# Kinematics, Singularity, and Workspaces of a Planar 4-Bar Tensegrity Mechanism

Authors: Zhifei Ji; Tuanjie Li; Min Lin
Journal: Journal of Robotics
 (2014)

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## Abstract

Compared with conventional mechanisms, tensegrity mechanisms have many attractive characteristics such as light weight, high ratio of strength to weight, and accuracy of modeling. In this paper, the kinematics, singularity, and workspaces of a planar 4-bar tensegrity mechanism have been investigated. Firstly, the analytical solutions to the forward and inverse kinematic problems are found by using an energy based method. Secondly, the definition of a tensegrity mechanism’s Jacobian is introduced. As a consequence, the singularity analysis of the planar 4-bar tensegrity mechanism has been completed. Thirdly, the actuator and output workspaces are mapped. Finally, some attractive characteristics of the mechanism are concluded.


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## Body

## 1. Introduction

As the complexity of robotic applications in space increases, new demands for lighter and quicker mechanisms arise. Tensegrity mechanisms can be viewed as one alternative solution to conventional mechanisms. For this reason, a planar 4-bar tensegrity mechanism is proposed in this paper and the kinematics and statics of the mechanism are studied.The term tensegrity was created by Fuller [1] as a combination of the words tensional and integrity. It seems that he was inspired by some novel sculptures completed by Snelson [2]. The detailed history of tensegrity systems was reviewed by Motro [3]. Tensegrity systems are formed by a set of compressive components and tensile components. Tensegrity systems have advantages of light weight, deployability, being easily tunable, and so forth. Due to these attractive characteristics, tensegrity systems have been used in several disciplines such as architecture, biology, aerospace, mechanics, and robotics during the last fifty years [4].The applications of tensegrity systems can be divided into two main branches. One application is used as structures and the other one is used as mechanisms. In addition, the research of tensegrity structures has two main issues, which are the form-finding problem and the behaviors under external loads. The form finding of a tensegrity structure corresponds to the computation of the structure’s equilibrium shape for a given set of parameters. This problem has been studied by many authors [5–7]. Moreover, a review of form-finding methods is given by Tibert and Pellegrino [5]. The behaviors of tensegrity structures under external loads have also been researched by many researchers [8, 9]. A static analysis of tensegrity structures was given by Juan and Mirats Tur [10]. When some components (rigid rods or springs) are actuated, tensegrity mechanisms can be obtained. In the past twenty years, several tensegrity mechanisms have been proposed [11–17]. The proposed applications of tensegrity mechanisms range from a flight simulator [18], a space telescope [19], and a robot [20] to a sensor [21]. For tensegrity mechanisms, the dynamics and open problems have been reviewed by Mirats Tur and Juan [22].During the past twenty years, considerable research has been performed on the control, statics, and dynamics of class-1 tensegrity mechanisms. However, there are few articles relating to class-2 tensegrity mechanisms, especially on the study of them. The main objective of this paper is to perform an analytical investigation of the kinematics, singularity, and workspaces of a planar 4-bar (class-2) tensegrity mechanism. The definitions of class-1 and class-2 tensegrity systems are given by Skelton and Oliveira [4].Marc Arsenault and Gosselin [23] introduced the conditions of static balancing of tensegrity mechanisms, which leads to important simplifications in the analysis of tensegrity mechanisms. These conditions of the static balancing of tensegrity mechanism were used in this paper to find the analytical solutions to the forward and inverse kinematic problems of the planar 4-bar tensegrity mechanism.This paper is organized as follows. In Section2, the planar 4-bar tensegrity mechanism was introduced. The forward and inverse kinematic analysis was performed in Section 3. Furthermore, the Jacobian was computed and the singularity configurations were obtained in Section 4. The actuator and output workspaces were mapped, respectively, in Section 5. Finally, conclusions were reported in Section 6.


## 2. Mechanism Description

A diagram of the planar 4-bar tensegrity mechanism considered here is shown in Figure1. It is composed of three springs, four bars, and two prismatic actuators. The bars of length L are joining node pairsDE,CF,BC, andAD while the springs are joining node pairsCE,EF, andDF. Moreover, the prismatic actuators are used to vary the distances between node pairsAC andBD.Figure 1
Planar 4-bar tensegrity mechanism.As shown in Figure1, the bars, springs, and prismatic actuators are connected to each other at each node by 2D frictionless rotational joints. Furthermore, nodes A and B are fixed to the ground and the whole mechanism lies in a horizontal plane. From Figure 1, it can be seen that joints A and B have restrained translational DOF in x- and y-axes but free rotational DOF. In addition, the angle between thex-axis and the bar joining nodesAD is defined as α while the angle between the horizontal and the bar joining nodesBC is defined as β. Moreover, the angle between thex-axis and the bar joining nodesCF is defined as γ while the angle between the horizontal and the bar joining nodesDE is defined as θ. From Figure 1, it can also be observed that the distance between nodes A and B is L.In Figure1, the stiffness of the springs of lengths ℓi (i=1,2,3) is denoted by K. Furthermore, it is assumed that the springs have zero free length. This hypothesis is not problematic since, as was explained by Gosselin [24] and Shekarforoush et al. [25], virtual zero-free-length spring can be created by extending the actual spring beyond its attachment point. The actuator lengths (ρ1 and ρ2) are chosen as the mechanism’s input variables while the Cartesian coordinates of node F (x and y) are chosen as the mechanism’s output variables. From Figure 1, it can be seen that the shape of the mechanism can be determined for the given actuator lengths. Moreover, the Y coordinates of nodes C and D are chosen to be nonnegative. For this reason, the ranges imposed to α and β can be obtained as follows:
(1)0≤α≤π,0≤β≤π.
Furthermore, the ranges imposed to θ and γ are chosen as
(2)0<γ<π,0<θ<π.From Figure1, it can be observed that the mechanism is unstable in the case ofDE andCF being parallel. In particular, in this case, the mechanism cannot be in equilibrium. For this reason, γ+θ≠π.


