A Survey of Wave Energy Converter Mechanisms Presented Under the Topological and Functional Viewpoints

Since the industrial revolution, humankind has been influencing and modifying the planet’s delicate ecosystem. In order to manufacture and energize the products and machines born from technological development, non-renewable energy sources such as oil, coal, and natural gas have been used, which are depletable. The relentless increase in the energy demand and the consequent high-level emission of carbon dioxide and other pollutants is causing more and more environmental damage. Thereby, energy extraction from renewable sources has to be significantly increased in order to reduce pollutant emissions. Nowadays, the most widely deployed renewable energy sources are hydro-power, solar, wind, and geothermal. Marine energy is a striking example of a renewable resource with great potential, but still accounts for a very small share with respect to the other deployed renewable resources. The major shortfall here lies in the extraction system which is often more complex and unprofitable if compared to other renewable systems. However, the energy potentially retrievable from oceans and seas is so huge that it can meet our future power needs, hence several countries are embracing the challenge of harnessing this vast reservoir of green energy. The worldwide potential of wave power is around 30,000 TWh/year but only a small fraction is efficiently extracted [1]. Furthermore, energy from the seas appear quite convenient to off-grid small off-shore systems.

As a matter of fact, wave energy converters (WECs) not only capture and convert efficiently the potential and/or kinetic energy of waves but also endure the marine environment, which is usually one of the highest cost headers. Indeed, maintenance and installation of the mooring system play a key role in both design and cost management.

The idea of harnessing wave energy to provide electrical, pneumatic or other kinds of power has attracted designers since the later half of the 19th century. The earliest WECs exploited articulated mechanisms to drive pumps, turbines, or electric generators, and this work includes also a substantial amount of patents from 1892 onwards.

Since the 1990s, a multitude of WEC devices have been devised and they widely vary in terms of structure, function and deployment site. This heterogeneity is a consequence of the lack of an established WEC technology; furthermore, it depends on local site characteristics such as the wave power density, local laws, and regulations, etc. that must be addressed in the design. Such a situation motivated the proposal of different WEC device classifications [2], Falcao [3], and Falnes [4].

The WEC devices are usually grouped according to the proximity of the plant from the coast:

• On-shore (water depth within 10–15 m),

• Near-shore (water depth within 20–25 m), and

• Off-shore (water depth greater than 50 m).

The harvesting working principle classification of WECs was proposed in Ref. [3] and entails five categories:

• Oscillating water column (OWC),

• Oscillating bodies,

• Oscillating wave surge converters (OWSC),

• Submerged pressure differential, and

• Overtopping devices.

OWC devices exploit the motion of incident waves to vary the water level inside a reservoir. The air, as a result of the fluctuating water level, is alternately compressed and suctioned. The pressure difference forces the air into a turbine coupled with an electric generator [5]. Both

• fixed or

• floating

OWCs can be found in the literature.

Devices that harvest energy as a result of the oscillations experienced by submerged or floating structures are generically referred to as oscillating bodies. Depending on their layout,

• one DoF (degrees of freedom) or

• more than one DoF

can be availed to convert the incoming wave energy.

Oscillating wave surge converters consist of a frame anchored on the seabed and a structure hinged to the frame that is swung by the pressure difference arising from the incoming wave.

In submerged pressure differential WECs the air pressure fluctuations in one or more plenums are used to blow air into a turbine. Such devices are often referable to as point absorbers planted near-shore and anchored to the seabed. The air-filled chambers may be flexible or rigid. In the latter case, the air volume fluctuation is accomplished by the relative motion between two interconnected structures.

Overtopping devices consist of a reservoir that is filled when the height of the incident waves overcomes the walls of the device. The collected water is then elaborated by a hydraulic turbine.

One of the most widespread classifications designates WEC devices according to the wave propagation direction namely:

• Attenuators,

• Terminators,

• Point absorbers, and

• Quasi-point absorbers.

Attenuators are composed of a set of articulated bodies and the relative motion among them is exploited for the generation of electric power. Such devices are deployed parallel to the traveling direction of the waves. Terminators, on the other hand, are activated by waves that impinge perpendicularly on their surface. Point absorbers are much smaller than a wavelength and are capable of harvesting energy regardless of the traveling direction of the waves. The floating body may have either single or multiple degrees of freedom. Quasi-point absorbers are characterized by a rather wide size with respect to a wavelength but are triggered independently of the traveling direction of the wave.

Finally, WEC devices can also be gathered according to their power take-off (PTO) system. The PTO system refers to the electrical conversion method, namely [6],

• air turbines,

• hydraulic converters,

• hydraulic turbines,

• direct mechanical drive systems, and

• direct electrical drive systems.

All the above-mentioned classification methods are quite universally accepted. However, they offer rather limited insight into the topological and functional traits of the devices. For example, there are many devices belonging to the same category that share the same topological family and may differ only in size or PTO system.

