Closed-Form Solutions for the Form-Finding of Regular Tensegrity Structures by Group Elements

: An analytical form-ﬁnding method for regular tensegrity structures based on the concept of force density is presented. The self-equilibrated state can be deduced linearly in terms of force densities, and then we apply eigenvalue decomposition to the force density matrix to calculate its eigenvalues. The eigenvalues are enforced to satisfy the non-degeneracy condition to fulﬁll the self-equilibrium condition. So the relationship between force densities can also be obtained, which is followed by the super-stability examination. The method has been developed to deal with planar tensegrity structure, prismatic tensegrity structure (triangular prism, quadrangular prism, and pentagonal prism) and star-shaped tensegrity structure by group elements to get closed-form solutions in terms of force densities, which satisﬁes the super stable conditions.


Introduction
Tensegrity structures refer to the stable structures that are based on the balance between their members in compressive or tensile states. This kind of structure is composed of struts carrying compressive forces and cables carrying tensile forces. The members are prestressed so that the structure gets its shape as well as rigidity in a self-equilibrated state in the absence of external loads [1]. Kennet Snelson designed and constructed the initial tensegrity model in 1947, the concept of which was then introduced by Fuller to define the very type of structure. In the last few decades, tensegrity structures have been widely studied and applied to various fields, such as civil engineering [2][3][4][5], aerospace [6,7], biology [8][9][10], mathematics [11], and robotics [12]. The process of determining the self-equilibrated configuration is known as form-finding [13], which is a necessity for tensegrity structures. The difficulties in solving the form-finding problems of tensegrity structures always lie in finding a self-equilibrated configuration, which satisfies the required specific properties of the structure [14]. A variety of studies about form-finding have been conducted. The existing form-finding methods can be divided into force design and shape design. Theories about force design have been developed a lot, such as searching feasible regions and optimizing internal forces with constraint conditions or given shape [7,15]. Shape design is usually based on force density method, dynamic relaxation method, or nonlinear structural analysis. Schek firstly developed the force density method for the form-finding of cable-nets in 1974 [16]. Following the track, many researchers have extended the concept of force density method [17][18][19][20][21][22]. The dynamic relaxation method has also been applied to tensile structures [23] and tensegrity structures [24]. Nonlinear structural analysis

Assumptions
The analytical method for form-finding bases on the equilibrium condition at the nodes. In this study, the fundamental assumptions for tensegrity structures are adopted as follows [18]: (a) Members are connected using pin joints; (b) No external loads are included, and the gravity is ignored; (c) Member failure is not considered: i.e., struts do not yield and cables do not bear compressive forces; (d) The structure is self-equilibrated and free-standing without any support; (e) The topology (connectivity between the nodes and members) of the tensegrity structure is known, and the geometrical configuration of the structure can be described in terms of nodal coordinates only.

Force Density Method
The force density of a structural member k is defined as where f k and l k are the force and length of member k, respectively.
We define E as the force density matrix, which can be written directly from the force densities q k [31,50] as −q k for nodes i and j are connected by member k 0 for other cases (2) where ϕ denotes the set of members connected to node i. It should be noted that sum of the entries in each row or column of the force density matrix in Equation (2) is zero, which is always true for free-standing structures. The equilibrium condition of tensegrity structures in terms of force density matrix E can be written as where x, y, and z are the coordinates of the structural system. In other words, the equilibrium condition of tensegrity structures can be expressed in terms of force densities.

Non-Degeneracy Condition
According to Schek [16], matrix E is always square and symmetric, and positive-definite if all members are in tension. So, for cable-nets, which only possess positive values of force densities, E is invertible, and nodal coordinates can be uniquely determined by solving the linear Equation (3) [18]. However, there are struts in tensegrity structures, which means negative elements of E. So, matrix E is invariably singular with rank deficiency. Define the rank deficiency h of E as where n is the number of free nodes. From the equilibrium equations Equation (3), the solution space is spanned by h independent vectors. If h = 1, all nodes degenerate into one node; and if h = 2, it turns into a line passing through the base node; h = 3 and h = 4 forms a two-dimensional and three-dimensional space, respectively [18]. Therefore, to obtain a non-degenerate d-dimensional (d = 2 or 3) tensegrity structure, the rank deficiency h of E should satisfy the following condition It should be noticed that Equation (5) is a necessary but not sufficient condition for non-degeneracy of condition tensegrity structures [27,29,44]. The sufficient conditions for tensegrities or self-equilibrated pin-jointed structures were investigated by Zhang and Ohsaki [57], and a rough explanation will be presented in the next section.