## 3. Kinematic and Static Analysis

For a tensegrity mechanism, the kinematics and statics should be considered simultaneously since the relationships between the input and output variables depend not only on the mechanism’s geometry but also on the internal forces in the springs. For this reason, it is always assumed that the planar 4-bar tensegrity mechanism is in equilibrium. Under this assumption, the explicit relationships between the input and output variables can be developed.


### 3.1. Forward Kinematic Analysis

For the mechanism considered here, the forward kinematic analysis consists in computing the Cartesian coordinates of nodeF for the given actuator lengths. From Figure 1, it can be seen that when the actuator lengths (ρ1 and ρ2) are specified, the triangle formed by nodes A, B, and D and the triangle formed by nodes A, B, and C are determined. As a consequence, the movement of node E is confined to a rotation centered on node D. Moreover, the movement of node F is also constrained to a rotation centered on node C. For the mechanism shown in Figure 1, the potential energy of the system will reach its minimum when the mechanism is in equilibrium. Therefore, the relationships between the input and output variables can be obtained by minimizing the potential energy with respect to a set of parameters, chosen here as θ and γ.From Figure1, the coordinates of nodes A, B, C, and D can be obtained as follows:
(3)PA=[00],PB=[L0],PC=[L(1-cos⁡β)Lsinβ],PD=[Lcos⁡αLsinα].
With the coordinates of nodes C and D now known, the coordinates of nodes E and F can be written in the following form:
(4)PE=[L(cos⁡α-cos⁡θ)L(sinα+sinθ)],PF=[L(1-cos⁡β+cos⁡γ)L(sinβ+sinγ)].Since the Cartesian coordinates of nodeF are chosen as the output variables, we therefore write
(5)x=L(1-cos⁡β+cos⁡γ),y=L(sinβ+sinγ).Furthermore, the lengths of the springsCE,EF, andFD can be easily calculated according to (3) and (4). Then, the potential energy of the mechanism takes the form
(6)U=∑i=1312Kℓi2=KL22[13+6(cos⁡αcos⁡β-sinαsinβ-cos⁡α-cos⁡β)hhhhhhh+4(1-cos⁡β-cos⁡α)hhhhhhhl·(cos⁡γ+cos⁡θ)+2cos⁡(θ+γ)hhhhhhh+(sinα-sinβ)(sinθ-sinγ)].As shown in Figure1, a cosine law for the triangle formed by nodes A, B, and D can be written as
(7)cos⁡α=2L2-ρ222L2.
Moreover, a cosine law for the triangle formed by nodes A, B, and C can be written as
(8)cos⁡β=2L2-ρ122L2.Due to the ranges imposed toα and β, the expressions for sin⁡α and sin⁡β can be derived as follows:
(9)sinα=1-cos⁡2α=ρ24L2-ρ222L2,(10)sinβ=1-cos⁡2β=ρ14L2-ρ122L2.Substituting (7), (8), (9), and (10) into (6), the potential energy of the mechanism can be expressed as
(11)U=KL22[32L4(ρ12ρ22-∏i=12ρi4L2-ρi2)+7hhhhh+2L2(ρ12+ρ22-2L2)(cos⁡γ+cos⁡θ)hhhhh+2L2(ρ24L2-ρ22-ρ14L2-ρ12)hhhhh×(sinθ-sinγ)+2cos⁡⁡(θ+γ)[32L4(ρ12ρ22-∏i=12ρi4L2-ρi2)+7].By differentiatingU with respect to θ and γ separately and equating the results to zero, the following equations are generated:
(12)∂U∂θ=KL22[2L2(ρ12+ρ22-2L2)(-sinθ)+2L2HHHhhlhH×(ρ24L2-ρ22-ρ14L2-ρ12)HHHjjjlhH×cos⁡θ-2sin⁡(θ+γ)2L2]=0,(13)∂U∂γ=KL22[2L2(ρ12+ρ22-2L2)(-sinγ)-2L2HhhHhHH×(ρ24L2-ρ22-ρ14L2-ρ12)HhHhhHH×(cos⁡γ)-2sin⁡(θ+γ)2L2]=0.Due toγ+θ≠π, the following equation can be obtained by eliminating the expression sin⁡(θ+γ) from (12) and (13):
(14)θ=γ+2tan-1×[(ρ24L2-ρ22-ρ14L2-ρ12)(ρ12+ρ22-2L2)].
Substituting (14) into (13) yields
(15)[-N1(N02+N12)-3N0N1]t14+2[N0(N02+N12)-(N02-N12)]t13-2N0N1t12+2[N0(N02+N12)+(N02-N12)]t1+N1(N02+N12)+N0N1=0,
where t1=tan(γ/2). The expressions for N0 and N1 are detailed in Appendix A. It is apparent that (15) is an equation of degree 4 in t1. Solving (15) for t1, we obtain
(16)t1j=14[(-1)j+1(E12-E2+E5)1/2-E1]+δ14hhhh×[((E12-E2+E5)1/2-8E5)-1(E1+(-1)j(E12-E2+E5)1/2hhhhhhlhh-8(E1E5-2M3)(-1)j(E1+(-1)j(E12-E2+E5)1/2)hhhhh×((E12-E2+E5)1/2-8E5)-1]1/2,
where δ1=±1. Moreover, the variables Eξ (ξ=1,…,5) are also detailed in Appendix A. t1j represents four solutions to (15) for j=1,2. Computing the arctangent of t1 generates a unique solution for γ due to the range imposed to γ. Moreover, negative solutions for t1 should be eliminated. With the solutions for γ now known, the solutions to the forward kinematic problem can be found by substituting (8) and (10) into (5).


