# Kinematics and Dynamics of a Tensegrity-Based Water Wave Energy Harvester

Authors: Min Lin; Tuanjie Li; Zhifei Ji
Journal: Journal of Robotics
 (2016)

---

## Abstract

A tensegrity-based water wave energy harvester is proposed. The direct and inverse kinematic problems are investigated by using a geometric method. Afterwards, the singularities and workspaces are discussed. Then, the Lagrangian method was used to develop the dynamic model considering the interaction between the harvester and water waves. The results indicate that the proposed harvester allows harvesting 13.59% more energy than a conventional heaving system. Therefore, tensegrity systems can be viewed as one alternative solution to conventional water wave energy harvesting systems.


---

## Body

## 1. Introduction

Tensegrity systems are formed by a combination of rigid elements (struts) under compression and elastic elements (cables or springs) under tension. The use of cables or springs as tensile components leads to an important reduction in the weight of the systems. Due to this attractive nature, tensegrity systems have been proposed to be used in many disciplines. Moreover, a detailed description of the history of tensegrity systems is provided in [1, 2].The first research work that deals with tensegrity systems was completed by Calladine [3]. Since then, tensegrity systems have been rapidly applied as structures in the architectural context. A tensegrity dome was proposed by Pellegrino [4]. Some design methods for tensegrity domes are proposed by Fu [5]. Afterwards, tensegrity structures have been also proposed to be served as bridges [6–9]. Moreover, the use of cables or springs in tensegrities allows them to be deployable [10, 11]. Due to this nature, some research works are found towards their use as antennas [12, 13]. For static applications, the subject of form-finding of tensegrities has attracted the attention of several researchers [14, 15]. Moreover, a review of form-finding methods was provided by Tibert and Pellegrino [16]. The basic issues about the statics of tensegrity structures were reviewed by Juan and Tur [17].From an engineering point of view, tensegrities are a special class of structures whose components may simultaneously perform the purposes of structural force, actuation, sense, and feedback control. For such kind of structure, pulleys or other kinds of actuators may stretch/shorten some of the constituting components in order to substantially change their forms with a little variation of the structure’s energy. Ingber [18] has demonstrated that tensegrity structures are very similar to cytoskeleton structures of unicellular organisms. Afterwards, the cellular tensegrity model is used to understand the cell structure, biological networks, and mechanoregulation [19, 20]. Tensegrity structures are also very similar to muscle-skeleton structures of high efficiency land animals whose speeds can reach up to 60 mph. The muscle-skeleton systems of these beings are composed of only tensional and compressional components. They thus have the ability to run with high speed [21].Another interesting application of tensegrities is their development for use as mechanisms. Oppenheim and Williams [22] were the first to consider the actuation of tensegrity systems by modifying the lengths of their components in order to obtain tensegrity mechanisms. Afterwards, several mechanisms based on tensegrity systems were proposed, such as a flight simulator [23], a space telescope [24], and a tensegrity walking robot [25–27]. For tensegrity mechanisms, an interesting topic named tensegrity parallel mechanism has been proposed recently. The concept of tensegrity parallel mechanism was introduced by Marshall [28]. Then, Shekarforoush et al. [29] presented the statics of a 3-3 tensegrity parallel mechanism. Afterwards, Crane III et al. [30] proposed a planar tensegrity parallel mechanism and completed its equilibrium analysis. Tensegrity systems have been identified as one of three main research trends in mechanisms and robotics for the second decade of the 21st century [31]. However, just a few references have stated the possibility of using tensegrity systems as water wave energy harvesters. Scruggs and Skelton [32] made a preliminary investigation on the potential use of controlled tensegrity structures to harvest energy. Sunny et al. [33] studied the feasibility of harvesting energy using polyvinylidene fluoride patches mounted on vibrating prestressed membrane. Vasquez et al. [34] stated the possibility of using a planar tensegrity mechanism in ocean applications. This application is attractive since it can play an important role in the expansion of clean energy technologies that help the world’s sustainable development.This work presents the analysis of a tensegrity-based water wave energy harvester. Since this is the first stage for the development of a new application for tensegrity systems, a simplified linear model of sea waves was used to analyze the proposed harvester. The analytical solutions to the direct and inverse kinematic problems are found using a geometric method. Based on the obtained relationships between the input and output variables, the singular configurations have been discussed. The workspaces of the proposed mechanism have subsequently been computed. Afterwards, the dynamics were investigated. Finally, the energy harvesting capabilities of the tensegrity-based harvester are compared with a conventional heaving system.


## 2. Geometry of the Water Wave Energy Harvester

A diagram of the tensegrity-based water wave energy harvester is shown in Figure1. It consists of a float, four springs, four linear generators, and one kinematic chain. The linear generators are joining node pairs A
i
B
i
(
i
=
1,2
,
3,4
) while the springs are joining node pairs A
1
B
4, A
2
B
1, A
3
B
2, and A
4
B
3. The float of height D is denoted by B
1
B
2
B
3
B
4. This harvester is obtained from a square tensegrity parallel prism [10] by connecting the top of the latter to a float.Figure 1
A tensegrity-based water wave energy harvester.From Figure1, it can be seen that the sides of the squares formed by nodes A
1
A
2
A
3
A
4, B
1
B
2
B
3
B
4, and B
1
′
B
2
′
B
3
′
B
4
′ have the same length L. Moreover, the length of the linear generator joining node pairs A
i
B
i is denoted by L
i. As illustrated in Figure 1, the springs and the linear generators are connected to the float and the sea bed at nodes A
i and B
i by spherical joints without friction. The sea bed is considered to be parallel to the horizontal plane. A fixed reference frame A
(
X
,
Y
,
Z
) is located at the center of the square A
1
A
2
A
3
A
4 with its X axis parallel to the line joining nodes A
4 and A
1 and its Z axis perpendicular to the sea bed, while a moving reference frame B
X
1
,
Y
1
,
Z
1 is located at the mass center of the float with its X
1 axis parallel to the line joining nodes B
4 and B
1 and its Z
1 axis perpendicular to the plane formed by nodes B
1, B
2, B
3, and B
4. Moreover, the vectors specifying the positions of nodes A
i and B
i in the fixed reference frame are defined as a
A
i and b
A
i, respectively. Also, the vectors specifying the positions of nodes B
i in the moving reference frame are defined as b
B
i.In order to obtain an appropriate kinematic model of the harvester, the following hypotheses are made:(i)
The springs are linear with stiffnessK and lengths l
j
j
=
1,2
,
3,4 and all the springs have the same free length L
0.
(ii)
The water waves are traveling along theY axis.In Figure1, a passive kinematic chain denoted by R
1
P
R
2 is used to connect nodes O and O
1. Nodes O and O
1 represent the centers of the squares A
1
A
2
A
3
A
4 and B
1
B
2
B
3
B
4, respectively. Considering the constraints introduced by this kinematic chain, the possible movements of the float driven by water waves are rotations about the X axis and translations along the Y and Z axes. Therefore, the harvester has three degrees of freedom.The Cartesian coordinates of the mass center of the float in the fixed reference frame are defined as (x
,
y
,
z). From Figure 1, it can be seen that x
=
0 is always satisfied. Moreover, the angle θ is used to specify the rotation of the float about X axis. Meanwhile, the range of θ is assumed to be -
π
/
2
π
/
2. The variables y, z, and θ are driven by the water waves. As a consequence, they are thus chosen as the inputs of the system. Furthermore, only three of the four linear generators’ lengths are independent. For this reason, the lengths of the generators joining nodes A
1
B
1, A
2
B
2, and A
3
B
3 are chosen as the outputs of the system. It follows that the harvester’s output vector is O
=
L
1
,
L
2
,
L
3
T while its input vector is I
=
y
,
z
,
θ
T.