Stable Conditions
Self-equilibrium analysis and stability properties are the two critical points in the design of tensegrity structures, both of which can be dealt with by investigating the eigenvalues of the force density matrix [37].
A structure is regarded as stable if it satisfies [57] where d is any nodal displacement vector, and K is the tangent stiffness matrix. It should be notable that the rigid-body motions are not included in d. The tangent stiffness matrix K of a tensegrity structure can be divided into two parts where K E is the linear stiffness matrix, and K G is the geometrical stiffness matrix. Substitute Equation An adaptive way to determine whether the stability condition is achieved was proposed [57]. A tensegrity structure is said to be super-stable if it keeps stable with any level of force densities satisfying self-equilibrium without inducing material failure [11,28]. Thus increased force densities only stiffen and stabilize the structure [38]. According to Zhang and Ohsaki, the sufficient super-stability conditions of a tensegrity structure are [54]: (i) the geometry matrix is full rank; (ii) the matrix E has (d+1) nonzero eigenvalues; (iii) the matrix E is positive semi-definite; where d is the number of the dimension of the tensegrity structure. For condition (iii), if the matrix E is not positive semi-definite, geometrical stiffness matrix K G will have negative eigenvalues and become dominant over the linear stiffness matrix K E at sufficiently large force densities in the elements. So the tangent stiffness matrix K fails to be positive definite, and the structure becomes unstable. Condition (ii) guarantees the self-equilibrium condition. Conditions (ii) and (iii) can be examined by the total number of zero-eigenvalues and the sign of the minimum eigenvalue respectively after eigenvalue decomposition of the force density matrix. According to Zhang [38], condition (i) will be satisfied in most cases if the structure is divisible.
Since the stable conditions of the prestress-stable structure relate to the level of force densities, we consider the super-stability condition for tensegrity structures in this paper.

Form-Finding Method
In this section, we are going to describe the form-finding method in detail and take a few examples to make it brighter. The process of the analytical form-finding method can be divided into assumptions (Section 4.1), equilibrium analysis (Section 4.2) and super-stability examination (Section 4.3) in terms of force densities. It should be noticed that the method is only suitable for regular and symmetric tensegrity structures. A brief flowchart of this method is presented in Figure 1. Assume that the topology and the type of each member are given. symmetric tensegrity structures. A brief flowchart of this method is presented in Figure 1. Assume that the topology and the type of each member are given.

Assumptions
Set up force density variables for all members. If the member is a cable, we use numbers as the subscripts, i.e., q1, q2, q3, ...; otherwise, we use letters, i.e., qs, qb, ... Moreover, the number of variables of force densities can be reduced by making use of symmetry properties.

Equilibrium Analysis
As shown in Figure 1, the equilibrium analysis includes three main steps, the construction, decomposition, and ensuring the non-degenerate condition analysis of the force density matrix E. With the force densities assumed, the force density matrix E can be obtained according to Equation (2). Then the eigenvalues λ1, λ2, λ3, ... λn of matrix E can be determined by solving the equation [38] det( ) 0   IE (9) where I is the unit matrix. The expanded form of Equation (9) is shown as where A0(·), A1(·), ..., An-1(·) denote the polynomial functions in terms of the force densities and Equation (10) can be obtained by eigenvalue decomposition. According to Section 2.3, the non-degeneracy condition has to be achieved to ensure self-equilibrium. Therefore, Equation (5) should be fulfilled. To achieve that, there should exist enough zero eigenvalues. For example, if there is a three-dimensional tensegrity structure, at least four of the eigenvalues should be zero. With the obtained expressions of eigenvalues, we should notice the number of zero eigenvalues before deciding expressions that are definitely greater than zero. Then, for the rest of eigenvalues, the value of which we cannot determine directly by the expressions set one or a few of them equal to zero so that a relationship between force densities can be obtained. The judging process is summarized as follows:

Assumptions
Set up force density variables for all members. If the member is a cable, we use numbers as the subscripts, i.e., q 1 , q 2 , q 3 , ...; otherwise, we use letters, i.e., q s , q b , ... Moreover, the number of variables of force densities can be reduced by making use of symmetry properties.