The present review proposes an atlas of WEC mechanisms obtained by an intensive investigation on the scientific and technical literature. The authors made their best to assemble the most complete and comprehensive collection of WEC mechanisms as possible, while any accidental omission cannot be excluded due to the complexity, variety, and hugeness of the different sources. In this case, it should be considered a totally unintentional exclusion.

The presented WEC mechanism have been categorized and grouped by means of a hybrid classification, which is based not only on traditional viewpoints but also on the topological and functional aspects. This allows designers to obtain detailed information on the complexity of the mechanical structure of each WEC system. Furthermore, this also allows them to identify possible correspondences and similarities between the mechanisms from the atlas, where redundancies have been reduced. In addition, details on the nominal power generated by the devices and the wave power density of the deployment site have been reported to offer a thorough overview of the wave energy converters panorama.

Some efforts have been made to extrapolate not only the functional representation of the WEC mechanisms from the original drawings but also to determine their kinematic chain (KC) and corresponding graph representations. The KC and graph representation are particularly useful to identify the topological families that have been used for grouping the mechanisms in the atlas. Since the functional, KC, and graph representations are not unique, some standard esthetic criteria for their representation have been established, based on previous similar investigations [7–10]. The grouping style based on the topological viewpoint will be helpful to make the atlas more systematic in the presentation of WEC mechanisms. Therefore, this atlas could represent a useful tool for the achievement of the following objectives:

• to develop an efficient way of WEC mechanisms classification;

• to support the patent offices in priority searches and patentability opinions and decisions;

• to design, develop, and fabricate original WEC systems;

• to support WEC system size and costs analysis;

• to assess WEC systems applicable power density;

• to support the calculation of the applicable wave frequency range for a WEC system;

• to evaluate the WEC systems energy efficiency;

• to identify the best methods for the evaluation of WEC mechanism scalability;

• to support the dissemination of the culture of renewable energy.

In this adopted classification of wave energy resource, the lowest category is class 1 which corresponds to wave power below 10 kW/m and is typical of the Baltic, Mediterranean, Red, and Black seas. Class 2 is characterized by wave power in the range 10–20 kW/m which is typical of tropical seas such as the Caribbean and Arabian seas, the Gulf of Guinea, and Bengal. Higher energetic conditions belong to class 3 which can be found in open ocean areas located at low and middle-lower latitudes. Class 4 wave power range (40–80 kW/m) still corresponds to open ocean areas but its wave conditions appear at middle-upper latitudes (i.e., North Pacific and North Atlantic areas). Mean wave power greater than 80 kW/m is associated with class 5, which uniquely occurs in deep waters of the Antarctic Ocean. Despite this class being characterized by the highest energetic source, it remains of limited practical relevance. Hereafter, the applicable wave power class will be associated with each presented WEC device, according to the available literature data.

The developed atlas makes use of a hybrid categorization that considers three levels.

• The first level highlights the working principle of devices and identifies a class of WECs, that is the functional classification.

• The second level subdivides each class according to the proximity from the coast.

• The third level gathers devices according to their topological family. All devices belonging to the same topological family share the same number of links and degrees of freedom.

The problem of finding any profitable correlation between the kinematic structure of a mechanism and its ability to perform a certain task has been investigated mainly in two decades, namely, in the seventies and the eighties, but some results have been obtained only for specific classes of mechanisms [13–16]. This endeavor is to be considered somewhat ambitious and difficult because typically, the topological characteristics of a mechanical system are completely detached from its physical characteristics, such as its geometric, kinematic, and dynamic properties. This concept has been reconsidered in 2015 [17] where some possible path to a solution has been explained by investigating new methods based on creative synthesis procedures.

From Section 3.1, it is clear that families are built by using topological conditions only, so it is very difficult to predict a family general ability to accomplish certain tasks, except for specific mechanisms that have been assigned all the geometrical, kinematic, and dynamic properties by designers. The authors of the present investigation believe that the topological-to-functional mapping of WEC system, albeit with some limitations, will be more feasible in the future, but this goal will be achieved only after adequate experience on different classes of WEC mechanisms will be acquired from a large both experimental and modeling work. In the present investigation, a brief attempt to find a (rather) weak correlative topology to function bound will be presented in Table 8 which lists the number of mechanisms obtained from any specific considered class, namely, point absorber, terminator, attenuator, OWC, and overtopping mechanisms.

The extensive number of mechanisms for wave energy converter which currently can be found in the state-of-the-art entails a highly arbitrary and nonuniform style of representation. Thus, the task of comparing different WEC mechanisms can be unbearably time-consuming and cumbersome. This atlas aims to provide a systematic and effective set of esthetic standards to represent such mechanisms, with the purpose of facilitating the comparison among devices which belong to the same and other classes. The following general guidelines establish the procedure by which the mechanisms have been presented:

• a proper way of illustrating the functional and geometrical characteristics of the mechanism,

• a systematic representation of the KC,

• a standard way for displaying the corresponding graph.