## 3. Kinematic Analysis

For the harvester, the linear generators are used to convert wave motion cleanly into electricity. Generally, the efficiency of electricity generation of the system is highly dependent on the motions of linear generators. To provide great insight into the kinematics of the harvester, the relationship between the input and output vectors is developed in this section.


### 3.1. Direct Kinematic Analysis

The direct kinematic analysis consists in computing the output vectorO for the given input vector I. According to [35], the most convenient approach to set an algebraic equation system for kinematic problem of a parallel mechanism is to use the rotation matrix parameters and the position vector of the moving platform. This approach is used in this work to deal with the kinematic problems of the harvester. The position and orientation of the float are described by the position vector P
=
O
O
′
=
0
,
y
,
z
T and the rotation matrix R
A
B with respect to the fixed reference frame. From Figure 1, it can be seen that the rotation matrix R
A
B can be defined by rotating the moving reference frame π
/
2 about Z
1 axis followed by θ about Y
1 axis. R
A
B thus takes the following form:(1)
R
A
B
=
0
-
1
0
cos
⁡
θ
0
sin
⁡
θ
-
sin
⁡
θ
0
cos
⁡
θ
.Then, the position vectors of pointsB
i
i
=
1,2
,
3,4 with respect to the fixed reference frame can be obtained:(2)
b
A
i
=
P
+
R
A
B
b
B
i
,
i
=
1,2
,
3,4
.The vectors specifying the positions of nodesB
i in the moving reference frame can be easily derived:(3)
b
B
1
=
L
2
-
L
2
-
D
2
,
b
B
2
=
L
2
L
2
-
D
2
,
b
B
3
=
-
L
2
L
2
-
D
2
,
b
B
4
=
-
L
2
-
L
2
-
D
2
.Substituting (3) into (2), we have(4)
b
A
1
=
L
2
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
,
b
A
2
=
-
L
2
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
,
b
A
3
=
-
L
2
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
,
b
A
4
=
L
2
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
.From Figure1, it can also be seen that a
A
1
=
[
L
/
2
,
-
L
/
2,0
]
T, a
A
2
=
[
L
/
2
,
L
/
2,0
]
T, a
A
3
=
[
-
L
/
2
,
L
/
2,0
]
T, and a
A
4
=
[
-
L
/
2
,
-
L
/
2,0
]
T. With the position vectors of points A
i and B
i now known, the vector equation of the ith linear generator can be written as(5)
L
i
=
b
A
i
-
a
A
i
,
i
=
1,2
,
3,4
.By using (5), the solution to the direct kinematic problem is found as follows: (6)
L
1
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
,
(7)
L
2
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
+
L
2
1
/
2
,
(8)
L
3
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
.Here, for the latter use, the length of the linear generator joining nodesA
4
B
4 is also presented:(9)
L
4
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
+
L
2
1
/
2
.


### 3.2. Inverse Kinematic Analysis

The inverse kinematic problem corresponds to the computation of the input vectorI for the given output vector O. The solution to this problem can be found by solving (6)–(8) for the input variables y, z, and θ. Subtracting the square of (7) from that of (6) yields (10)
2
L
y
+
L
2
cos
⁡
θ
-
L
2
-
L
1
2
+
L
2
2
-
D
L
sin
⁡
θ
=
0
.Subtracting the square of (8) from that of (6), we obtain(11)
2
L
y
cos
⁡
θ
+
1
-
D
L
sin
⁡
θ
-
L
1
2
+
L
3
2
-
2
L
z
sin
⁡
θ
=
0
.From (10) and (11), the following expressions can be derived:(12)
y
=
1
2
L
L
2
+
L
1
2
-
L
2
2
-
L
2
cos
⁡
θ
+
D
L
sin
⁡
θ
,
z
=
1
2
L
sin
⁡
θ
L
2
sin
2
⁡
θ
+
D
L
sin
⁡
θ
cos
⁡
θ
+
L
1
2
-
L
2
2
cos
⁡
θ
-
L
2
2
+
L
3
2
.By substituting (12) into (6), the following equation is obtained:(13)
4
L
2
-
L
2
2
L
2
cos
2
⁡
θ
-
L
1
2
-
L
2
2
2
+
L
2
2
-
L
3
2
2
-
L
1
2
-
L
3
2
2
cos
⁡
θ
-
4
L
2
L
2
-
L
2
2
-
L
1
2
-
L
2
2
2
-
L
2
2
-
L
3
2
2
=
0
.Because of the range imposed onθ, four solutions for θ can be arrived at by solving (13). Furthermore, by substituting these results into (12), the solutions to the inverse kinematic problem are found.


## 3.1. Direct Kinematic Analysis

The direct kinematic analysis consists in computing the output vectorO for the given input vector I. According to [35], the most convenient approach to set an algebraic equation system for kinematic problem of a parallel mechanism is to use the rotation matrix parameters and the position vector of the moving platform. This approach is used in this work to deal with the kinematic problems of the harvester. The position and orientation of the float are described by the position vector P
=
O
O
′
=
0
,
y
,
z
T and the rotation matrix R
A
B with respect to the fixed reference frame. From Figure 1, it can be seen that the rotation matrix R
A
B can be defined by rotating the moving reference frame π
/
2 about Z
1 axis followed by θ about Y
1 axis. R
A
B thus takes the following form:(1)
R
A
B
=
0
-
1
0
cos
⁡
θ
0
sin
⁡
θ
-
sin
⁡
θ
0
cos
⁡
θ
.Then, the position vectors of pointsB
i
i
=
1,2
,
3,4 with respect to the fixed reference frame can be obtained:(2)
b
A
i
=
P
+
R
A
B
b
B
i
,
i
=
1,2
,
3,4
.The vectors specifying the positions of nodesB
i in the moving reference frame can be easily derived:(3)
b
B
1
=
L
2
-
L
2
-
D
2
,
b
B
2
=
L
2
L
2
-
D
2
,
b
B
3
=
-
L
2
L
2
-
D
2
,
b
B
4
=
-
L
2
-
L
2
-
D
2
.Substituting (3) into (2), we have(4)
b
A
1
=
L
2
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
,
b
A
2
=
-
L
2
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
,
b
A
3
=
-
L
2
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
,
b
A
4
=
L
2
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
.From Figure1, it can also be seen that a
A
1
=
[
L
/
2
,
-
L
/
2,0
]
T, a
A
2
=
[
L
/
2
,
L
/
2,0
]
T, a
A
3
=
[
-
L
/
2
,
L
/
2,0
]
T, and a
A
4
=
[
-
L
/
2
,
-
L
/
2,0
]
T. With the position vectors of points A
i and B
i now known, the vector equation of the ith linear generator can be written as(5)
L
i
=
b
A
i
-
a
A
i
,
i
=
1,2
,
3,4
.By using (5), the solution to the direct kinematic problem is found as follows: (6)
L
1
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
,
(7)
L
2
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
+
L
2
1
/
2
,
(8)
L
3
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
.Here, for the latter use, the length of the linear generator joining nodesA
4
B
4 is also presented:(9)
L
4
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
+
L
2
1
/
2
.