### 3.2. Inverse Kinematic Analysis

The inverse kinematic analysis of the mechanism corresponds to the computation of the actuator lengths for the given Cartesian coordinates of nodeF.Eliminating the parameter,γ, from (5) yields
(17)[(x-L)2+y2-2L(x-L)]t22-4yLt2+(x-L)2+y2+2L(x-L)=0,
where t2=tan(β/2).Solving (17) for t2, we obtain
(18)t2=([(x-L)2+y2+2L(x-L)]}1/22yL±{4y2L2-[(x-L)2+y2-2L(x-L)]HHHHHHH×[(x-L)2+y2+2L(x-L)]}1/2)×((x-L)2+y2-2L(x-L))-1.
Substituting the expression t2=tan⁡(β/2) into (8) yields
(19)ρ1=2Lt21+t22.
Therefore, substituting (18) into (19), two solutions for ρ1 are found. In the following paragraphs, we will find the solutions for ρ2.By substituting (8) and (10) into (5), we obtain
(20)sinγ=2Ly-ρ14L2-ρ122L2,cos⁡γ=2Lx-ρ122L2.Moreover, substituting (7), (8), (9), and (10) into (14) yields
(21)θ=γ+2tan-1[(2L2sinα-ρ14L2-ρ12)(ρ12-2L2cos⁡α)].
Substituting (20) and (21) into (13) and rearranging yields
(22)λ1t34+λ2t33+λ3t32+λ4t3+λ5=0,
where t3=tan(α/2). The expressions for λζ (ζ=1,…,5) are detailed in Appendix B.It can be seen that (22) is an equation of degree 4 in t3. Therefore, solving (22), the following is obtained:
(23)t3μ=(-1)μ+1(4λ12η-4λ1λ3+λ22)1/2-λ22λ1+δ22{14λ12[λ2+(-1)μ(4λ12φ-4λ1λ3+λ22)-1/2]2hhhhh-(-1)μ2(λ2η-2λ4)hhhhh×(4λ12φ-4λ1λ3+λ22)-1/2-2η14λ12}1/2,
where
(24)η=λ33λ1+[-Q2+Q24-P327]1/3+[-Q2-Q24-P327]1/3,
and where
(25)P=1λ12[λ2λ4-4λ1λ5-13λ32],Q=1λ13[2λ3327+λ33(λ2λ4-4λ1λ5)hhhhhhh+(4λ1λ3λ5-λ22λ5-λ1λ42)2λ3327].
Substituting the expression t3=tan(α/2) into (7), we obtain
(26)ρ2=2Lt31+t32.In (23), it should be noted that δ2=±1. Moreover, t3μ represents four solutions to (22) for μ=1,2. Considering the range imposed to α, negative solutions for t3 should be eliminated. Substituting (23) into (26), the solutions for ρ2 are found.Two solutions forρ1 are given by (19). Four solutions for ρ2 can be found by (26). In particular, if these solutions for ρ1 and ρ2 are all nonnegative, eight solutions to the inverse kinematic problem are found.


## 3.1. Forward Kinematic Analysis

For the mechanism considered here, the forward kinematic analysis consists in computing the Cartesian coordinates of nodeF for the given actuator lengths. From Figure 1, it can be seen that when the actuator lengths (ρ1 and ρ2) are specified, the triangle formed by nodes A, B, and D and the triangle formed by nodes A, B, and C are determined. As a consequence, the movement of node E is confined to a rotation centered on node D. Moreover, the movement of node F is also constrained to a rotation centered on node C. For the mechanism shown in Figure 1, the potential energy of the system will reach its minimum when the mechanism is in equilibrium. Therefore, the relationships between the input and output variables can be obtained by minimizing the potential energy with respect to a set of parameters, chosen here as θ and γ.From Figure1, the coordinates of nodes A, B, C, and D can be obtained as follows:
(3)PA=[00],PB=[L0],PC=[L(1-cos⁡β)Lsinβ],PD=[Lcos⁡αLsinα].
With the coordinates of nodes C and D now known, the coordinates of nodes E and F can be written in the following form:
(4)PE=[L(cos⁡α-cos⁡θ)L(sinα+sinθ)],PF=[L(1-cos⁡β+cos⁡γ)L(sinβ+sinγ)].Since the Cartesian coordinates of nodeF are chosen as the output variables, we therefore write
(5)x=L(1-cos⁡β+cos⁡γ),y=L(sinβ+sinγ).Furthermore, the lengths of the springsCE,EF, andFD can be easily calculated according to (3) and (4). Then, the potential energy of the mechanism takes the form
(6)U=∑i=1312Kℓi2=KL22[13+6(cos⁡αcos⁡β-sinαsinβ-cos⁡α-cos⁡β)hhhhhhh+4(1-cos⁡β-cos⁡α)hhhhhhhl·(cos⁡γ+cos⁡θ)+2cos⁡(θ+γ)hhhhhhh+(sinα-sinβ)(sinθ-sinγ)].As shown in Figure1, a cosine law for the triangle formed by nodes A, B, and D can be written as
(7)cos⁡α=2L2-ρ222L2.
Moreover, a cosine law for the triangle formed by nodes A, B, and C can be written as
(8)cos⁡β=2L2-ρ122L2.Due to the ranges imposed toα and β, the expressions for sin⁡α and sin⁡β can be derived as follows:
(9)sinα=1-cos⁡2α=ρ24L2-ρ222L2,(10)sinβ=1-cos⁡2β=ρ14L2-ρ122L2.Substituting (7), (8), (9), and (10) into (6), the potential energy of the mechanism can be expressed as
(11)U=KL22[32L4(ρ12ρ22-∏i=12ρi4L2-ρi2)+7hhhhh+2L2(ρ12+ρ22-2L2)(cos⁡γ+cos⁡θ)hhhhh+2L2(ρ24L2-ρ22-ρ14L2-ρ12)hhhhh×(sinθ-sinγ)+2cos⁡⁡(θ+γ)[32L4(ρ12ρ22-∏i=12ρi4L2-ρi2)+7].By differentiatingU with respect to θ and γ separately and equating the results to zero, the following equations are generated:
(12)∂U∂θ=KL22[2L2(ρ12+ρ22-2L2)(-sinθ)+2L2HHHhhlhH×(ρ24L2-ρ22-ρ14L2-ρ12)HHHjjjlhH×cos⁡θ-2sin⁡(θ+γ)2L2]=0,(13)∂U∂γ=KL22[2L2(ρ12+ρ22-2L2)(-sinγ)-2L2HhhHhHH×(ρ24L2-ρ22-ρ14L2-ρ12)HhHhhHH×(cos⁡γ)-2sin⁡(θ+γ)2L2]=0.Due toγ+θ≠π, the following equation can be obtained by eliminating the expression sin⁡(θ+γ) from (12) and (13):
(14)θ=γ+2tan-1×[(ρ24L2-ρ22-ρ14L2-ρ12)(ρ12+ρ22-2L2)].
Substituting (14) into (13) yields
(15)[-N1(N02+N12)-3N0N1]t14+2[N0(N02+N12)-(N02-N12)]t13-2N0N1t12+2[N0(N02+N12)+(N02-N12)]t1+N1(N02+N12)+N0N1=0,
where t1=tan(γ/2). The expressions for N0 and N1 are detailed in Appendix A. It is apparent that (15) is an equation of degree 4 in t1. Solving (15) for t1, we obtain
(16)t1j=14[(-1)j+1(E12-E2+E5)1/2-E1]+δ14hhhh×[((E12-E2+E5)1/2-8E5)-1(E1+(-1)j(E12-E2+E5)1/2hhhhhhlhh-8(E1E5-2M3)(-1)j(E1+(-1)j(E12-E2+E5)1/2)hhhhh×((E12-E2+E5)1/2-8E5)-1]1/2,
where δ1=±1. Moreover, the variables Eξ (ξ=1,…,5) are also detailed in Appendix A. t1j represents four solutions to (15) for j=1,2. Computing the arctangent of t1 generates a unique solution for γ due to the range imposed to γ. Moreover, negative solutions for t1 should be eliminated. With the solutions for γ now known, the solutions to the forward kinematic problem can be found by substituting (8) and (10) into (5).