The adopted reference system for all the forthcoming drawings is shown in Fig. 1. The standard functional representation adopted for the lower kinematic pairs is depicted in Fig. 2. It is worth pointing out that the notation F has been introduced to represent the contact of a link with water, when a link is directly activated by the wave motion.

In each graph, numbers refer to links while letters identify a unique kinematic pair. The following notations were used for the revolute and prismatic pairs (see Fig. 3):

• x, y, z, f identify R-pairs with axis parallel to X, Y, Z respectively, while f denotes pairs with floating axis;

• X, Y, Z, P identify P-pairs with axis parallel to X, Y, Z respectively, while P denotes pairs with floating axis.

The class of higher kinematic pairs are reported in Figs. 4 and 5 and include the following:

• friction wheels,

• gears,

• rack and cogwheels,

• wheels rolling on a surface,

• belt drives,

• pin-on-a-slot.

All these pairs remove a single DoF and are denoted with the letter g, apart from wheels rolling on a surface and pin-on-a-slot pairs. In fact, wheels rolling on a surface and pin-on-a-slot (pairs 4 and 6) are denoted with x, y, z respectively whether the axis of rotation is parallel to the X, Y, Z axis. If the rotation axis is floating it is denoted with r.

Special kinematic pairs encountered in several patents or devices consist of ratchets, one-way clutches, bevel gears, and chain drives. For such pairs, an ad hoc functional and topological representation is illustrated in Fig. 6.

Figure 6(a) presents a special pair known as ratchet and pawl. The ratchet consists of a wheel with teeth similar to circular saw blades, while the pawl may rotate in order to fit the ratchet teeth slots (called keyways) and lock the ratchet wheel rotation. The pawls, supported by springs that keep them in contact with the wheel teeth, are essential to the wheel motion capability. In fact, they can be arranged in such a way to allow the wheel to rotate in one direction only. This means that the degree of constraint is not uniquely assigned because it depends on the pawls status, namely, locked in one direction and free to rotate in the other. Therefore, this unconventional pair is characterized by 3 or 1 degrees of constraints depending on whether the relative rotation among links i and j is locked (j₀ kinematic pair) or relative rotation is allowed (j₂ kinematic pair).

Similarly, the clutches (Fig. 6(b)) remove 3 DoF when the relative rotation between a pair of adjacent links is restrained (j₀ kinematic pair) and 2 DoF when the relative rotation is allowed, in such case it can be treated as a j₁ kinematic pair. Bevel gears as represented in Fig. 6(c) remove 3 DoF in the depicted configuration. Chain drives configured as in Fig. 6(d) introduce 2 degrees of constraints.

For each topological family, the functional, KC, and graph representations of at least one device are presented along with a brief description, hence resulting in an atlas of WEC mechanisms. The proposed atlas is based on similar works that exploited the topological features as a classification method [9,18,19]. During the last decades, topological analysis has been widely adopted to analyze planetary gear trains [20], epicyclic gear trains [21], variable kinematic joints [22], and to simulate the dynamic of articulated machinery [23]. Since graph theory was crucial to separate the structure and function concepts, it has been widely used as the primary tool in topological classification but also in automatic sketching [24,25] as well as computer-aided design [26]. An overview of all the WEC devices analyzed in this review is hereinafter provided. As can be seen in Tables 2–7, a hybrid classification was adopted that takes into account topological family, proximity from the coast and installation method. Wherever available, nominal power and wave power density ratings of the proposed devices have also been included so that the overview could be as exhaustive as possible.

As a careful analysis of Tables 2–7 suggests, point absorbers are by all means the most widely adopted WEC devices. For this class of WECs, 21 topological families have been identified among which the most recurrent are J1, U1, and E1. The peculiarity of point absorbers is their ability to convert energy regardless of the wave propagation direction. They usually show 1 or 2 DoF but point absorbers up to 4 DoF have been designed (Q1 in Table 3). The solutions adopted to convert the wave energy into useful energy are manifold:

• the oscillations of a floating body drive a mechanism which, in turn, runs an electric generator,

• the mechanism driven by the floating body may operate a pump to provide pressurized fluid to a turbine coupled with a generator,

• the floating body is used as the moving element of a linear electric generator,

• the hull swing perturbs a gyroscopic device whose spinning flywheel is used to extract power since it is coupled to an electric generator.

Figure 7 summarizes the functional and topological characteristics of the device proposed in 1892 [27]. The mechanism can be decomposed into two fundamental KC’s that are repeated several times. Hull 1 can rotate around the Y and Z axes and drives the sub-KC’s 1–18–19–20–21–22 and 1,2,3,4,5,14,15, respectively (Fig. 7). Pistons 5 and 22 pressurize a fluid that is delivered to a turbine coupled to an electric generator. Such a mechanism belongs to the A1 family and it consists of 31 links, 40 pairs j₁, and 8 pairs j₂ and, since a bevel gear is involved, $F=3$⁠.