## 3.2. Inverse Kinematic Analysis

The inverse kinematic problem corresponds to the computation of the input vectorI for the given output vector O. The solution to this problem can be found by solving (6)–(8) for the input variables y, z, and θ. Subtracting the square of (7) from that of (6) yields (10)
2
L
y
+
L
2
cos
⁡
θ
-
L
2
-
L
1
2
+
L
2
2
-
D
L
sin
⁡
θ
=
0
.Subtracting the square of (8) from that of (6), we obtain(11)
2
L
y
cos
⁡
θ
+
1
-
D
L
sin
⁡
θ
-
L
1
2
+
L
3
2
-
2
L
z
sin
⁡
θ
=
0
.From (10) and (11), the following expressions can be derived:(12)
y
=
1
2
L
L
2
+
L
1
2
-
L
2
2
-
L
2
cos
⁡
θ
+
D
L
sin
⁡
θ
,
z
=
1
2
L
sin
⁡
θ
L
2
sin
2
⁡
θ
+
D
L
sin
⁡
θ
cos
⁡
θ
+
L
1
2
-
L
2
2
cos
⁡
θ
-
L
2
2
+
L
3
2
.By substituting (12) into (6), the following equation is obtained:(13)
4
L
2
-
L
2
2
L
2
cos
2
⁡
θ
-
L
1
2
-
L
2
2
2
+
L
2
2
-
L
3
2
2
-
L
1
2
-
L
3
2
2
cos
⁡
θ
-
4
L
2
L
2
-
L
2
2
-
L
1
2
-
L
2
2
2
-
L
2
2
-
L
3
2
2
=
0
.Because of the range imposed onθ, four solutions for θ can be arrived at by solving (13). Furthermore, by substituting these results into (12), the solutions to the inverse kinematic problem are found.


## 4. Singularity Analysis

### 4.1. Jacobian Matrix

The Jacobian matrix of the harvester is defined as the relationships between a set of infinitesimal changes of its input vectors and the corresponding infinitesimal changes of its output vectors. The Jacobian matrix,J, relates δ
I to δ
O such that δ
O
=
J
δ
I. J can be rewritten in terms of matrices C and D such that C
δ
O
=
D
δ
I. From (6)–(8), the elements of C and D can be computed and written in terms of the input variables as follows:(14)
C
11
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
,
C
22
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
+
L
2
1
/
2
,
C
33
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
,
C
12
=
C
13
=
C
21
=
C
23
=
C
31
=
C
32
=
0
,
D
11
=
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
,
D
12
=
D
22
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
,
D
13
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
D
2
sin
⁡
θ
-
L
2
cos
⁡
θ
-
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
L
2
sin
⁡
θ
-
D
2
cos
⁡
θ
,
D
21
=
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
,
D
31
=
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
,
D
32
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
,
D
23
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
D
2
sin
⁡
θ
-
L
2
cos
⁡
θ
-
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
L
2
sin
⁡
θ
+
D
2
cos
⁡
θ
,
D
33
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
D
2
sin
⁡
θ
+
L
2
cos
⁡
θ
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
L
2
sin
⁡
θ
-
D
2
cos
⁡
θ
.For (14), it is noted that C
i
j and D
i
j are the elements located on the ith line and jth column of C and D, respectively.


### 4.2. Singular Configurations

The singular configurations of the harvester consist in finding the situations where the relationships between infinitesimal changes in its input and output variables degenerate. When such a situation occurs, the harvester will gain or lose one or more degrees of freedom, thus leading to a loss of control. As a consequence, such configurations are usually avoided when possible. Generally, the singular configurations of the harvester can be obtained by settingdet
⁡
C
=
0, det
⁡
D
=
0, or both. The determinants of C and D can be expressed as follows:(15)
det
⁡
C
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
+
L
2
1
/
2
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
=
0
,
det
⁡
D
=
L
2
8
D
L
-
2
D
y
-
2
L
z
sin
⁡
2
θ
+
2
L
y
-
2
D
z
-
L
2
cos
⁡
2
θ
+
8
z
2
+
2
D
2
cos
⁡
θ
+
8
y
z
-
4
L
z
+
2
D
L
sin
⁡
θ
+
L
2
-
6
D
z
-
2
L
y
=
0
.By examining (15), it is possible to extract the expressions corresponding to singular configurations. The following is a list of these expressions as well as their descriptions with respect to the mechanism’s behaviors:(16)
i
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
=
0
.
(a)
The length of the linear generator joining nodesA
1 and B
1 is equal to zero. Node A
1 is thus coincident with node B
1. Moreover, node A
4 is also coincident with node B
2.
(b)
The movement of the float is reduced to a rotation about the axis joining nodesA
1 and A
4. When this is the case, only one variable is needed to define the system. The harvester thus loses two degrees of freedom.
(c)
Infinitesimal movements of nodeO
′ in a direction perpendicular to the line joining nodes A
1 and B
4 are possible without deforming the springs and the linear generators.
(d)
External forces parallel to the lineB
1
B
4 are resisted by the harvester.(17)
ii
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
=
0
.
(a)
The length of the linear generator joining nodesA
3 and B
3 is equal to zero. Node A
2 is coincident with node B
4.
(b)
The movement of the float is reduced to a rotation about the axis joining nodesA
2 and A
3. When this case occurs, only one variable can be used to describe the rotation of the float. The harvester thus loses two degrees of freedom.
(c)
Infinitesimal movements of nodeO
′ in a direction perpendicular to the line joining nodes A
2 and B
1 are possible without deforming the springs and the linear generators.
(d)
External forces parallel to the lineB
1
B
4 are resisted by the harvester.(18)
iii
D
L
-
2
D
y
-
2
L
z
sin
⁡
2
θ
+
2
L
y
-
2
D
z
-
L
2
cos
⁡
2
θ
+
8
z
2
+
2
D
2
cos
⁡
θ
+
8
y
z
-
4
L
z
+
2
D
L
sin
⁡
θ
+
L
2
-
6
D
z
-
2
L
y
=
0
.
(a)
Actually, it is impossible to extract the behaviors of the harvester from (18). This case corresponds to the boundaries of the input workspace and will be mapped in Section 5.2. Generally speaking, when this is the case, infinitesimal movements of the input variables along a direction perpendicular to a certain surface cannot be generated.From (16) and (17), it can be seen that the singular configuration (i) corresponds to the situation where the length of the linear generator A
1
B
1 is equal to zero while configuration (ii) corresponds to the situation where the length of the linear generator A
3
B
3 is equal to zero. From an engineering point of view, the linear generators are generally limited to operate within a range of nonzero lengths. However, from the aspect of mechanism’s analysis, the lengths of prismatic actuators can be set to be zero. This case belongs to one kind of the singular configurations of the proposed mechanism.