## 3.2. Inverse Kinematic Analysis

The inverse kinematic analysis of the mechanism corresponds to the computation of the actuator lengths for the given Cartesian coordinates of nodeF.Eliminating the parameter,γ, from (5) yields
(17)[(x-L)2+y2-2L(x-L)]t22-4yLt2+(x-L)2+y2+2L(x-L)=0,
where t2=tan(β/2).Solving (17) for t2, we obtain
(18)t2=([(x-L)2+y2+2L(x-L)]}1/22yL±{4y2L2-[(x-L)2+y2-2L(x-L)]HHHHHHH×[(x-L)2+y2+2L(x-L)]}1/2)×((x-L)2+y2-2L(x-L))-1.
Substituting the expression t2=tan⁡(β/2) into (8) yields
(19)ρ1=2Lt21+t22.
Therefore, substituting (18) into (19), two solutions for ρ1 are found. In the following paragraphs, we will find the solutions for ρ2.By substituting (8) and (10) into (5), we obtain
(20)sinγ=2Ly-ρ14L2-ρ122L2,cos⁡γ=2Lx-ρ122L2.Moreover, substituting (7), (8), (9), and (10) into (14) yields
(21)θ=γ+2tan-1[(2L2sinα-ρ14L2-ρ12)(ρ12-2L2cos⁡α)].
Substituting (20) and (21) into (13) and rearranging yields
(22)λ1t34+λ2t33+λ3t32+λ4t3+λ5=0,
where t3=tan(α/2). The expressions for λζ (ζ=1,…,5) are detailed in Appendix B.It can be seen that (22) is an equation of degree 4 in t3. Therefore, solving (22), the following is obtained:
(23)t3μ=(-1)μ+1(4λ12η-4λ1λ3+λ22)1/2-λ22λ1+δ22{14λ12[λ2+(-1)μ(4λ12φ-4λ1λ3+λ22)-1/2]2hhhhh-(-1)μ2(λ2η-2λ4)hhhhh×(4λ12φ-4λ1λ3+λ22)-1/2-2η14λ12}1/2,
where
(24)η=λ33λ1+[-Q2+Q24-P327]1/3+[-Q2-Q24-P327]1/3,
and where
(25)P=1λ12[λ2λ4-4λ1λ5-13λ32],Q=1λ13[2λ3327+λ33(λ2λ4-4λ1λ5)hhhhhhh+(4λ1λ3λ5-λ22λ5-λ1λ42)2λ3327].
Substituting the expression t3=tan(α/2) into (7), we obtain
(26)ρ2=2Lt31+t32.In (23), it should be noted that δ2=±1. Moreover, t3μ represents four solutions to (22) for μ=1,2. Considering the range imposed to α, negative solutions for t3 should be eliminated. Substituting (23) into (26), the solutions for ρ2 are found.Two solutions forρ1 are given by (19). Four solutions for ρ2 can be found by (26). In particular, if these solutions for ρ1 and ρ2 are all nonnegative, eight solutions to the inverse kinematic problem are found.


## 4. Singularity Analysis

The singularity analysis of a mechanism can be completed by analyzing its Jacobian. The objective of this section is to obtain singular configurations of the planar 4-bar tensegrity mechanism.