In 1894, Singer and Wood [28] proposed a mechanism which belongs to B1 family and it consists of ℓ = 13, j₁ = 16, j₂ = 3, $F=1$⁠. The motion of the floating body 1 spins wheel 2 linked to a bevel gear that drives pistons 10,11,12,13 (Fig. 8). The pistons compress air that is delivered to a reservoir installed on the coast.

Martin [29] proposed a mechanism which belongs to C1 family and it consists of ℓ = 8, j₁ = 10, $F=1$⁠. The motion of the floating body 2 actuates link 3 which in turn swings the rocker 7. As a consequence, pistons 10 and 11 are driven. The rack and pinion wheel 4–5 allow to adjust the height of link 4 (Fig. 9). The pistons motion could be used to pressurize a fluid.

In 1899, Norton [30] designed a mechanism which belongs to D1 family and it consists of ℓ = 16, j₁ = 17, j₂ = 6, j₀ = 1, $F=2$⁠. The floating bodies 2 and 3 are both coupled to the frame link and one to each other by means of pin-on-a-slots. The motion of link 2 drives racks 6–7 and sliders 13–14, thereby pistons 15–16 may pump water into a reservoir. The racks 6–7 also actuate the cogwheels 8–11. ratchets 12 and 9 allow to transfer the motion to shaft 10 from one of the wheels 8 and 11 at a time. Shaft 10 drives an electric generator (Fig. 10).

Williams [31] proposed a device which belongs to E1 family with ℓ = 2, j₁ = 1, $F=1$⁠. The advantage of this family of mechanisms lies in their simpler architecture. Float 2 can slide inside rail 1 thrusting air to a reservoir (Fig. 11).

Rich [32] proposed a device which falls within the E1 family. This device is basically a linear generator in which member 1 represents the stator while the floating body 2 is the moving element on which the permanent magnets are placed (Fig. 12).

Filipenco [33] proposed a device which belongs to E1 family. This device is similar to the one proposed by Williams [31] (Fig. 11).

The pontoon power converter [1,34] belongs to the E1 family. The floating body 2 can slide on the prismatic guide of link 1. Link 2 acts as a hydraulic pumping cylinder (Fig. 13).

The Archimede wave swing [35,36] has similar functional characteristics and belongs to the E1 family too. Referring to Fig. 13, link 1 is anchored to the seabed and together with link 2, they can be intended as a linear generator.

Nutt [37] proposed in 1908 a F1 family device that consists of ℓ = 20, j₁ = 19, j₂ = 14, j₀ = 4, $F=1$⁠. The device can exploit roll and pitch motions to generate electrical energy. By means of the ratchets, 3–20 and 8–19 the roll motion of floating body 2 drives the bevel gear 4 or 7. Links 6 and 19 are coupled to the generator by means of gears 6–17 and 5–15–17. Pitch motions (along X-direction) actuate link 18 which turns gear 9 or 10 that transmits motion to the generator by means of gears 9–16–15–17 and 10–15–17 (Fig. 14). The presence of three degenerate structures such as ratchets 3–20 and 11–14 and gear train 4–5–6–7–9–10–15–16–17 determines a single degree of freedom.

Casella and Reynolds [38] designed a G1 family device that consists of ℓ = 11, j₁ = 9, j₂ = 9, j₀ = 1, $F=1$⁠. The floating bodies 1 and 2 are coupled by a revolute pair and their relative motion tightens the ropes coupled to links 3–5 and 4–6 and rigidly fixed to link 2. The ropes turn the drums 5–6 in such a way that wheel 9 turns always in the same direction. A gear train consisting of members 9–10–11 transfers the motion to an alternator (Fig. 15).

Taylor [39] proposed in 1922 the H1 family device that consists of ℓ = 11, j₁ = 8, j₂ = 6, j₀ = 2, $F=2$⁠. The mechanism is symmetric as can be seen in Fig. 16, thereby only half will be described. The gear 6 is coupled to the frame link 1 and to the wheels 7–8 by means of a chain. The waves activate the paddlewheel 8 thereby hull 2 can rotate around Z-axis actuating piston 4. The cogwheels 5–9 engage link 4 and are coupled to shaft 10 by means of a one-way clutch wheel. The same kind of clutch wheel couples links 7–11 so that shaft 11 spin always in one direction and actuates a generator.

Pasquariello [40] proposed in 1924 the I1 family device that consists of ℓ = 21, j₁ = 18, j₂ = 12, j₀ = 3, $F=3$⁠. Such mechanism includes coaxial cylinders thereby the typical schematization adopted for robotic wrist has been exploited [7]. By means of the floating bodies 4–5–12, the wheel 18 and piston 2 mechanism can adapt to the tides (Fig. 17). Only one of the two floats 4 and 5 can engage, via a unidirectional clutch, the sleeve 7 which is coaxial with connecting rod 3. Sleeve 7 engages gear 10 which in turn is coupled to sleeve 13 (coaxial with links 11 and 12). Gear 12 engages 14 that transmits motion to link 13 by means of a one-way clutch. Links 11 and 15 are coupled to frame link 1. Gear 15 engages 13 and 16, and the latter turns gear 17 coupled both to the frame link and to link 18 via a one-way clutch. The gear train 16–19–20–21 runs two electric generators.

