i, θ i, a i, α i ] T, the change of the reference frame with respect to the (i 1)-th reference frame is obtained as follows: Ti i 1 (ξ i ) = Ti i 1 (d i, θ i, a i, α i ) = cos θ i cos α i sin θ i sin α i sin θ i a i cos θ i sin θ i cos α i cos θ i sin α i cos θ i a i sin θ i 0 sin α i cos α i d i 0 0 0 1 (1) A North-East-Down (NED)-like base reference frame O b = [x b, y b, z b ] was used. Fig. 2 shows the different reference frames used and the geometric parameters involved. Table 1 summarises the corresponding DH parameters. p n is the translation along the north-axis of the base reference frame x b, p d is the translation along the down-axis z b, q 1 is the pitch angle of the vehicle, q 2 is the angle between the vehicle main body and the first link and q 3 is the angle between the link and the wing. Fig. 2: Reference frames of the UWG. Tab. 1: DH parameters for the UWG. d θ a α 0 π/2 0 π/2 p n 0 0 π/2 p d π/2 0 π/2 0 q 1 a 1 0 0 q 2 a 2 0 0 q 3 a 3 0 0 0 0 π/2 6 There is an additional transformation to recover the end-effector reference frame. This system has an origin coinciding with the wing aerodynamic centre, Fig. 3, which is the point where all
hydrodynamic forces are assumed to be applied. This reference frame was chosen for modelling convenience to express drag and lift forces in the (x e, z e ) plane and the torque around the y e axis; other choices are also possible. Fig. 3: End-effector reference frame. Using the described modelling conventions, it is possible to describe the end-effector position pe b and orientation Re b with respect to the base frame b by combining all DH relative transformations Ti i 1 (ξ i ) into [ Te b (q) = T1 b T2 1 (p n )T3 2 (p d )T4 3 (q 1 )T5 4 (q 2 )T6 5 (q 3 )Te 6 R b = e (q) pe b ] (q) (2) 0 1 q = [p n, p d, q 1, q 2, q 3 ] T is the vector of the generalised positions and the final roto-translation T 6 e is obtained by an extra DH row with d 6 = 0, θ 6 = 0, a 6 = 0 and α 6 = π/2. 4 Differential kinematics A geometric Jacobi matrix can be used to relate the velocities q of the (virtual) joints to the velocity of a particular point p b, [( ) ( )] J b (q, p b JP1 JPn ) = (3) J O1 J On with [ JPi J Oi ] (q, p b ) = From the definition of a Jacobi matrix, ( zi 1 ) ( 0 zi 1 (p b pi 1 b ) ) v b (q, q) = ( ṗb ω b z i 1 if joint i is P if joint i is R (4) ) = J b (q, p b ) q (5) ṗ b and ω b are the linear and the angular velocities of the point p b, respectively. Since the hydrodynamic forces act on the individual parts of the multi-body system, local velocities are required. In a general frame u, they are given by [ ] v u R u (q, q) = b 0 0 Rb u J b (q, p b ) q = Jb u (q)j b (q, p b ) q (6) Note the duality property valid for a generalised force f b applied at the point p b, that corresponds to the joint generalised force τ(q, p b ) = J b (q, p b ) T f b (7) Ship Technology Research Schiffstechnik VOL. / NO. - 20 7
5 Vehicle dynamics via Lagrangian modelling Motion equations are derived via a standard Lagrangian approach based on the differential equation ( ) ( ) ( ) d L L Fd + = τ T (8) dt q q q L = T U is the Lagrangian function, T is the kinetic energy, U is the potential energy, q is the configuration vector, τ is the vector of the generalised forces and F d is a Rayleigh-like dissipation function used to model viscous effects acting in the serial-chain joints. To model the dynamics of a vehicle moving in a fluid, added mass, hydrodynamic damping and restoring forces have to be determined. The hydrodynamic forces and the generalised forces generated by the wings are included in the term τ. The forces acting on each body part are considered as applied in each local reference frame and then related to the whole body. 5.1 Kinetic energy Let p b mi be the centre of mass of the link i, which has mass m i and moment of inertia I j mi with respect to the reference frame j. The kinetic energy of the link i is Ji T i (q, q) = 1 2 m i q T i J Pi (q, p b mi ) J Pi (q, p b mi ) q + 1 2 qt i J Oi (q, p b mi ) T R b i (q)i i mi R b i (q) T J Pi (q, p b mi ) q (9) indicates that the Jacobian takes into account the contributions up to the link i of the serial chain. The first two joints are modelled as virtual; they have zero masses and moments of inertia. The total kinetic energy is obtained by summing the contributions over all single links, 5.2 Potential energy T (q, q) = 5 i=1 T i (q, q) (10) The potential energy describes the effects of restoring forces, such as buoyancy and gravity, and additional terms due to waves. Potential energy of a link i is U i (q) = (ρv i m i )g bt 0 z b mi (11) g bt 0 = [0, 0, g] T, g is the gravity acceleration, V i is the volume of the link i and ρ is the fluid density. The wave potential energy is modelled as that of a two-spring system, taking into account pitch and heave motions. Therefore the potential energy of the waves can be modelled as follows: U w (f w ) = 1 2 k d (p b d f b d ) 2 + 1 2 k 1 (q 1 f 1 ) 2 (12) k d, k 1 are positive constants to be identified and f w = [ f b d, f 1 ] T are functions describing wave characteristics. Sinusoidal wave profile of amplitude A w and circular frequency ω w /(2π) was modelled, f b d (t) = A w sin(ω w t), f 1 (t) = tan 1 [A w ω w cos(ω w t)] (13) The total potential energy is given by the sum U(q, f w ) = U w (f w ) + 5 i=1 U i (q) (14) 8