### 4.1. Mechanism Jacobian

For conventional mechanisms, Jacobian is used to describe the relations between input and output velocities. However, for tensegrity mechanisms, these relationships cannot be established since there are more degrees of freedom than actuators. When a tensegrity mechanism is in equilibrium, its Jacobian can be defined as(27)δφ=Jδψ,
where φ=[x,y]T and ψ=[ρ1,ρ2]T. For the 4-bar tensegrity mechanism considered here, J can be written as follows:
(28)J=[∂x∂ρ1∂x∂ρ2∂y∂ρ1∂y∂ρ2].
Substituting (8) and (10) into (5), we obtain
(29)x=ρ122L+Lcos⁡γ,y=ρ12L4L2-ρ12+Lsinγ.
The elements of the Jacobian matrix, J, can be computed as follows:
(30)∂x∂ρ1=ρ1L-Lsinγ∂γ∂ρ1,∂x∂ρ2=-Lsinγ∂γ∂ρ2,∂y∂ρ1=2L2-ρ12L4L2-ρ12+Lcos⁡γ∂γ∂ρ1,∂y∂ρ2=Lcos⁡γ∂γ∂ρ2.
Then, in the following paragraphs, we will derive the expression for ∂γ/∂ρi for i=1,2. Substituting (7), (8), (9), and (10) into (12) and (13) yields
(31)2(1-cos⁡α-cos⁡β)sinθ-2(sinα-sinβ)cos⁡θ+sin(θ+γ)=0,2(1-cos⁡α-cos⁡β)(-sinγ)-2(sinα-sinβ)cos⁡γ-sin(θ+γ)=0.
By differentiating both sides of (31) and with respect to ρi, for i=1,2, we obtain
(32)[2cos⁡θ+cos⁡(θ+γ)-2cos⁡(β-θ)-2cos⁡(α-θ)]∂θ∂ρih+cos⁡(θ+γ)∂γ∂ρi-2cos⁡(α+θ)∂α∂ρih+2cos⁡(β-θ)∂β∂ρi=0,(33)[2cos⁡(β+γ)+2cos⁡(α-γ)-cos⁡(θ+γ)-2cos⁡γ]∂γ∂ρih-cos⁡(θ+γ)∂θ∂ρi-2cos⁡(α-γ)∂α∂ρih+2cos⁡(β+γ)∂β∂ρi=0.
Furthermore, the following equations can be derived from (7) and (8):
(34)∂α∂ρ1=0,∂β∂ρ2=0.
Considering the condition shown in (34) and combining (32) with (33), we have
(35)∂γ∂ρ1=Φ1Φ2·∂β∂ρ1,(36)∂γ∂ρ2=Φ3Φ2·∂α∂ρ2,
where
(37)Φ1=-{2cos⁡(β-θ)cos⁡(θ+γ)+2cos⁡(β+γ)HHHHh×[2cos⁡θ-2cos⁡(β-θ)HHHHhhH+cos⁡(θ+γ)-2cos⁡(α-θ)]},Φ2=cos⁡2(θ+γ)Φ2=+[2cos⁡θ+cos⁡(θ+γ)-2cos⁡(β-θ)-2cos⁡(α-θ)]Φ2=·[2cos⁡(β+γ)+2cos⁡(α-γ)-cos⁡(θ+γ)-2cos⁡γ],Φ3=2cos⁡(α+θ)cos⁡(θ+γ)hhhh+2cos⁡(α-γ)[2cos⁡θ+cos⁡(θ+γ)hhhhhhhhhhhhhhhh-2cos⁡(β-θ)-2cos⁡(α-θ)].
Moreover, the expressions for ∂α/∂ρ2 and ∂β/∂ρ1 can be computed from (7) and (8) as follows:
(38)∂α∂ρ2=-11-cos⁡2α·∂cos⁡α∂ρ2=24L2-ρ22,∂β∂ρ1=-11-cos⁡2β·∂cos⁡β∂ρ1=24L2-ρ12.
By substituting (35) and (36) into (30), the elements of the mechanism’s Jacobian can be computed.The Jacobian,J, describes the relationships between the infinitesimal movements of the actuators and the end-effector (node F) of the mechanism. In addition, the detailed discussion of the Jacobian will be presented in Section 4.2.


### 4.2. Singular Configurations

The singular configurations of a mechanism correspond to situations where the determinant ofJ is zero, goes to infinity, or is indeterminate.The determinant ofJ can be derived from (28) as follows:
(39)det⁡(J)=L2sin(β+γ)∂β∂ρ1·∂γ∂ρ2.
Substituting (36) and (38) into (39), we obtain
(40)det⁡(J)=4L2sin⁡(β+γ)(4L2-ρ12)(4L2-ρ22)·Φ3Φ2,
where the expressions for Φ2 and Φ3 are shown in (37). Furthermore, the singular configurations can be obtained by examining (37) and (40). The expressions of these singular configurations and the corresponding behaviors of the mechanism are described as follows.(i)
ρ1=0. One has the following.
(1)
NodeA is coincident with node C.(2)
Finite movements of nodeF in a direction perpendicular to the line joining nodes C and F are possible while finite movements of node E in a direction perpendicular to the line joining nodes D and E are also possible with actuators being locked.(3)
Infinitesimal movements of nodeF along a direction parallel to the line joining nodes C and F cannot be generated. Moreover, infinitesimal movements of node E in a direction parallel to the line that joins nodes D and E cannot be generated either with actuators being locked.(4)
External forces applied in a direction perpendicular to the line joining nodesE and D cannot be resisted by the actuators.(ii)
ρ2=0. One has the following.
(1)
NodeB is coincident with node D.(2)
Finite movements of nodeE along a direction perpendicular to the line joining nodes D and E are possible with actuators being locked.(3)
Infinitesimal movements of nodeE in a direction parallel to the line that joins nodes D and E cannot be generated with actuators being locked.(4)
External forces applied along a direction perpendicular to the line joining nodesC and F cannot be resisted by the actuators.(iii)
ρ1=2L. One has the following.
(1)
NodeC is located on the x-axis.(2)
Finite movements of NodeF along a direction perpendicular to the line joining nodes C and F are possible with actuators locked. Moreover, finite movements of node C in a direction parallel to the y-axis are also possible with actuators being locked.(3)
Infinitesimal movements of nodeC along a line parallel to the x-axis cannot be generated.(4)
External forces applied in a direction perpendicular to the line joining nodesD and E cannot be resisted by the actuators.(iv)
ρ2=2L. One has the following.
(1)
NodeD is located on the x-axis.(2)
Finite movements of nodeE along a direction perpendicular to the line joining nodes D and E are possible while finite movements of node D in a direction parallel to the y-axis are also possible with actuators being locked.(3)
Infinitesimal movements of nodeD along a line parallel to the x-axis cannot be generated.(4)
External forces applied in a direction perpendicular to the line joining nodesC and F cannot be resisted by the actuators.(v)
ρ1=ρ2=2L. One has the following.
(1)
All the nodes of the mechanism are located on thex-axis.(2)
NodeF is coincident with node A while node E is coincident with nodeB.(3)
Finite movements of nodesC,D, E, and F along a direction parallel to the y-axis are possible.(4)
External forces applied in a direction parallel to they-axis cannot be resisted by the actuators. Moreover, external forces applied along a direction parallel to the x-axis are resisted by the mechanism with no forces generated in the actuators.(vi)
ρ1=ρ2=L. One has the following.
(1)
NodeC is coincident with node D while node E is coincident with node F.(2)
Finite movements of nodeE in a direction perpendicular to the line joining nodes D and E are possible. Moreover, the compressive element joining node pairs E and D can gain an arbitrary rotation with respect to node D with actuators being locked.(3)
Infinitesimal movements of nodeE in a direction parallel to the line joining nodes Dand E cannot be generated.(4)
External forces applied along a direction perpendicular to the line joining nodesD and E cannot be resisted by the actuators.(vii)
sin⁡(β+γ)=0,Φ2=0andΦ3=0. One has the following.
(1)
Finite movements of nodeE in a direction perpendicular to the line joining nodes D and E are possible while finite movements of node F in a direction perpendicular to the line joining nodes C and F are also possible with actuators being locked.It can be noted that the mechanism will reach its dead point when the configuration described in (v) occurs. In such situations, the mechanism cannot be operated by actuators since the actuators cannot provide forces along a direction parallel to they-axis with all the mechanism’s nodes located on the x-axis. Moreover, an external force parallel to the y-axis is needed to put the mechanism to use in its normal state. Furthermore, this characteristic brings an advantage to the mechanism. The mechanism can be folded in a small volume for transportation purposes.