Kriegel [41] designed in 1967 a J1 family device that consists of ℓ = 3, j₁ = 2, $F=2$⁠. Referring to Fig. 18, link 1 is anchored to the seabed while the floating body 2 can slide along it. The relative motion among links 1 and 2 allows the water to be suctioned and pumped toward a hydraulic turbine 3 which is coupled to an electric generator.

The Ceto device [42] and Aqua Buoy [43] share the same topological family J1 and can be represented as in Fig. 18.

Weills [44] designed in 1955 a J1 family device. Referring to Fig. 19, link 2 slides within the guide of hull 1. As link 2 oscillates the water is alternatively suctioned and pumped toward a hydraulic turbine coupled to the hull by means of a revolute pair. The electric power is produced with a generator coupled to the turbine. The topological representation in Fig. 19 is represented also by the fluid-device coupling between links 2 and 3 but it is not taken into account in the DoF count.

The Oceanus 2 device [45] shares the same functional and topological characteristics of the device illustrated in Fig. 19.

Orlando [46,47] proposed a J1 family gyroscopic device in 2014. Referring to Fig. 20, flywheel 3 spins around Y-axis and is coupled to hull 1. When the hull oscillates around Z-axis, link 2 undergoes a precession torque along X-axis, so that the alternator is operated.

Kanki [48,49] also proposed a J1 family gyroscopic device in 2006. The working principle is analogous to Refs. [46,47] and the functional and topological characteristics are reported in Fig. 21.

Masuda [50] designed in 1965 a K1 family device that consists of ℓ = 4, j₁ = 6, j₂ = 2, $F=1$⁠. Referring to Fig. 22, the floating body 1 carries the sun gear while carrier 2 is coupled to link 1 by means of a revolute pair. The carrier 2 is coupled to the planetary gear 3 which engages both the sun 1 and link 4 by means of a friction wheel. A generator is coupled to link 4 and the generated power is stored in an electrochemical accumulator.

Masuda [51] proposed also a L1 family device that includes ℓ = 7, j₁ = 5, j₂ = 4, j₀ = 1, $F=1$⁠. Referring to Fig. 23, the relative motion between floating body 2 and the frame link 1 engages gear 3 which is coupled to link 1 via revolute pair. link 3 is coupled with bevel gears 4 and 5 that engage and spin gear 6 in a single direction, by means of one-way clutches. Gear 7 engages with gear 6 and runs the electric generator. As can be seen from Fig. 24 the author reports a further version of his invention where link 3 is actuated by the relative rotations between buoy 1 and link 2.

Hinck [52] in 1966 proposed also a M1 family device that includes ℓ = 3, j₁ = 2, j₂ = 1, $F=1$⁠. Referring to Fig. 25, thanks to the relative motion between link 2 and floating body 1 gear 3 engages the rack of body 1. The electric generator is directly driven by gear 3 and the generated power is stored in an electrochemical accumulator.

Filipenco [33] in 1975 proposed also a M1 family device. Referring to Fig. 26, hull 1 is anchored to the seabed and its vertical oscillation drives a hydraulic turbine coupled to link 2. Gear 3 operates a hydraulic pump to deliver pressurized water toward a reservoir.

Filipenco [33] in 1975 designed a device belonging to the N1 family that includes ℓ = 6, j₁ = 7, $F=1$⁠. Referring to Fig. 27, the relative motion among link 2 and hull 1 actuates pistons 5 and 6 by means of the couplers 3 and 4, respectively. The pistons are intended to aspirate and pump water to a hydraulic turbine.

The Searev device [43] shares the same family of Filipenco N1 device Referring to Fig. 28, links 3 and 4 are coupled to hull 1 via revolute pairs and links 5–6 by means of prismatic guides. The relative motion between hulls 1 and 2 allows pumping oil to operate a hydraulic motor coupled with an electric generator.

Tsubota [53] in 1980 proposed a device belonging to the O1 family that includes ℓ = 7, j₁ = 8, $F=2$⁠. Referring to Fig. 29, a hydraulic turbine 7 is activated by the waves that overcome the boundary of hull 2. Frame link 1 is anchored to the seabed and includes the guides for links 5 and 6. Pistons 5 and 6 are actuated by the relative motion among link 1 and hull 2. The mechanism may produce electric power through a generator coupled to the turbine and pressurized water via pistons motion.

Ng [54] in 1981 patented a device belonging to the P1 family that includes ℓ = 5, j₁ = 4, j₂ = 3, $F=1$⁠. Referring to Fig. 30, wheels 2 and 6 roll on their guides thanks to the hull 1 motion. Ratchets 3 and 7 ensure the rolling of the two wheels in opposite directions. Shafts 4 and 8 are respectively coupled to links 5 and 9 by means of bevel gear. The electric generators are driven by links 5 and 9.