Equilibrium Analysis
As shown in Figure 1, the equilibrium analysis includes three main steps, the construction, decomposition, and ensuring the non-degenerate condition analysis of the force density matrix E. With the force densities assumed, the force density matrix E can be obtained according to Equation (2). Then the eigenvalues λ 1 , λ 2 , λ 3 , ... λ n of matrix E can be determined by solving the equation [38] det(λI − E) = 0 (9) where I is the unit matrix. The expanded form of Equation (9) is shown as where A 0 (·), A 1 (·), ..., A n-1 (·) denote the polynomial functions in terms of the force densities and Equation (10) can be obtained by eigenvalue decomposition. According to Section 2.3, the non-degeneracy condition has to be achieved to ensure self-equilibrium. Therefore, Equation (5) should be fulfilled. To achieve that, there should exist enough zero eigenvalues. For example, if there is a three-dimensional tensegrity structure, at least four of the eigenvalues should be zero. With the obtained expressions of eigenvalues, we should notice the number of zero eigenvalues before deciding expressions that are definitely greater than zero. Then, for the rest of eigenvalues, the value of which we cannot determine directly by the expressions set one or a few of them equal to zero so that a relationship between force densities can be obtained. The judging process is summarized as follows: Symmetry 2020, 12, 374 6 of 17 Step 1: Calculate the rank deficiency h of the force density matrix E (Equation (4)).
Step 2: Notice the number of expressions that are equal to zero and record it as t 1 . Usually, there is only one expression of eigenvalue equal to zero after the decomposition. We assume this eigenvalue as λ 1 .
Step 3: Find all the expressions of eigenvalues that are definitely greater than zero and denote the number of eigenvalues as t 2 .
Step 4: Select a few of the rest expressions of eigenvalues equal to zero. Record the number of expressions that are assumed to zero as t 3 , and t 3 + t 1 ≥ h should be satisfied.
Step 5: Get the relationship between force densities from the equations in Step 4.

Super-Stability Examination
According to Section 3, the super-stability condition is usually satisfied if conditions (ii) and (iii) are fulfilled. It should be noticed that the non-degeneracy condition is identical to the condition (ii). So only condition (iii) remains to be examined.
The number of all eigenvalues is m, which includes (t 3 + t 1 ) zero eigenvalues, t 2 positive eigenvalues and (m-t 1 -t 2 -t 3 ) uncertain eigenvalues. If all of the uncertain eigenvalues are not less than zero, the structure is super stable. N1, N2, N3, ... represent nodes while M1, M2, ... denote members. A planar tensegrity structure, three types of prismatic tensegrity structures (triangular prism, quadrangular prism, pentagonal prism) and a star-shaped tensegrity structure will be analyzed in the next section.

Examples
In this section, examples are presented based on the proposed method. A planar tensegrity structure, three types of prismatic tensegrity structure (triangular prism, quadrangular prism, pentagonal prism) and a star-shaped tensegrity structure are selected as examples. The detailed descriptions of prismatic and star-shaped tensegrity structures can be found in chapters 7 and 8 of reference [57].

Planar Tensegrity
For a simple planar tensegrity structure with six cables and three struts, an explicit study is carried out to demonstrate the capability of the investigated method.
Suppose that the tensegrity structure, which is shown in Figure 2, has three kinds of force densities: q 1 , q 2 , and q s . q 1 represents the force density of cable elements M1, M3, and M5; q 2 denotes the force density of cable elements M2, M4, and M6; q s indicates the force density of struts. Then the force density matrix E can be given as Eigenvalues of the matrix E in Equation (11) are From Equation (12), it should be noticed that q s is a negative value. Assume that λ 2 is greater than zero. Therefore, λ 3 and λ 5 are always greater than zero. Then assume λ 4 and λ 6 equal to zero, so relationship among q 1 , q 2 , and q s is obtained as To ensure that q 1 is positive, q 2 must be greater than -q s . q 1 is known by Equation (13), and therefore λ 2 is greater than 0. In conclusion, λ 2 , λ 3 , and λ 5 in Equation (12) are greater than zero while the other three are equal to zero, which satisfies the super-stability condition of tensegrity structures. To ensure that q1 is positive, q2 must be greater than -qs. q1 is known by Equation (13), and therefore λ2 is greater than 0. In conclusion, λ2, λ3, and λ5 in Equation (12) are greater than zero while the other three are equal to zero, which satisfies the super-stability condition of tensegrity structures.