## 4.1. Mechanism Jacobian

For conventional mechanisms, Jacobian is used to describe the relations between input and output velocities. However, for tensegrity mechanisms, these relationships cannot be established since there are more degrees of freedom than actuators. When a tensegrity mechanism is in equilibrium, its Jacobian can be defined as(27)δφ=Jδψ,
where φ=[x,y]T and ψ=[ρ1,ρ2]T. For the 4-bar tensegrity mechanism considered here, J can be written as follows:
(28)J=[∂x∂ρ1∂x∂ρ2∂y∂ρ1∂y∂ρ2].
Substituting (8) and (10) into (5), we obtain
(29)x=ρ122L+Lcos⁡γ,y=ρ12L4L2-ρ12+Lsinγ.
The elements of the Jacobian matrix, J, can be computed as follows:
(30)∂x∂ρ1=ρ1L-Lsinγ∂γ∂ρ1,∂x∂ρ2=-Lsinγ∂γ∂ρ2,∂y∂ρ1=2L2-ρ12L4L2-ρ12+Lcos⁡γ∂γ∂ρ1,∂y∂ρ2=Lcos⁡γ∂γ∂ρ2.
Then, in the following paragraphs, we will derive the expression for ∂γ/∂ρi for i=1,2. Substituting (7), (8), (9), and (10) into (12) and (13) yields
(31)2(1-cos⁡α-cos⁡β)sinθ-2(sinα-sinβ)cos⁡θ+sin(θ+γ)=0,2(1-cos⁡α-cos⁡β)(-sinγ)-2(sinα-sinβ)cos⁡γ-sin(θ+γ)=0.
By differentiating both sides of (31) and with respect to ρi, for i=1,2, we obtain
(32)[2cos⁡θ+cos⁡(θ+γ)-2cos⁡(β-θ)-2cos⁡(α-θ)]∂θ∂ρih+cos⁡(θ+γ)∂γ∂ρi-2cos⁡(α+θ)∂α∂ρih+2cos⁡(β-θ)∂β∂ρi=0,(33)[2cos⁡(β+γ)+2cos⁡(α-γ)-cos⁡(θ+γ)-2cos⁡γ]∂γ∂ρih-cos⁡(θ+γ)∂θ∂ρi-2cos⁡(α-γ)∂α∂ρih+2cos⁡(β+γ)∂β∂ρi=0.
Furthermore, the following equations can be derived from (7) and (8):
(34)∂α∂ρ1=0,∂β∂ρ2=0.
Considering the condition shown in (34) and combining (32) with (33), we have
(35)∂γ∂ρ1=Φ1Φ2·∂β∂ρ1,(36)∂γ∂ρ2=Φ3Φ2·∂α∂ρ2,
where
(37)Φ1=-{2cos⁡(β-θ)cos⁡(θ+γ)+2cos⁡(β+γ)HHHHh×[2cos⁡θ-2cos⁡(β-θ)HHHHhhH+cos⁡(θ+γ)-2cos⁡(α-θ)]},Φ2=cos⁡2(θ+γ)Φ2=+[2cos⁡θ+cos⁡(θ+γ)-2cos⁡(β-θ)-2cos⁡(α-θ)]Φ2=·[2cos⁡(β+γ)+2cos⁡(α-γ)-cos⁡(θ+γ)-2cos⁡γ],Φ3=2cos⁡(α+θ)cos⁡(θ+γ)hhhh+2cos⁡(α-γ)[2cos⁡θ+cos⁡(θ+γ)hhhhhhhhhhhhhhhh-2cos⁡(β-θ)-2cos⁡(α-θ)].
Moreover, the expressions for ∂α/∂ρ2 and ∂β/∂ρ1 can be computed from (7) and (8) as follows:
(38)∂α∂ρ2=-11-cos⁡2α·∂cos⁡α∂ρ2=24L2-ρ22,∂β∂ρ1=-11-cos⁡2β·∂cos⁡β∂ρ1=24L2-ρ12.
By substituting (35) and (36) into (30), the elements of the mechanism’s Jacobian can be computed.The Jacobian,J, describes the relationships between the infinitesimal movements of the actuators and the end-effector (node F) of the mechanism. In addition, the detailed discussion of the Jacobian will be presented in Section 4.2.