## 4.1. Jacobian Matrix

The Jacobian matrix of the harvester is defined as the relationships between a set of infinitesimal changes of its input vectors and the corresponding infinitesimal changes of its output vectors. The Jacobian matrix,J, relates δ
I to δ
O such that δ
O
=
J
δ
I. J can be rewritten in terms of matrices C and D such that C
δ
O
=
D
δ
I. From (6)–(8), the elements of C and D can be computed and written in terms of the input variables as follows:(14)
C
11
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
,
C
22
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
+
L
2
1
/
2
,
C
33
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
,
C
12
=
C
13
=
C
21
=
C
23
=
C
31
=
C
32
=
0
,
D
11
=
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
,
D
12
=
D
22
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
,
D
13
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
D
2
sin
⁡
θ
-
L
2
cos
⁡
θ
-
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
L
2
sin
⁡
θ
-
D
2
cos
⁡
θ
,
D
21
=
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
,
D
31
=
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
,
D
32
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
,
D
23
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
D
2
sin
⁡
θ
-
L
2
cos
⁡
θ
-
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
L
2
sin
⁡
θ
+
D
2
cos
⁡
θ
,
D
33
=
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
D
2
sin
⁡
θ
+
L
2
cos
⁡
θ
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
L
2
sin
⁡
θ
-
D
2
cos
⁡
θ
.For (14), it is noted that C
i
j and D
i
j are the elements located on the ith line and jth column of C and D, respectively.


## 4.2. Singular Configurations

The singular configurations of the harvester consist in finding the situations where the relationships between infinitesimal changes in its input and output variables degenerate. When such a situation occurs, the harvester will gain or lose one or more degrees of freedom, thus leading to a loss of control. As a consequence, such configurations are usually avoided when possible. Generally, the singular configurations of the harvester can be obtained by settingdet
⁡
C
=
0, det
⁡
D
=
0, or both. The determinants of C and D can be expressed as follows:(15)
det
⁡
C
=
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
+
L
2
1
/
2
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
=
0
,
det
⁡
D
=
L
2
8
D
L
-
2
D
y
-
2
L
z
sin
⁡
2
θ
+
2
L
y
-
2
D
z
-
L
2
cos
⁡
2
θ
+
8
z
2
+
2
D
2
cos
⁡
θ
+
8
y
z
-
4
L
z
+
2
D
L
sin
⁡
θ
+
L
2
-
6
D
z
-
2
L
y
=
0
.By examining (15), it is possible to extract the expressions corresponding to singular configurations. The following is a list of these expressions as well as their descriptions with respect to the mechanism’s behaviors:(16)
i
z
-
D
2
cos
⁡
θ
-
L
2
sin
⁡
θ
2
+
y
+
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
+
L
2
2
1
/
2
=
0
.
(a)
The length of the linear generator joining nodesA
1 and B
1 is equal to zero. Node A
1 is thus coincident with node B
1. Moreover, node A
4 is also coincident with node B
2.
(b)
The movement of the float is reduced to a rotation about the axis joining nodesA
1 and A
4. When this is the case, only one variable is needed to define the system. The harvester thus loses two degrees of freedom.
(c)
Infinitesimal movements of nodeO
′ in a direction perpendicular to the line joining nodes A
1 and B
4 are possible without deforming the springs and the linear generators.
(d)
External forces parallel to the lineB
1
B
4 are resisted by the harvester.(17)
ii
z
-
D
2
cos
⁡
θ
+
L
2
sin
⁡
θ
2
+
y
-
L
2
cos
⁡
θ
-
D
2
sin
⁡
θ
-
L
2
2
1
/
2
=
0
.
(a)
The length of the linear generator joining nodesA
3 and B
3 is equal to zero. Node A
2 is coincident with node B
4.
(b)
The movement of the float is reduced to a rotation about the axis joining nodesA
2 and A
3. When this case occurs, only one variable can be used to describe the rotation of the float. The harvester thus loses two degrees of freedom.
(c)
Infinitesimal movements of nodeO
′ in a direction perpendicular to the line joining nodes A
2 and B
1 are possible without deforming the springs and the linear generators.
(d)
External forces parallel to the lineB
1
B
4 are resisted by the harvester.(18)
iii
D
L
-
2
D
y
-
2
L
z
sin
⁡
2
θ
+
2
L
y
-
2
D
z
-
L
2
cos
⁡
2
θ
+
8
z
2
+
2
D
2
cos
⁡
θ
+
8
y
z
-
4
L
z
+
2
D
L
sin
⁡
θ
+
L
2
-
6
D
z
-
2
L
y
=
0
.
(a)
Actually, it is impossible to extract the behaviors of the harvester from (18). This case corresponds to the boundaries of the input workspace and will be mapped in Section 5.2. Generally speaking, when this is the case, infinitesimal movements of the input variables along a direction perpendicular to a certain surface cannot be generated.From (16) and (17), it can be seen that the singular configuration (i) corresponds to the situation where the length of the linear generator A
1
B
1 is equal to zero while configuration (ii) corresponds to the situation where the length of the linear generator A
3
B
3 is equal to zero. From an engineering point of view, the linear generators are generally limited to operate within a range of nonzero lengths. However, from the aspect of mechanism’s analysis, the lengths of prismatic actuators can be set to be zero. This case belongs to one kind of the singular configurations of the proposed mechanism.


## 5. Workspaces

Since the input variablesy, z, and θ are driven by water waves, the ranges of the input variables can be used to describe the strengths of the water waves. Moreover, the amount of the electricity produced by the harvester depends on the movements of the linear generators. The ranges of the output variables can be considered as an indicator of the efficiency of energy harvesting. In this section, the ranges of the input vectors are referred to as the input workspace while the ranges of the output vectors are referred to as the output workspace. The boundaries of the input and output workspaces usually correspond to singular configurations described in Section 4.2. From (16)–(18), it can be seen that the singular configurations are expressed in terms of the input variables. According to these expressions, the boundaries of the input workspace can be computed. Afterwards, these boundaries will be mapped from the input domain into the output domain in order to generate the output workspace.


### 5.1. Input Workspace

The input workspace of the harvester is a volume whose boundaries correspond to singular configurations discussed in Section4.2. An example of such a workspace with L
=
1 m and D
=
1 m is shown in Figure 2.Figure 2
Input workspace of the harvester withL
=
1 m and D
=
0.1 m.In Figure2, the surface corresponding to the singular configuration (iii) is identified by surface (iii). From this figure, it can be seen that the input workspace can be divided into three parts. The first part is defined by -
1
≤
y
≤
0 and 0
≤
θ
≤
π
/
2. It is bounded by surface (iii) and the planes corresponding to y
=
-
1, θ
=
π
/
2, and z
=
2. Moreover, the second part is defined by 0
≤
y
≤
1 and 0
≤
θ
≤
π
/
2. It is bounded by the planes corresponding to θ
=
π
/
2, y
=
1, and surface (iii). Finally, the third part is defined by 0
≤
y
≤
1 and -
π
/
2
≤
θ
≤
0. It is bounded by the planes corresponding to θ
=
-
π
/
2, y
=
-
1, and surface (iii). Furthermore, from Figure 2, it can also be observed that curves (i) and (ii) correspond to the singular configurations (i) and (ii), respectively. Since the harvester will be uncontrolled when it reaches a singular configuration, the boundaries of the input workspace and the singular curves (i) and (ii) should be avoided during the use of such a harvester.