Figure 2.
Two-dimensional rotary symmetry planar tensegrity structure with six cables (black lines) and three struts (green lines).

Triangular Prism
Suppose that the triangular prism tensegrity structure, shown in Figure 3, has four kinds of force densities: q1, q2, qb, and qs. q1 represents the force density of cable element M1, M2, M3; qb denotes the force density of M4, M5, M6; q2 denotes the force density of M7, M8, M9, and qs denotes the force density of struts. Then the force density matrix E can be given as Eigenvalues of the matrix E in Equation (14) are shown as where q1, q2, qb are all positive values, and qs is negative because of the compression in struts. Set λ2 equal to zero to satisfy the non-degeneracy condition. Then, we have bs qq  Thus, it should be noticed that λ3 and λ5 are always greater than zero. Set λ4 and λ6 also equal to zero. Thus, the relationship among q1, q2, qb, and qs is obtained as By substituting Equation (16) into Equation (17), we obtain

Triangular Prism
Suppose that the triangular prism tensegrity structure, shown in Figure 3, has four kinds of force densities: q 1 , q 2 , q b , and q s . q 1 represents the force density of cable element M1, M2, M3; q b denotes the force density of M4, M5, M6; q 2 denotes the force density of M7, M8, M9, and q s denotes the force density of struts. Then the force density matrix E can be given as Eigenvalues of the matrix E in Equation (14) are shown as where q 1 , q 2 , q b are all positive values, and q s is negative because of the compression in struts. Set λ 2 equal to zero to satisfy the non-degeneracy condition. Then, we have Thus, it should be noticed that λ 3 and λ 5 are always greater than zero. Set λ 4 and λ 6 also equal to zero. Thus, the relationship among q 1 , q 2 , q b , and q s is obtained as Symmetry 2020, 12, 374 8 of 17 By substituting Equation (16) into Equation (17), we obtain q s = − 3q 1 q 2 (18)  If both Equation (16) and Equation (17) are satisfied, the structure fulfills the non-degeneracy condition because four eigenvalues are zero. λ3 and λ5 in Equation (15) are greater than zero; the other eigenvalues in Equation (15) are all equal to zero so that the structure also satisfies the super-stability condition for tensegrity structures. The results are found the same with analytical solutions derived by Pellegrino and Tibert [27]. The relationship between variables q1, q2, and qs are plotted in the three-dimensional vector graph, shown in Figure 4.

Quadrangular Prism Tensegrity Structure
Suppose that the quadrangular prism tensegrity structure, which is shown in Figure 5, has four kinds of force densities: q1, q2, qb, and qs. q1 represents the force density of elements M1, M2, M3, M4; q2 represents the force density of M5, M6, M7, M8; qb represents the force density of M9, M10, M11, M12; and qs represents that of struts. Then the force density matrix E can be given as If both Equation (16) and Equation (17) are satisfied, the structure fulfills the non-degeneracy condition because four eigenvalues are zero. λ 3 and λ 5 in Equation (15) are greater than zero; the other eigenvalues in Equation (15) are all equal to zero so that the structure also satisfies the super-stability condition for tensegrity structures. The results are found the same with analytical solutions derived by Pellegrino and Tibert [27]. The relationship between variables q 1 , q 2 , and q s are plotted in the three-dimensional vector graph, shown in Figure 4. If both Equation (16) and Equation (17) are satisfied, the structure fulfills the non-degeneracy condition because four eigenvalues are zero. λ3 and λ5 in Equation (15) are greater than zero; the other eigenvalues in Equation (15) are all equal to zero so that the structure also satisfies the super-stability condition for tensegrity structures. The results are found the same with analytical solutions derived by Pellegrino and Tibert [27]. The relationship between variables q1, q2, and qs are plotted in the three-dimensional vector graph, shown in Figure 4.