Caputo and coworkers [55–57] designed a P1 family device. Referring to Fig. 31, the flywheel of the gyroscopic device (links 5–6–7) is driven by an electric motor coupled with hull 1. Link 5 and circular plate 4 are coupled via prismatic pair so that link 4 oscillation is transmitted to pistons 2 and 9 respectively by means of links 3 and 8. Link 5 undergoes a precession torque which is due to the hull 1 oscillation and flywheel 7 angular momentum. The motion of pistons 2 and 9 is used to pressurize a fluid which will be processed by a turbine.

Sachs [58] in 1982 designed a gyroscopic device belonging to the Q1 family that includes ℓ = 14, j₁ = 14, j₂ = 4, j₀ = 1, $F=4$⁠. Referring to Fig. 32, flywheel 14 is driven by an electric motor and spins at a high angular velocity around Y-axis. When hull 1 rolls around Z-axis, a torque is transmitted to frame link 13. Such torque originates a precession torque along the X-axis that rotates link 13 which engages gear 10 or 5 depending on the direction of rotation. The gear trains 8–9–10 and 3–4–5 operate the generators that are coupled to links 3 and 8. If the hull swings around X-axis the precession torque acts along Z-axis, as a result, the springs between links 11-12 and 6-7 will be compressed or elongated. In fact, wave motion in general will induce a combination of the previously described hull motions.

Sachs proposed also a device belonging to the R1 family [58] that includes ℓ = 9, j₁ = 8, j₂ = 4, j₀ = 1, $F=3$⁠. In this version, a single electric generator is exploited. Referring to Fig. 33, when hull 1 oscillates around Z-axis the link 6 is subjected to a precession torque around X-axis. Depending on the direction of such torque the links 2-9 or 2-8 will be engaged. Hence, the gear trains 9-4-5 or 8-3-4-5 will be activated so that the generator can generate electric power. If the precession torque acts along Z-axis, the elastic elements that join links 2 and 6 will be compressed or elongated.

The Triton device belongs to the R1 family [59–61]. Referring to Fig. 34, body 2 is submerged and is coupled to pulley 3 by means of a rope. Gear 4 engages gear 3 and is coupled to link 1. The electric generator is driven by link 4.

Minguela [62,63] proposed a gyroscopic device belonging to the S1 family that includes ℓ = 5, j₁ = 4, j₂ = 2, $F=2$⁠. The flywheel 3 is driven by an electric motor and spins around Y-axis. When hull 1 oscillates around Z-axis, link 2 undergoes a precession torque around X-axis. Thereby, the gear train 4–5 is engaged and can power the generator (Fig. 35).

Rutta [64] designed a device belonging to the T1 family that includes ℓ = 9, j₁ = 7, j₂ = 6, j₀ = 1, $F=1$⁠. Float 2 can slide along a prismatic guide within frame link 1. The relative motion between links 1 and 2 rotates pulley 3 which has a gear engaging with gears 4 and 5. Gears 4 and 5 are coupled to gears 6 and 7 by means of one-way clutches. Gear 9, which drives the generator, engages gears 6 and 8. The clutches ensure that the generator spins only in one direction (Fig. 36).

Ocean harvesting [65] designed the infinity WEC device belonging to the U1 family that includes ℓ = 4, j₁ = 3, j₂ = 1, $F=2$⁠. Piston 3 is activated by the motion of the floating body 2. Link 4 actuates the generator and is coupled with link 3 by means of steel balls.

Seacap [66], Centipod [67], Wave Rider [68], Neptune 5B [69], Sea Based [70], Wavebob [42], and Power Buoy [42] devices share the same functional and topological representation of the U1 family devices (Fig. 37).

Fanghella and coworkers [71–73] recently proposed Alettone WEC system that works as a double rocker four-bar linkage in two possible working configurations, namely, Alettone 1 and Alettone 2.

The former belongs to the V1 topological family: ℓ = 5, j₁ = 5, j₂ = 1, $F=1$⁠. As illustrated in Fig. 38, the incoming waves activate alternatively rotations of rocker 2 which, through to the action of coupler 3, transmits alternative rotations to rocker 4. A one-way clutch couples rocker 4 to the one-way generator shaft that returns electric power. When clutch C is not engaged, the motion between rocker 4 and shaft 5 is not transmitted and the corresponding graph is still connected but not bi-connected. Therefore, shaft 5 is independent of rocker 4 when the clutch is disengaged and vertex 1 becomes a cut vertex.

Alettone 2 configuration exploits both rocker 4 rotation directions and falls within W1 family: ℓ = 9, j₀ = 2, j₁ = 7, j₂ = 3, $F=1$⁠. A planar representation of this version is given in Fig. 39. In this case, the transmission line to the electric generator is different with respect to the first configuration. Two one-way clutches are connected to rocker 4. If rocker 4 rotates clockwise, clutch 5 is engaged by means of selector 9 and the motion of gear 5 is transmitted to gear 8 which is rigidly coupled to the generator’s shaft. When rocker 4 rotates counterclockwise, clutch 6 is engaged by selector 9 so that link 8 is driven by means of gear train 6–7–8.