### 5.2. Output Workspace

In order to obtain the output workspace, the singular configurations detailed in Section4.2 should be rewritten in terms of the output variables firstly. From (16) and (17), it can be concluded that the singular configuration (i) in the output domain corresponds to L
1
=
0 while the singular configuration (ii) corresponds to L
2
=
0. Generally speaking, by substituting the solutions to the inverse kinematic problem into (18), an expression for singular configuration (iii) in terms of the output variables can be arrived at. However, this procedure is rather tedious. Here, Bezout’s method [36] was used to derive the expression corresponding to singular configuration (iii) in the output domain due to its simplicity.Equation (13) is firstly rewritten as(19)
M
1
cos
2
⁡
θ
+
M
2
cos
⁡
θ
+
M
3
=
0
,where(20)
M
1
=
4
L
2
-
L
2
2
L
2
,
M
2
=
-
L
1
2
-
L
2
2
2
+
L
2
2
-
L
3
2
2
-
L
1
2
-
L
3
2
2
,
M
3
=
-
4
L
2
L
2
-
L
2
2
-
L
1
2
-
L
2
2
2
-
L
2
2
-
L
3
2
2
.Moreover, by substituting (12) into (18), the following equation is obtained:(21)
N
1
cos
2
⁡
θ
+
N
2
cos
⁡
θ
+
N
3
=
0
,where (22)
N
1
=
L
1
2
-
L
2
2
L
2
2
-
L
3
2
,
N
2
=
-
L
1
2
-
L
2
2
2
-
L
2
2
-
L
3
2
2
,
N
3
=
L
2
2
-
L
3
2
L
2
2
-
L
3
2
.It should be noted that (21) represents singular configuration (iii) expressed by L
1, L
2, L
3, and θ. Moreover, (19) is used to compute θ for the given values of L
1, L
2, and L
3. Generally, the solutions to θ obtained by solving (21) should satisfy (19). Furthermore, both (19) and (21) can be considered as two quadratics with respect to cos
⁡
θ. According to Bezout’s method, the condition that (19) and (21) have a comment root for cos
⁡
θ is as follows:(23)
M
1
M
2
N
1
N
2
M
2
M
3
N
2
N
3
-
M
1
M
3
N
1
N
3
2
=
0
.Simplifying (23) yields(24)
L
1
2
-
L
3
2
L
1
2
-
2
L
2
2
+
L
3
2
2
4
L
4
-
4
L
2
L
2
2
+
L
1
4
-
2
L
1
2
L
2
2
+
L
1
4
4
L
4
-
4
L
2
L
2
2
+
L
2
4
-
2
L
2
2
L
3
2
+
L
3
4
=
0
.Equation (24) represents the surfaces corresponding to singular configuration (iii) in the output workspace. By plotting these surfaces, the output workspace of the harvester can be obtained. An example of such plots is shown in Figure 3 with L
=
1 m and D
=
0.1 m.Figure 3
Output workspace of the water wave energy harvester withL
=
1 m and D
=
0.1 m.From Figure3, it can be seen that the singular configuration (iii) determined by (24) corresponds to four surfaces (surfaces (iv)–(vii)) in the output workspace. Moreover, surfaces (iv), (v), (vi), and (vii) correspond to expressions L
1
-
L
3
=
0, L
1
2
-
2
L
2
2
+
L
3
2
=
0, 4
L
4
-
4
L
2
L
2
2
+
L
1
4
-
2
L
1
2
L
2
2
+
L
1
4
=
0, and 4
L
4
-
4
L
2
L
2
2
+
L
2
4
-
2
L
2
2
L
3
2
+
L
3
4
=
0, respectively. It can also be observed that the output workspace of the harvester can be divided into two parts. The first part is bounded by surface (v), surface (vi), plane L
1
=
0, and plane L
3
=
0 while the second part is bounded by surfaces denoted by (iv), (vi), and (vii) and planes denoted by L
2
=
0 and L
1
=
10. This output workspace should be considered during the use and design of such a harvester.It is noted that the forward and inverse kinematics, Jacobian matrix, and workspaces should be considered when such harvester is being designed. Moreover, when the harvester is put to use, the singular configurations should be avoided. The kinematics and Jacobian matrix are used to find the singular configurations.


## 5.1. Input Workspace

The input workspace of the harvester is a volume whose boundaries correspond to singular configurations discussed in Section4.2. An example of such a workspace with L
=
1 m and D
=
1 m is shown in Figure 2.Figure 2
Input workspace of the harvester withL
=
1 m and D
=
0.1 m.In Figure2, the surface corresponding to the singular configuration (iii) is identified by surface (iii). From this figure, it can be seen that the input workspace can be divided into three parts. The first part is defined by -
1
≤
y
≤
0 and 0
≤
θ
≤
π
/
2. It is bounded by surface (iii) and the planes corresponding to y
=
-
1, θ
=
π
/
2, and z
=
2. Moreover, the second part is defined by 0
≤
y
≤
1 and 0
≤
θ
≤
π
/
2. It is bounded by the planes corresponding to θ
=
π
/
2, y
=
1, and surface (iii). Finally, the third part is defined by 0
≤
y
≤
1 and -
π
/
2
≤
θ
≤
0. It is bounded by the planes corresponding to θ
=
-
π
/
2, y
=
-
1, and surface (iii). Furthermore, from Figure 2, it can also be observed that curves (i) and (ii) correspond to the singular configurations (i) and (ii), respectively. Since the harvester will be uncontrolled when it reaches a singular configuration, the boundaries of the input workspace and the singular curves (i) and (ii) should be avoided during the use of such a harvester.


## 5.2. Output Workspace

In order to obtain the output workspace, the singular configurations detailed in Section4.2 should be rewritten in terms of the output variables firstly. From (16) and (17), it can be concluded that the singular configuration (i) in the output domain corresponds to L
1
=
0 while the singular configuration (ii) corresponds to L
2
=
0. Generally speaking, by substituting the solutions to the inverse kinematic problem into (18), an expression for singular configuration (iii) in terms of the output variables can be arrived at. However, this procedure is rather tedious. Here, Bezout’s method [36] was used to derive the expression corresponding to singular configuration (iii) in the output domain due to its simplicity.Equation (13) is firstly rewritten as(19)
M
1
cos
2
⁡
θ
+
M
2
cos
⁡
θ
+
M
3
=
0
,where(20)
M
1
=
4
L
2
-
L
2
2
L
2
,
M
2
=
-
L
1
2
-
L
2
2
2
+
L
2
2
-
L
3
2
2
-
L
1
2
-
L
3
2
2
,
M
3
=
-
4
L
2
L
2
-
L
2
2
-
L
1
2
-
L
2
2
2
-
L
2
2
-
L
3
2
2
.Moreover, by substituting (12) into (18), the following equation is obtained:(21)
N
1
cos
2
⁡
θ
+
N
2
cos
⁡
θ
+
N
3
=
0
,where (22)
N
1
=
L
1
2
-
L
2
2
L
2
2
-
L
3
2
,
N
2
=
-
L
1
2
-
L
2
2
2
-
L
2
2
-
L
3
2
2
,
N
3
=
L
2
2
-
L
3
2
L
2
2
-
L
3
2
.It should be noted that (21) represents singular configuration (iii) expressed by L
1, L
2, L
3, and θ. Moreover, (19) is used to compute θ for the given values of L
1, L
2, and L
3. Generally, the solutions to θ obtained by solving (21) should satisfy (19). Furthermore, both (19) and (21) can be considered as two quadratics with respect to cos
⁡
θ. According to Bezout’s method, the condition that (19) and (21) have a comment root for cos
⁡
θ is as follows:(23)
M
1
M
2
N
1
N
2
M
2
M
3
N
2
N
3
-
M
1
M
3
N
1
N
3
2
=
0
.Simplifying (23) yields(24)
L
1
2
-
L
3
2
L
1
2
-
2
L
2
2
+
L
3
2
2
4
L
4
-
4
L
2
L
2
2
+
L
1
4
-
2
L
1
2
L
2
2
+
L
1
4
4
L
4
-
4
L
2
L
2
2
+
L
2
4
-
2
L
2
2
L
3
2
+
L
3
4
=
0
.Equation (24) represents the surfaces corresponding to singular configuration (iii) in the output workspace. By plotting these surfaces, the output workspace of the harvester can be obtained. An example of such plots is shown in Figure 3 with L
=
1 m and D
=
0.1 m.Figure 3
Output workspace of the water wave energy harvester withL
=
1 m and D
=
0.1 m.From Figure3, it can be seen that the singular configuration (iii) determined by (24) corresponds to four surfaces (surfaces (iv)–(vii)) in the output workspace. Moreover, surfaces (iv), (v), (vi), and (vii) correspond to expressions L
1
-
L
3
=
0, L
1
2
-
2
L
2
2
+
L
3
2
=
0, 4
L
4
-
4
L
2
L
2
2
+
L
1
4
-
2
L
1
2
L
2
2
+
L
1
4
=
0, and 4
L
4
-
4
L
2
L
2
2
+
L
2
4
-
2
L
2
2
L
3
2
+
L
3
4
=
0, respectively. It can also be observed that the output workspace of the harvester can be divided into two parts. The first part is bounded by surface (v), surface (vi), plane L
1
=
0, and plane L
3
=
0 while the second part is bounded by surfaces denoted by (iv), (vi), and (vii) and planes denoted by L
2
=
0 and L
1
=
10. This output workspace should be considered during the use and design of such a harvester.It is noted that the forward and inverse kinematics, Jacobian matrix, and workspaces should be considered when such harvester is being designed. Moreover, when the harvester is put to use, the singular configurations should be avoided. The kinematics and Jacobian matrix are used to find the singular configurations.