Quadrangular Prism Tensegrity Structure
Suppose that the quadrangular prism tensegrity structure, which is shown in Figure 5, has four kinds of force densities: q1, q2, qb, and qs. q1 represents the force density of elements M1, M2, M3, M4; q2 represents the force density of M5, M6, M7, M8; qb represents the force density of M9, M10, M11, M12; and qs represents that of struts. Then the force density matrix E can be given as

Quadrangular Prism Tensegrity Structure
Suppose that the quadrangular prism tensegrity structure, which is shown in Figure 5, has four kinds of force densities: q 1 , q 2 , q b , and q s . q 1 represents the force density of elements M1, M2, M3, M4; q 2 represents the force density of M5, M6, M7, M8; q b represents the force density of M9, M10, M11, M12; and q s represents that of struts. Then the force density matrix E can be given as Symmetry 2020, 12, 374 Symmetry 2020, 12, x FOR PEER REVIEW 10 of 17 Bring Equation (21) and Equation (25) into Equation (20), we obtain λ6 and λ8 q q q q q q q q q q q q q q q q Therefore, there are only three eigenvalues equal to zero while others are greater than zero. The non-degeneracy condition for tensegrity structures is not fulfilled. The relationship between variables q1, q2, and qs are plotted in the three-dimensional vector graph, shown in Figure 6. Eigenvalues of the matrix E in Equation (19) are q 1 , q 2 and q b , the force densities of cable members, are all positive values, and q s , the force density of struts, is negative. Presume that λ 2 equals to zero. Then It should be noticed that λ 3 , λ 5 and λ 7 are always greater than zero. Thus, the sign of λ 1 , λ 2 , λ 3 , λ 5 , and λ 7 are determined. Set λ 6 and λ 8 or λ 4 equal to zero.
Case 2: If we set λ 4 equal to zero, the relationship between variables by substituting Equation (21) is shown as Bring Equation (21) and Equation (25) into Equation (20), we obtain λ 6 and λ 8 Therefore, there are only three eigenvalues equal to zero while others are greater than zero. The non-degeneracy condition for tensegrity structures is not fulfilled. The relationship between variables q 1 , q 2 , and q s are plotted in the three-dimensional vector graph, shown in Figure 6.
Bring Equation (21) and Equation (25) into Equation (20), we obtain λ6 and λ8 q q q q q q q q q q q q q q q q Therefore, there are only three eigenvalues equal to zero while others are greater than zero. The non-degeneracy condition for tensegrity structures is not fulfilled. The relationship between variables q1, q2, and qs are plotted in the three-dimensional vector graph, shown in Figure 6.

Quadrangular Prism Tensegrity Structure
Suppose that the pentagonal prism tensegrity structure, as shown in Figure 7, has four kinds of force densities, q 1 , q 2 , q b , and q s ; q 1 represents the force density of elements M1, M2, M3, M4, M5; q 2 represents the force density of M11, M12, M13, M14, M15; q b represents the force density of M6, M7, M8, M9, M10; and q s represents the force density of struts. Then the force density matrix E can be given as force densities, q1, q2, qb, and qs; q1 represents the force density of elements M1, M2, M3, M4, M5; q2 represents the force density of M11, M12, M13, M14, M15; qb represents the force density of M6, M7, M8, M9, M10; and qs represents the force density of struts. Then the force density matrix E can be given as  The first two eigenvalues of the matrix E in Equation (25) are shown as Set λ2 equal to zero. We obtain Substitute Equation (29) into the other eigenvalues From Equation (30), since q 1 , q 2 , q b are all positive values, λ 3 , λ 5 , λ 7 , and λ 9 are always greater than zero. However, the sign of λ 4 , λ 6 , λ 8 and λ 10 cannot be determined yet.
Case 1: Set both λ 4 and λ 8 equal to zero, the relationship among q 1 , q 2 , q s is obtained as Substitute Equation (31) into Equation (30) so the structure does not meet the requirement of the super-stability condition. Case 2: Set λ 6 and λ 10 equal to zero, the relationship among q 1 , q 2 , q s is obtained as Substitute Equation (33) into Equation (30), λ 4 and λ 8 are shown as If Equation (32) are satisfied, the structure achieves the non-degeneracy condition because four eigenvalues, λ 1 , λ 2 , λ 6 , and λ 10 , are zero while the other eigenvalues are greater than zero. So the super-stability condition for tensegrity structures is satisfied. The relationship between variables q 1 , q 2 and q s are plotted in the three-dimensional vector graph, shown in Figure 8.
Case 1: Set both λ4 and λ8 equal to zero, the relationship among q1, q2, qs is obtained as Substitute Equation (31) If Equation (32) are satisfied, the structure achieves the non-degeneracy condition because four eigenvalues, λ1, λ2, λ6, and λ10, are zero while the other eigenvalues are greater than zero. So the super-stability condition for tensegrity structures is satisfied. The relationship between variables q1, q2 and qs are plotted in the three-dimensional vector graph, shown in Figure 8.