Devices pertaining to the topological families A2–E2 can be classified as terminators. Such devices exploit the oscillations of a float to produce electrical power. The most widely adopted technological solutions for electric power generation in terminators are reported below:

• the float and the electric generator are coupled via a gear train. In such a way, the generator shaft undergoes alternating direction turns, thereby a current rectifier circuit should be used to produce direct current,

• the gear trains coupled to one-way clutches in order to provide generator’s shaft rotation around the same direction,

• the float oscillations may actuate one or more pistons that pressurize a fluid. The fluid is then sent to a turbine coupled with a generator for electric power production.

Tidwell [74] in 1930 proposed a terminator device which falls within the A2 family and includes ℓ = 20, j₁ = 15, j₂ = 14, j₀ = 4, $F=1$⁠. Referring to Fig. 40, the floating bodies 9–10–11–12 slide along the guides located on link 1. Each floating body engages a rack-pinion gear (19–20, 17–18, 15–16, 13–14). The gears 9,10,11,12 are coupled respectively with shafts 5,6,7,8 via a one-way clutch. The gears on shafts 5–6 and 7–8 engage the gears respectively mounted on shafts 4 and 2. All the shafts are coupled to link 1 via revolute pairs.

Salter [75] in 1975 proposed the popular Salter’s duck device which falls within the B2 family and includes ℓ = 2, j₁ = 1, $F=1$⁠. Referring to Fig. 41, the floating body 2 is allowed to rotate around the frame link 1 thanks to properly designed conjugate surfaces. The relative motion between links 1 and 2 allows water to be pumped toward a hydraulic turbine.

Several devices exist that share the same functional and topological representation as Salter’s invention (Fig. 41): Oyster 2 [42], EB Frond [76], Waveroller [43,77], Oswec [78], Biowave [79], Langlee [80,81], Wave 2O [82], Cyc WEC [83].

The Wavestar device [34,43] belongs to the C2 family and includes ℓ = 5, j₁ = 5, $F=2$⁠. Referring to Fig. 42, when the floating body 3 is activated by wave motion, it induces the link 2 rotation that drives the piston 4. Link 4 and link 5 are coupled via prismatic pair and their relative motion is used to obtain a fluid compression. The high-pressure fluid is sent to a turbine coupled with an electric generator.

Analogous functional and topological representation of Fig. 42 applies for Seahorse and Ecowave devices [34,42,43].

The Waveclapper device [84] belongs to the D2 family and includes ℓ = 7, j₁ = 8, $F=2$⁠. Referring to Fig. 43, the submerged body 3 is activated by wave motion and causes the link 2 rotation that drives the pistons 7 and 4. Link 5 and link 6 are coupled both to link 3 and the frame link 1 via revolute pairs. The relative motions between links 4–5 and 6–7 are used to obtain to pressurize the water that is sent to a hydraulic turbine.

The Gelsystem device [85,86] belongs to the E2 family and includes ℓ = 4, j₁ = 4, $F=1$⁠. Referring to Fig. 44, the floating body 2 induces the piston 3 motion. Piston 3 represents the moving element of a linear electric generator with link 4 being the stator component.

Devices pertaining to the topological families A3–E3 can be classified as attenuators. The most widely adopted technological solutions for electric power generation in such devices are as follows:

• an elastic membrane or bladder, anchored to the seabed, that exploits the pressure differential establishing within the membrane due to the passing waves to push fluid through a turbine,

• an array of floats interconnected one to each other is used to pressurize a fluid. A turbine coupled to a generator exploits the previously pressurized fluid.

Thorsheim [87] in 1979 proposed a device belonging to the A3 family and includes ℓ = 2, j₁ = 1, $F=1$⁠. Referring to Fig. 45, membrane 1 is anchored to the seabed and half-filled with water. A hydraulic turbine is located in the mid-section of a convergent-divergent duct embedded within body 1. The wave passing over body 1 compress the membrane and force the water to flow into the duct. The turbine is coupled with a generator for electric power production.

Similar devices that share the same topological and functional features are Anaconda [88], S3 [89], MWave [90], and Etymol [90].

Meyerand [91] in 1986 proposed a device belonging to the A3 family. Referring to Fig. 46, a gas-filled constant pressure reservoir positioned on land is connected by a hose to a submerged case. The submerged case consists of a water-filled outer housing 1 and a fluid-filled inner flexible bladder. The housing includes one opening through a wall thereof communicating with the sea. The opening includes turbine 2 which is connected to an electrical generator. When a wave crest flows above the case, the water pressure over the housing and inner reservoir increases thereby collapsing the bladder. The collapse of the bladder forces water to flow into the outer housing and gas will flow into the constant pressure reservoir. The incoming water into the housing flows through the turbine generating electrical power.