## 6. Dynamic Analysis

The efficiency of the water wave harvesting is highly dependent on the dynamics of the harvester. Therefore, it is of utmost importance to research the dynamics of the harvester. In this section, the dynamic model of the harvester is developed. Furthermore, in order to compare the efficiency of a conventional heaving system with that of the proposed harvester, the dynamic model of the conventional heaving system is firstly introduced. Before introducing the dynamic models of the two systems, it is assumed that the linear water waves are applied on the two systems.


### 6.1. Dynamic Model of a Conventional Heaving System

A diagram of the conventional heaving wave energy harvester [37] composed of a float, a bar magnet, and a battery is shown in Figure 4. In order to compare the efficiency of the conventional heaving system with the proposed harvester, the floats of both systems are assumed to have the same size. Moreover, in this paper, the weight of the bar magnet was neglected.Figure 4
A conventional heaving wave energy harvester [37].According to [38], the motion equation of the float, driven by linear water waves, in a conventional heaving system is given by(25)
m
+
a
w
z
d
2
z
d
t
2
+
b
r
z
+
b
v
z
+
b
p
z
d
z
d
t
+
ρ
g
A
w
p
+
N
k
s
z
=
F
z
0
cos
⁡
ω
t
+
α
z
.The coefficients in (25) are given as follows:m is the mass of the float.
a
w
z is the added mass.
b
r
z is the damping coefficient.
b
v
z is the viscous damping coefficient.
b
p
z is the power take-off coefficient.
A
w
p is the waterplane area when the body is at rest.
ρ is the density of seawater.
g is the acceleration due to gravity.
k
s is the spring constant of mooring lines and N is the number of lines (mooring restoring force).
F
z
0 is the water-induced vertical force amplitude and ω
=
2
π
/
T is the circular wave frequency (T is the wave period).
α
z is the phase angle between the wave and force.Finally, it should be noted that the computations of the above coefficients in (25) can be found in [38].


### 6.2. Dynamic Model of the Tensegrity-Based Water Wave Energy Harvester

As stated in Section2, the harvester has three degrees of freedom. Therefore, three generalized coordinates, chosen as q
=
q
1
q
2
q
3
T
=
y
z
θ
T, are needed to develop the dynamic model.In order to derive an appropriate dynamic model of the harvester, the following hypotheses are made:(i)
The links of the mechanism, except for the float, are massless.
(ii)
The springs are massless.
(iii)
There is no friction in the harvester’s revolute, prismatic, and spherical joints.The equations of motion of the harvester are developed using the Lagrangian approach; namely,(26)
d
d
t
∂
T
∂
q
˙
-
∂
T
∂
q
+
∂
E
∂
q
=
Q
k
,where T and E are the kinetic and potential energies of the harvester and Q
k is the vector of nonconservative forces acting on the system. In [37], the translation of the float along Y axis is defined as surge, the translation of the float along Z axis is defined as heave, and the rotation of the float with respect to X axis is defined as pitch. The kinetic energy, due only to the surge, heave, and pitch movements of the float, can be expressed as(27)
T
=
1
2
q
˙
T
M
q
˙
,where(28)
M
=
m
+
a
w
y
0
0
0
m
+
a
w
y
0
0
0
I
y
+
A
w
.I
y is the mass moment of inertia with respect to Y axis and A
w is added-mass moment of inertia due to pitching. The potential energies due to heaving and pitching motions of the top platform are described by McCormick [37] as(29)
E
p
z
=
1
2
ρ
g
A
w
p
q
2
2
,
E
p
θ
=
1
2
C
q
3
2
,where C is the restoring moment constant, defined for a bottom-flat body in terms of the draft. The total potential energy of the harvester becomes(30)
E
=
U
+
E
p
z
+
E
p
θ
=
1
2
ρ
g
A
w
p
q
2
2
+
1
2
C
q
3
2
+
K
σ
1
2
+
σ
2
2
-
l
0
2
+
K
σ
3
2
+
σ
4
2
-
l
0
2
.The nonconservative forces, which correspond to the radiation damping force, viscous damping force, and water wave induced forces, can be expressed as(31)
Q
k
=
-
b
v
y
q
˙
1
F
z
0
cos
⁡
ω
t
-
b
r
z
+
b
v
z
q
˙
2
M
θ
0
c
o
s
ω
t
-
b
r
θ
q
˙
3
.Substituting (27), (30), and (31) into (26), the dynamic model of the harvester can be rewritten as(32)
M
q
¨
+
B
q
˙
+
G
=
F
,where(33)
B
=
b
v
y
0
0
0
b
r
z
+
b
v
z
+
b
p
z
0
0
0
b
r
θ
,
G
=
∂
E
∂
q
1
∂
E
∂
q
2
∂
E
∂
q
3
T
,
F
=
0
F
z
o
cos
⁡
ω
t
M
θ
0
sin
⁡
ω
t
T
.The elements of G are detailed in the Appendix. For (32), it should be noted that b
v
y is viscous damping coefficient corresponding to the surge movements of the float while b
r
θ is the radiation damping coefficient due to pitching motion. M
θ
0 is the water-induced torque amplitude (applied on the float). The computations of b
v
y, b
r
θ, M
θ
0, and C can also be found in [37]. These computations are also not repeated here.