Star-Shaped Tensegrity Structure
Suppose that the star-shaped tensegrity structure, as shown in Figure 9, has four kinds of force densities: q 1 , q 2 , q b , and q s. q 1 represents the force density of elements M1, M2, M3; q 2 represents the force density of elements M4, M5, M6; q b represents the force density of elements M7, M8, M9, M10; and q s represents that of struts. Then the force density matrix E can be given as Symmetry 2020, 12, x FOR PEER REVIEW 14 of 17 Figure 9. Star-shaped rotary symmetry tensegrity structure with nine cables (black lines) and three struts (green lines).

Figure 10.
Vector graph of the solutions for the relationship between variables q1, q2, and qb of the star-shaped rotary symmetry tensegrity structure.

Conclusions
An analytical method based on the force density method was proposed in this paper to get closed-form solutions for regular tensegrity structures with symmetry. The necessary condition for non-degeneracy of a tensegrity structure is that three or four eigenvalues of the matrix E are zero (two-dimensional tensegrities and three-dimensional tensegrities, respectively). A closed-form solution of the force densities is induced by setting the necessary number of eigenvalues to zero. The first two eigenvalues of the matrix E in Equation (33) are shown as Set eigenvalue λ 2 equal to zero. We obtain Substitute Equation (37) into the other eigenvalues From Equation (36), since q 1 , q 2 , q b are all positive value, λ 3 , λ 4 , λ 5 , and λ 7 are always greater than zero. Set both λ 6 and λ 8 equal to zero, the relationship among q 1 , q 2 , q s is obtained as If both Equations (35) and (37) are satisfied, the structure achieves the non-degeneracy condition because four eigenvalues are zero while the others are greater than zero in Equation (36) so that the super-stability condition for tensegrity structures is satisfied. The results of structures in Section 5.2 are found to be in good agreement with analytical solutions conjectured by Estrada, which used a unitary value [58]. The relationship between variables q 1 , q 2 , and q s are plotted in the three-dimensional vector graph, shown in Figure 10.

Conclusions
An analytical method based on the force density method was proposed in this paper to get closed-form solutions for regular tensegrity structures with symmetry. The necessary condition for non-degeneracy of a tensegrity structure is that three or four eigenvalues of the matrix E are zero (two-dimensional tensegrities and three-dimensional tensegrities, respectively). A closed-form solution of the force densities is induced by setting the necessary number of eigenvalues to zero. Then, the super-stability conditions for tensegrity structures are examined.
Several examples were investigated comprehensively in this paper to verify the capability of the proposed technique. A planar tensegrity structure, three types of prismatic tensegrity structure (triangular prism, quadrangular prism, pentagonal prism) and a star-shaped tensegrity structure are included as examples, all of which have regular shapes. After the eigenvalue decomposition of the matrix force density E, a closed-form solution can be conjectured according to the non-degeneracy condition. The closed-form solution can help in understanding the design of regular tensegrity structures in many fields, both in robotics and architecture, as well as deployable structures. Though the proposed method is efficient enough to solve the examples mentioned above, more advanced methods should be developed for form-finding of more complex tensegrities, which will be studied in the future. Figure 10. Vector graph of the solutions for the relationship between variables q 1 , q 2 , and q b of the star-shaped rotary symmetry tensegrity structure.

Conclusions
An analytical method based on the force density method was proposed in this paper to get closed-form solutions for regular tensegrity structures with symmetry. The necessary condition for non-degeneracy of a tensegrity structure is that three or four eigenvalues of the matrix E are zero (two-dimensional tensegrities and three-dimensional tensegrities, respectively). A closed-form solution of the force densities is induced by setting the necessary number of eigenvalues to zero. Then, the super-stability conditions for tensegrity structures are examined.
Several examples were investigated comprehensively in this paper to verify the capability of the proposed technique. A planar tensegrity structure, three types of prismatic tensegrity structure (triangular prism, quadrangular prism, pentagonal prism) and a star-shaped tensegrity structure are included as examples, all of which have regular shapes. After the eigenvalue decomposition of the matrix force density E, a closed-form solution can be conjectured according to the non-degeneracy condition. The closed-form solution can help in understanding the design of regular tensegrity structures in many fields, both in robotics and architecture, as well as deployable structures. Though the proposed method is efficient enough to solve the examples mentioned above, more advanced methods should be developed for form-finding of more complex tensegrities, which will be studied in the future.