## 4.2. Singular Configurations

The singular configurations of a mechanism correspond to situations where the determinant ofJ is zero, goes to infinity, or is indeterminate.The determinant ofJ can be derived from (28) as follows:
(39)det⁡(J)=L2sin(β+γ)∂β∂ρ1·∂γ∂ρ2.
Substituting (36) and (38) into (39), we obtain
(40)det⁡(J)=4L2sin⁡(β+γ)(4L2-ρ12)(4L2-ρ22)·Φ3Φ2,
where the expressions for Φ2 and Φ3 are shown in (37). Furthermore, the singular configurations can be obtained by examining (37) and (40). The expressions of these singular configurations and the corresponding behaviors of the mechanism are described as follows.(i)
ρ1=0. One has the following.
(1)
NodeA is coincident with node C.(2)
Finite movements of nodeF in a direction perpendicular to the line joining nodes C and F are possible while finite movements of node E in a direction perpendicular to the line joining nodes D and E are also possible with actuators being locked.(3)
Infinitesimal movements of nodeF along a direction parallel to the line joining nodes C and F cannot be generated. Moreover, infinitesimal movements of node E in a direction parallel to the line that joins nodes D and E cannot be generated either with actuators being locked.(4)
External forces applied in a direction perpendicular to the line joining nodesE and D cannot be resisted by the actuators.(ii)
ρ2=0. One has the following.
(1)
NodeB is coincident with node D.(2)
Finite movements of nodeE along a direction perpendicular to the line joining nodes D and E are possible with actuators being locked.(3)
Infinitesimal movements of nodeE in a direction parallel to the line that joins nodes D and E cannot be generated with actuators being locked.(4)
External forces applied along a direction perpendicular to the line joining nodesC and F cannot be resisted by the actuators.(iii)
ρ1=2L. One has the following.
(1)
NodeC is located on the x-axis.(2)
Finite movements of NodeF along a direction perpendicular to the line joining nodes C and F are possible with actuators locked. Moreover, finite movements of node C in a direction parallel to the y-axis are also possible with actuators being locked.(3)
Infinitesimal movements of nodeC along a line parallel to the x-axis cannot be generated.(4)
External forces applied in a direction perpendicular to the line joining nodesD and E cannot be resisted by the actuators.(iv)
ρ2=2L. One has the following.
(1)
NodeD is located on the x-axis.(2)
Finite movements of nodeE along a direction perpendicular to the line joining nodes D and E are possible while finite movements of node D in a direction parallel to the y-axis are also possible with actuators being locked.(3)
Infinitesimal movements of nodeD along a line parallel to the x-axis cannot be generated.(4)
External forces applied in a direction perpendicular to the line joining nodesC and F cannot be resisted by the actuators.(v)
ρ1=ρ2=2L. One has the following.
(1)
All the nodes of the mechanism are located on thex-axis.(2)
NodeF is coincident with node A while node E is coincident with nodeB.(3)
Finite movements of nodesC,D, E, and F along a direction parallel to the y-axis are possible.(4)
External forces applied in a direction parallel to they-axis cannot be resisted by the actuators. Moreover, external forces applied along a direction parallel to the x-axis are resisted by the mechanism with no forces generated in the actuators.(vi)
ρ1=ρ2=L. One has the following.
(1)
NodeC is coincident with node D while node E is coincident with node F.(2)
Finite movements of nodeE in a direction perpendicular to the line joining nodes D and E are possible. Moreover, the compressive element joining node pairs E and D can gain an arbitrary rotation with respect to node D with actuators being locked.(3)
Infinitesimal movements of nodeE in a direction parallel to the line joining nodes Dand E cannot be generated.(4)
External forces applied along a direction perpendicular to the line joining nodesD and E cannot be resisted by the actuators.(vii)
sin⁡(β+γ)=0,Φ2=0andΦ3=0. One has the following.
(1)
Finite movements of nodeE in a direction perpendicular to the line joining nodes D and E are possible while finite movements of node F in a direction perpendicular to the line joining nodes C and F are also possible with actuators being locked.It can be noted that the mechanism will reach its dead point when the configuration described in (v) occurs. In such situations, the mechanism cannot be operated by actuators since the actuators cannot provide forces along a direction parallel to they-axis with all the mechanism’s nodes located on the x-axis. Moreover, an external force parallel to the y-axis is needed to put the mechanism to use in its normal state. Furthermore, this characteristic brings an advantage to the mechanism. The mechanism can be folded in a small volume for transportation purposes.


## 5. Workspace

The actuator workspace of a mechanism is defined as the region that the actuators can operate while the output workspace is defined as the region that the end-effectors can reach. The boundaries and singular curves of a workspace usually correspond to the mechanism’s singular configurations since, in such situations, the mechanism cannot be controlled or cannot generate certain displaces of its actuators and end-effectors.


### 5.1. Actuator Workspace

For the 4-bar tensegrity mechanism researched here, the actuator workspace consists of the ranges of variables,ρ1 and ρ2. Generally, the boundary of the actuator workspace and singular curves inside the actuator workspace always correspond to the singular configurations. Moreover, by plotting the curves of singular configurations described in Section 4.2, the actuator workspace of the mechanism can be obtained as shown in Figure 2.Figure 2
Actuator workspace and singular curves for the planar 4-bar tensegrity mechanism withL=10 m.From Figure2, it can be seen that each curve is identified according to the singular configurations listed in Section 4.2. In addition, the singular configuration described in (v) corresponds to a point (ρ1=ρ2=20 m) of the actuator workspace boundary. Moreover, the configuration described in (vi) corresponds to a singular point (ρ1=ρ2=10 m) inside the actuator workspace.