5.3 Drag modelling Viscous friction acting on the joints is modelled as linear with respect to the velocities at the joints, D q q = ( F d / q) T (15) The hydrodynamic forces acting on the i-th body part can be represented in a linear form as τ h = J i q T (p b mi, q)d hi J i b J (p b mi, q) q = D hi (q) q (16) The description of the hydrodynamic drag D hi (q) of the i-th link is provided in its local reference frame. This technique simplifies the modelling effort, since the single hydrodynamic contribution can be efficiently computed using a standard finite-element-analysis. Since it is expected that the UWG moves at low speeds, it is assumed that the relative motion between the body and fluid is described by a laminar flow, which simplifies drag description of each link. The shape of each vehicle part was simplified as ellipsoidal. As a result of these simplifications, the analytical results of Chwang and Wu (1975) were used to describe the corresponding viscous drag forces. 5.4 Dynamic model Considering the contributions shown in the previous subsections, the following dynamic multi-body model can be derived: B(q) q + C(q, q) q + D(q, q) q + g(q, f w ) = τ + J b (q, p b e ) T f b e (17) The drag matrix has constant and linear terms, D(q, q) = D q + D h (q). The end-effector wrench generated by the wing f b e (R b e, R b w, v b e, v w ) arises due to the end-effector orientation R b e and velocity v b e in addition to the wave front orientation R b e and velocity v w. 6 Simulation The control aimed at a clean energy-powered vehicle. It is passively controlled during the wave gliding mode, when the energy can be collected to recharge the batteries, Manley and Willcox (2010). The use of solar panels and the conversion of the wave potential energy provide ideally unlimited endurance for the UWG. The dimensions of the model a 1, a 2 and a 3 are 1.0, 1.0 and 0.3 m, respectively; the corresponding masses are respectively 10.0, 5.0 and 2.0 kg. The passive control strategy for the wave glider configuration is simulated for typical wave parameters of the Mediterranean sea, Cavalieri (2004). In the case shown in Fig. 4 the UWG navigates over a wave profile of amplitude of 0.5 m and frequency of 0.1 Hz. The vehicle floating part is chosen of an ellipsoidal shape with the major axis 1.0 m. A two-wing configuration based on NACA0009 profiles, Raymer (1992), Bertram (2000), having a dimension of 0.3 m 0.5 m for each wing is used. The numerical simulation shows a surge speed of about 1.0 knot, which is comparable with the declared speeds of the Wave Glider in similar seaway conditions, Manley and Willcox (2010). 7 Conclusions A Lagrangian modelling approach is described for a novel class of hybrid Underwater Wave Gliders. The benefit of the proposed vehicle design is that it can accomplish autonomously both surface and underwater missions with an ideally unlimited endurance. Within this concept, the exploitation of the wave potential force is fundamental for achieving full autonomy. This work may be continued in many directions. An ongoing line of research is focused on the optimisation of the shape parameters of the vehicle, e.g. to maximise the nominal speed in wave Ship Technology Research Schiffstechnik VOL. / NO. - 20 9
1 Mean Speed = 0.665810 [m/s] 0.5 Speed [m/s] 0 0.5 1 0 10 20 30 40 50 60 70 80 90 100 Time [s] 0.6 0.4 Vertical Position [m] 0.2 0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 60 70 80 90 100 Time [s] Fig. 4: Surge speed and heave motion. glider mode. In addition, the vehicle could be passively controlled in the wave gliding mode using flexible joints, and the energy could be collected to recharge the onboard batteries. References Alvarez, A., Caffaz, A., Caiti, A., Casalino, G., Gualdesi, L., Turetta, A. and Viviani, R. (2009). Folaga: A low-cost autonomous underwater vehicle combining glider and AUV capabilities. Ocean Eng., 36(1):34 38 Antonelli, G. (2006). Underwater Robots. Motion and Force Control of Vehicle-Manipulator Systems. Springer Bertram, V. (2000). Practical Ship Hydrodynamics. Butterworth-Heinemann Caiti, A. and Calabró, V. (2010). Control-oriented modelling of a hybrid AUV. In IEEE Conf. on Robotics and Automation, Anchorage Cavalieri, L. (2004). Atlas of the Waves and Wind of the Mediterranean Sea (in Italian). ISMAR-CNR Chwang, A. T. and Wu, T. Y.-T. (1975). Hydromechanics of low-reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech., 67:787 815 Denavit, J. and Hartenberg, R. (1955). A kinematic notation for lower-pair mechanisms based on matrices. J. Applied Mechanics, 23:215 221 Fossen, T. (2002). Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics Jones, W. D. (2008). Update. Ocean power catches a wave. IEEE Spectrum, 45:14 Manley, J. and Willcox, S. (2010). The Wave Glider: a persistent platform for ocean science. In IEEE OCEANS Conf., Sydney N.N. (2008). Wave Power Boat. Popular Science, March Raymer, D. (1992). Aircraft design - A conceptual approach. AIAA Educational Series Willcox, S., Meinig, C., Sabine, C. L., Lawrence-Slavas, N., Richardson, T., Hine, R. and Manley, J. (2009). An autonomous mobile platform for underway surface carbon measurements in open-ocean and coastal waters. In IEEE OCEANS Conf., Biloxi 10