## 6.1. Dynamic Model of a Conventional Heaving System

A diagram of the conventional heaving wave energy harvester [37] composed of a float, a bar magnet, and a battery is shown in Figure 4. In order to compare the efficiency of the conventional heaving system with the proposed harvester, the floats of both systems are assumed to have the same size. Moreover, in this paper, the weight of the bar magnet was neglected.Figure 4
A conventional heaving wave energy harvester [37].According to [38], the motion equation of the float, driven by linear water waves, in a conventional heaving system is given by(25)
m
+
a
w
z
d
2
z
d
t
2
+
b
r
z
+
b
v
z
+
b
p
z
d
z
d
t
+
ρ
g
A
w
p
+
N
k
s
z
=
F
z
0
cos
⁡
ω
t
+
α
z
.The coefficients in (25) are given as follows:m is the mass of the float.
a
w
z is the added mass.
b
r
z is the damping coefficient.
b
v
z is the viscous damping coefficient.
b
p
z is the power take-off coefficient.
A
w
p is the waterplane area when the body is at rest.
ρ is the density of seawater.
g is the acceleration due to gravity.
k
s is the spring constant of mooring lines and N is the number of lines (mooring restoring force).
F
z
0 is the water-induced vertical force amplitude and ω
=
2
π
/
T is the circular wave frequency (T is the wave period).
α
z is the phase angle between the wave and force.Finally, it should be noted that the computations of the above coefficients in (25) can be found in [38].


## 6.2. Dynamic Model of the Tensegrity-Based Water Wave Energy Harvester

As stated in Section2, the harvester has three degrees of freedom. Therefore, three generalized coordinates, chosen as q
=
q
1
q
2
q
3
T
=
y
z
θ
T, are needed to develop the dynamic model.In order to derive an appropriate dynamic model of the harvester, the following hypotheses are made:(i)
The links of the mechanism, except for the float, are massless.
(ii)
The springs are massless.
(iii)
There is no friction in the harvester’s revolute, prismatic, and spherical joints.The equations of motion of the harvester are developed using the Lagrangian approach; namely,(26)
d
d
t
∂
T
∂
q
˙
-
∂
T
∂
q
+
∂
E
∂
q
=
Q
k
,where T and E are the kinetic and potential energies of the harvester and Q
k is the vector of nonconservative forces acting on the system. In [37], the translation of the float along Y axis is defined as surge, the translation of the float along Z axis is defined as heave, and the rotation of the float with respect to X axis is defined as pitch. The kinetic energy, due only to the surge, heave, and pitch movements of the float, can be expressed as(27)
T
=
1
2
q
˙
T
M
q
˙
,where(28)
M
=
m
+
a
w
y
0
0
0
m
+
a
w
y
0
0
0
I
y
+
A
w
.I
y is the mass moment of inertia with respect to Y axis and A
w is added-mass moment of inertia due to pitching. The potential energies due to heaving and pitching motions of the top platform are described by McCormick [37] as(29)
E
p
z
=
1
2
ρ
g
A
w
p
q
2
2
,
E
p
θ
=
1
2
C
q
3
2
,where C is the restoring moment constant, defined for a bottom-flat body in terms of the draft. The total potential energy of the harvester becomes(30)
E
=
U
+
E
p
z
+
E
p
θ
=
1
2
ρ
g
A
w
p
q
2
2
+
1
2
C
q
3
2
+
K
σ
1
2
+
σ
2
2
-
l
0
2
+
K
σ
3
2
+
σ
4
2
-
l
0
2
.The nonconservative forces, which correspond to the radiation damping force, viscous damping force, and water wave induced forces, can be expressed as(31)
Q
k
=
-
b
v
y
q
˙
1
F
z
0
cos
⁡
ω
t
-
b
r
z
+
b
v
z
q
˙
2
M
θ
0
c
o
s
ω
t
-
b
r
θ
q
˙
3
.Substituting (27), (30), and (31) into (26), the dynamic model of the harvester can be rewritten as(32)
M
q
¨
+
B
q
˙
+
G
=
F
,where(33)
B
=
b
v
y
0
0
0
b
r
z
+
b
v
z
+
b
p
z
0
0
0
b
r
θ
,
G
=
∂
E
∂
q
1
∂
E
∂
q
2
∂
E
∂
q
3
T
,
F
=
0
F
z
o
cos
⁡
ω
t
M
θ
0
sin
⁡
ω
t
T
.The elements of G are detailed in the Appendix. For (32), it should be noted that b
v
y is viscous damping coefficient corresponding to the surge movements of the float while b
r
θ is the radiation damping coefficient due to pitching motion. M
θ
0 is the water-induced torque amplitude (applied on the float). The computations of b
v
y, b
r
θ, M
θ
0, and C can also be found in [37]. These computations are also not repeated here.


## 7. Energy Harvesting

In this section, two energy harvesting systems are researched, respectively. One is a conventional heaving system and the other is the tensegrity-based water wave harvester. Also, the powers of the two systems have been computed, respectively. The parameters of water waves are selected asH
=
0.2 m, T
=
6 s, and h
=
100 m. H is the wave height measured from the trough to the crest while T is the wave period. h denotes the water depth. Moreover, the floats used in the two energy harvesting systems are supposed to have the same dimensions as L
=
1 m, D
=
0.1 m, and d
=
0.05 m.


### 7.1. Conventional Heaving System

For a conventional heaving system, the motion of the float is expressed by (25). For the given water wave parameters and the dimensions of the float, the coefficients of (25) can be calculated according to [38]. The results are listed in Table 1.Table 1
Conventional heaving float coefficients.
Coefficient
Value
Unit
m
51.50
kg
a
w
z
454.19
kg
b
r
z
1065.50
N⋅s/m
b
v
z
114.91
N⋅s/m
b
p
z
0
N⋅s/m
A
w
p
1
m2
N
0
—
k
s
0
N/m
F
z
0
2012.10
N
α
z
0
rad
ω
n
z
4.47
Rad/s
b
c
z
4518.60
N⋅s/m
Z
0
0.21
mSolving (25) yields the position and velocity of the float which are shown in Figure 5. The power of the heaving body is given by [37] (34)
P
z
t
=
F
z
t
d
z
t
d
t
,where F
z
t is the wave introduced heaving force on the float.Figure 5
Motion of the conventional heaving system.The power for take-off,P
(
t
), is given by the difference between the available power P
z
t and the power dissipated due to radiation P
r
z
t and viscous effects P
v
z
t: (35)
P
t
=
P
z
t
-
P
r
z
t
-
P
v
z
t
.The average power for take-off over one period of time is given by(36)
P
a
v
e
=
1
T
∫
T
P
t
d
t
.The water wave energy and power are [37](37)
E
=
ρ
g
2
H
2
T
2
b
16
π
,
(38)
P
=
ρ
g
2
H
2
T
b
32
π
i
.Applying (36) over two wave periods of the function shown in Figure 6 gives an average power P
a
v
e
=
0.154 kW. Since the float’s breadth is 1 m, then we can compare this result with the power contained in one meter of wave front. The maximum available power per meter of wave front is P
=
0.236 kW (computed by (38)). Therefore, 65.17% of the wave energy can be harvested with electrical generators.Figure 6
Power for take-off of the conventional heaving system.