### 5.2. Output Workspace

The output workspace corresponds to the ranges of Cartesian coordinates of nodeF. In most cases, the output workspace can be obtained by mapping the actuator workspace into the output domain. However, the boundaries of the actuator workspace are not always corresponding to the boundaries of the output workspace. Generally, the output workspace can be obtained by analyzing the singular configurations and the corresponding behaviors of the mechanism. An example of the mechanism’s output workspace and singular curves is shown in Figure 3.Figure 3
Output workspace and singular curves for the planar 4-bar tensegrity mechanism withL=10 m.From Figure3, it can be seen that curve i corresponds to the singular configuration (i) described in Section 4.2. In addition, curve i takes the form
(41)x2+y2=L2.Furthermore, curve iii corresponding to the singular configuration (iii) can be described by(42)(x-2L)2+y2=L2.From Section4.2, we know that the mechanism becomes uncontrollable when the singular configuration (vi) occurred. The node C is coincident with node D while node E is coincident with node F. Moreover, the force in the spring joining nodes E and F is equal to zero. The only possible deformation of the mechanism is a rotation with respect to node C of the strut joining nodes C and F. Therefore, the expression for the singular configuration (vi) in the output workspace can be written as
(43)(x-xc)2+(y-yc)2=L2,
where
(44)xc=L2,yc=3L2.In (43), xc and yc are the Cartesian coordinates of node C with ρ1=ρ2=L.From Figure3, it can be observed that the curve viii does not correspond to any singular configuration described in Section 4.2. Moreover, curve viii is obtained by analyzing the behaviors of the mechanism. As illustrated in Figure 1, the movement of node C is a rotation with respect to node B while the movement of node F is a rotation with respect to node C. Therefore, when the node F generates a rotation centered on node B with 2L in radius, the end-effector of the mechanism, node F, will reach the boundaries of output workspace. Furthermore, the curve viii can be described by
(45)(x-L)2+y2=(2L)2.Finally, the actuator workspace and output workspace are both obtained by analyzing the singular configurations and corresponding behaviors of the mechanism. The actuator and output workspaces of the planar tensegrity mechanism should be considered when such a mechanism is put to use or being designed.


## 5.1. Actuator Workspace

For the 4-bar tensegrity mechanism researched here, the actuator workspace consists of the ranges of variables,ρ1 and ρ2. Generally, the boundary of the actuator workspace and singular curves inside the actuator workspace always correspond to the singular configurations. Moreover, by plotting the curves of singular configurations described in Section 4.2, the actuator workspace of the mechanism can be obtained as shown in Figure 2.Figure 2
Actuator workspace and singular curves for the planar 4-bar tensegrity mechanism withL=10 m.From Figure2, it can be seen that each curve is identified according to the singular configurations listed in Section 4.2. In addition, the singular configuration described in (v) corresponds to a point (ρ1=ρ2=20 m) of the actuator workspace boundary. Moreover, the configuration described in (vi) corresponds to a singular point (ρ1=ρ2=10 m) inside the actuator workspace.


## 5.2. Output Workspace

The output workspace corresponds to the ranges of Cartesian coordinates of nodeF. In most cases, the output workspace can be obtained by mapping the actuator workspace into the output domain. However, the boundaries of the actuator workspace are not always corresponding to the boundaries of the output workspace. Generally, the output workspace can be obtained by analyzing the singular configurations and the corresponding behaviors of the mechanism. An example of the mechanism’s output workspace and singular curves is shown in Figure 3.Figure 3
Output workspace and singular curves for the planar 4-bar tensegrity mechanism withL=10 m.From Figure3, it can be seen that curve i corresponds to the singular configuration (i) described in Section 4.2. In addition, curve i takes the form
(41)x2+y2=L2.Furthermore, curve iii corresponding to the singular configuration (iii) can be described by(42)(x-2L)2+y2=L2.From Section4.2, we know that the mechanism becomes uncontrollable when the singular configuration (vi) occurred. The node C is coincident with node D while node E is coincident with node F. Moreover, the force in the spring joining nodes E and F is equal to zero. The only possible deformation of the mechanism is a rotation with respect to node C of the strut joining nodes C and F. Therefore, the expression for the singular configuration (vi) in the output workspace can be written as
(43)(x-xc)2+(y-yc)2=L2,
where
(44)xc=L2,yc=3L2.In (43), xc and yc are the Cartesian coordinates of node C with ρ1=ρ2=L.From Figure3, it can be observed that the curve viii does not correspond to any singular configuration described in Section 4.2. Moreover, curve viii is obtained by analyzing the behaviors of the mechanism. As illustrated in Figure 1, the movement of node C is a rotation with respect to node B while the movement of node F is a rotation with respect to node C. Therefore, when the node F generates a rotation centered on node B with 2L in radius, the end-effector of the mechanism, node F, will reach the boundaries of output workspace. Furthermore, the curve viii can be described by
(45)(x-L)2+y2=(2L)2.Finally, the actuator workspace and output workspace are both obtained by analyzing the singular configurations and corresponding behaviors of the mechanism. The actuator and output workspaces of the planar tensegrity mechanism should be considered when such a mechanism is put to use or being designed.


## 6. Conclusion

Compared with conventional mechanisms, tensegrity mechanisms can be modeled with greater accuracy since all of their components are axially loaded. Furthermore, the use of springs in tensegrity allows them to have the advantage of being deployable. For this reason, tensegrity mechanisms can be viewed as one alternative solution to conventional mechanisms in some applications. In this paper, the kinematics, singularity, and workspaces of a planar 4-bar tensegrity mechanism were presented.The analytical solutions to the forward and inverse kinematic problems were found by using an energy based method. Unlike conventional mechanisms, the shape of the 4-bar tensegrity mechanism depends not only on its geometry but also on the internal forces in the springs. As a consequence, the kinematic analysis should consider the constraint that the potential energy of the mechanism will reach its minimum when the mechanism is in equilibrium. Afterwards, a Jacobian was developed and the singular configurations were discussed. It was demonstrated that the finite movements of the actuators can be generated when the end-effector reached the boundaries of the output workspace. Moreover, the external loads exerted on the end-effector cannot be resisted by the actuators when the singular configurations corresponding to the singular curves inside the actuator workspace occurred. Furthermore, an attractive characteristic was found; that is, the mechanism can be folded in a small volume for transportation purposes. Finally, according to the singular configurations and the corresponding behaviors of the mechanism, the actuator and output workspaces were mapped. The singular configurations and workspaces of the mechanism should be considered when such a mechanism is put to use or being designed.In future work, the authors wish to research the control of the 4-bar tensegrity mechanism.



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*Source: 967251-2014-03-25.xml*