Gordon [92] in 1988 designed a device belonging to the B3 family and includes ℓ = 8, j₁ = 9, $F=3$⁠. Referring to Fig. 47, when hull 2 is activated by the incoming waves, the piston 4 slide on its guide 2. Such motion is exploited to force water toward a turbine coupled to hull 2. The chain 1–2–3–4 is kinematically equivalent to a swinging glyph. By interconnecting several hull-glyph substructures an array configuration can be obtained. Each glyph-hull structure should be connected to each other by using a universal joint. Spider 5 allows the relative rotation around Y-axis between two hulls (i.e., 2 and 6 in Fig. 47). Each 5–6–7–8 chain is kinematically equivalent to the chain 1–2–3–4.

The Pelamis device [93] falls within the C3 family and includes ℓ = 11, j₁ = 14, $F=2$⁠. Referring to Fig. 48, links 2,3,4 are coupled via revolute pairs to hull 1. Links 5,6,7,8 are prismatically coupled to link 2. Thanks to the relative motion between hulls 11 and 1, oil could be pressurized and provided to a turbine coupled to a generator for electric power generation.

The Sea Power device [94] shares similar features with Pelamis device reported in Fig. 48.

The Stingray [95] belongs to the D3 family and includes ℓ = 4, j₁ = 3, j₂ = 1, $F=2$⁠. Referring to Fig. 49, the floating body 2 is linked to frame 1 and to hull 3. The oscillations of hull 3 actuate the electric generator coupled to link 4.

The Weptos device [96] belongs to the E3 family and includes ℓ = 3, j₁ = 2, j₂ = 1, $F=1$⁠. Referring to Fig. 50, the floating body 2 is linked to pulley 3 and to hull 1. The oscillations of float 2 lead to the rotation of shaft 3 which is coupled to the electric generator.

In 1962, Corbett et al. [97] proposed an OWC device that belongs to the A4 family and includes ℓ = 2, j₁ = 1, $F=1$⁠. Referring to Fig. 51, gas turbine 2 exploits the air pressure oscillations in a duct caused by the waves flowing through a vessel. The air duct connects the turbine with the water tank thereby the waves motion leads to an alternate airflow direction. In order to get a single airflow direction, a valve system must be implemented. A different option involves the use of a Wells turbine that rotates continuously in one direction regardless of the direction of the airflow.

Analogous devices pertaining to the same topological family A4 and sharing the same representation in Fig. 51 are Masuda’s second version of its patented device [50], Rho-Cee [98], Havkraft [99], and Leancon [100].

The Mighty Whale [42,101,102], OE Buoy [34,81], Rewec 3 [42,103], Limpet [42,104–106], Pico [42,106], Mutriku Plant [42,107], Yongsoo Plant [42,105,108], Wave Swell [42,109], Pressure Differential [110,111], and Hace [112] pertain to the A4 family but have a different functional representation.

Referring to Fig. 52, gas turbine 2 exploits the air pressure oscillations in a duct caused by the waves at the waterline. The turbine is coupled with an electric generator.

Systems belonging to the A5 topological family are referred to as Overtopping devices. These devices entail a hydraulic turbine that is driven by sea waves that are able to overcome the devices’ walls. To ensure a smooth operation, the turbine is placed below a water reservoir.

Wirt in 1979 proposed an Overtopping device [113] that belongs to the A5 family and includes ℓ = 2, j₁ = 1, $F=1$⁠. Referring to Fig. 53, hydraulic turbine 2 is coupled to frame 1 via revolute pair. The frame is anchored to the seabed and the Kaplan is the most adopted turbine for such systems.

Similar functional and topological features reported in Fig. 53 are shared by Drakoo [114], Wave Dragon [36,115,116], Tapchan [42,117,3], SSG [42,118], and Wave Plane [36,115] devices.

As mentioned in Sec. 3.1.1 Table 8 presents a simple quantitative correlation between the considered classes, namely, point absorbers, terminators, attenuators, OWCs, and overtoppings, and the number of their representative mechanisms that have been found in scientific or technical literature. There is a topological equivalence between some classes that have been differently labeled. However, by using a topological code composed of $F$⁠, ℓ, j₁, j₂, and j₀, the equivalence can be clearly identified, as follows:

• E1 ≡ B2 ≡ A3 ≡ A4 ≡ A5,

• M1 ≡ E3,

• O1 ≡ D2, and

• U1 ≡ D3.

Of course, the most prolific families are also the ones that have a limited number of links. For example,

• 41 different mechanisms have been found belonging to the class E1 (or equivalent B2, A3, A4, and A5) that have two links only;

• nine different mechanisms belongs to class U1 (or equivalent D3), which have four links only;

• seven mechanisms found belonging to class J1 with three links.

Table 8 can be interpreted as the matrix whole columns represent physical characteristics (working principle and then the function) and the rows correspond to different topological families (topology). Therefore, an element (i–j) with a higher value could be related to a stronger correlation between that specific type j of mechanism (working principle) and the topological family i. The presence of several null elements in the (sparse) matrix could be a symptom that the field of application is not fully mature yet and that there could be several solutions that remain unexplored.