### 7.2. Tensegrity-Based Wave Energy Harvester

For the harvester considered here, it has infinitesimal mechanisms inherent of many tensegrity systems. This means that there are infinitesimal deformations of the mechanism that do not require any changes in the lengths of the harvester’s components. It follows that some wave energy would not be harvested as the mechanism could deform some degree without the deformation being felt by the linear generators. However, since the deformations are infinitesimal, the effects of infinitesimal mechanisms are negligible.Let the dimensions of the float in the tensegrity-based water wave harvester be the same as the conventional heaving float. The additional constant physical parameters areL
0
=
4 m and K
=
10 N/m. Table 2 contains the values of the coefficients (computed according to [38]) for the equation of motion (see (32)).Table 2
Tensegrity-based harvester coefficients.
Coefficient
Value
Unit
m
51.50
kg
a
w
y
2.02
kg
a
w
z
454.19
kg
A
w
15.03
kg⋅m2
I
x
4.33
Kg⋅m2
C
849.58
N⋅m/rad
b
r
z
1065.50
N⋅s/m
b
r
θ
88.79
N⋅m⋅s/rad
b
v
y
114.91
N⋅s/m
b
v
z
114.91
N⋅s/m
b
p
z
0
N⋅s/m
A
w
p
1
M2
F
z
0
2012.10
N
M
θ
0
18.77
N⋅m
α
z
0
radThe simulation is performed over two wave periods, that is, 12 seconds. Figures7–9 show the position and velocity response of the float: surge, heave, and pitch.Figure 7
Surge motions of the tensegrity-based water wave energy harvester.Figure 8
Heave motions of the tensegrity-based water wave energy harvester.Figure 9
Pitch motions of the tensegrity-based water wave energy harvester.Figure10 shows the instantaneous power for take-off. The average power over two wave periods is P
a
v
e
=
0.186 kW. The power contained in one meter of wave front is P
=
0.236 kW (computed by (38)). Therefore, 78.76% of the available energy could be harvested by electrical generators. By comparing Figures 6 with 10, it is found that the proposed tensegrity-based harvester allows harvesting 13.59% more energy than a conventional heaving device under linear water wave conditions. For the conventional heaving device, the movement of the float is translation along the Z axis. It is proper to say that the conventional heaving device has one degree of freedom. However, the possible movements of the proposed harvester are rotations about the X axis and translations along the Y and Z axes (see Section 2). It is thus proper to say that the proposed harvester has three degrees of freedom. That is why the harvester can harvest more energy than a conventional device.Figure 10
Tensegrity-based harvester: power for take-off.


## 7.1. Conventional Heaving System

For a conventional heaving system, the motion of the float is expressed by (25). For the given water wave parameters and the dimensions of the float, the coefficients of (25) can be calculated according to [38]. The results are listed in Table 1.Table 1
Conventional heaving float coefficients.
Coefficient
Value
Unit
m
51.50
kg
a
w
z
454.19
kg
b
r
z
1065.50
N⋅s/m
b
v
z
114.91
N⋅s/m
b
p
z
0
N⋅s/m
A
w
p
1
m2
N
0
—
k
s
0
N/m
F
z
0
2012.10
N
α
z
0
rad
ω
n
z
4.47
Rad/s
b
c
z
4518.60
N⋅s/m
Z
0
0.21
mSolving (25) yields the position and velocity of the float which are shown in Figure 5. The power of the heaving body is given by [37] (34)
P
z
t
=
F
z
t
d
z
t
d
t
,where F
z
t is the wave introduced heaving force on the float.Figure 5
Motion of the conventional heaving system.The power for take-off,P
(
t
), is given by the difference between the available power P
z
t and the power dissipated due to radiation P
r
z
t and viscous effects P
v
z
t: (35)
P
t
=
P
z
t
-
P
r
z
t
-
P
v
z
t
.The average power for take-off over one period of time is given by(36)
P
a
v
e
=
1
T
∫
T
P
t
d
t
.The water wave energy and power are [37](37)
E
=
ρ
g
2
H
2
T
2
b
16
π
,
(38)
P
=
ρ
g
2
H
2
T
b
32
π
i
.Applying (36) over two wave periods of the function shown in Figure 6 gives an average power P
a
v
e
=
0.154 kW. Since the float’s breadth is 1 m, then we can compare this result with the power contained in one meter of wave front. The maximum available power per meter of wave front is P
=
0.236 kW (computed by (38)). Therefore, 65.17% of the wave energy can be harvested with electrical generators.Figure 6
Power for take-off of the conventional heaving system.


## 7.2. Tensegrity-Based Wave Energy Harvester

For the harvester considered here, it has infinitesimal mechanisms inherent of many tensegrity systems. This means that there are infinitesimal deformations of the mechanism that do not require any changes in the lengths of the harvester’s components. It follows that some wave energy would not be harvested as the mechanism could deform some degree without the deformation being felt by the linear generators. However, since the deformations are infinitesimal, the effects of infinitesimal mechanisms are negligible.Let the dimensions of the float in the tensegrity-based water wave harvester be the same as the conventional heaving float. The additional constant physical parameters areL
0
=
4 m and K
=
10 N/m. Table 2 contains the values of the coefficients (computed according to [38]) for the equation of motion (see (32)).Table 2
Tensegrity-based harvester coefficients.
Coefficient
Value
Unit
m
51.50
kg
a
w
y
2.02
kg
a
w
z
454.19
kg
A
w
15.03
kg⋅m2
I
x
4.33
Kg⋅m2
C
849.58
N⋅m/rad
b
r
z
1065.50
N⋅s/m
b
r
θ
88.79
N⋅m⋅s/rad
b
v
y
114.91
N⋅s/m
b
v
z
114.91
N⋅s/m
b
p
z
0
N⋅s/m
A
w
p
1
M2
F
z
0
2012.10
N
M
θ
0
18.77
N⋅m
α
z
0
radThe simulation is performed over two wave periods, that is, 12 seconds. Figures7–9 show the position and velocity response of the float: surge, heave, and pitch.Figure 7
Surge motions of the tensegrity-based water wave energy harvester.Figure 8
Heave motions of the tensegrity-based water wave energy harvester.Figure 9
Pitch motions of the tensegrity-based water wave energy harvester.Figure10 shows the instantaneous power for take-off. The average power over two wave periods is P
a
v
e
=
0.186 kW. The power contained in one meter of wave front is P
=
0.236 kW (computed by (38)). Therefore, 78.76% of the available energy could be harvested by electrical generators. By comparing Figures 6 with 10, it is found that the proposed tensegrity-based harvester allows harvesting 13.59% more energy than a conventional heaving device under linear water wave conditions. For the conventional heaving device, the movement of the float is translation along the Z axis. It is proper to say that the conventional heaving device has one degree of freedom. However, the possible movements of the proposed harvester are rotations about the X axis and translations along the Y and Z axes (see Section 2). It is thus proper to say that the proposed harvester has three degrees of freedom. That is why the harvester can harvest more energy than a conventional device.Figure 10
Tensegrity-based harvester: power for take-off.


## 8. Conclusion

A tensegrity-based water wave energy harvester was proposed in this work. The geometry of the harvester was described. The solutions to the direct and inverse kinematic problems were found by using a geometric method. The Jacobian matrix and singular configurations were subsequently computed. Then, the input and output workspaces were computed on the basis of the analysis of the obtained singular configurations. Afterwards, the dynamic analysis was performed considering the interaction with linear water waves, considering added mass, radiation damping, and viscous damping phenomena. It was shown that the proposed tensegrity-based water wave energy harvester allows harvesting 13.59% more energy than a conventional heaving system.



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*Source: 2190231-2016-06-14.xml*