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

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 4bar 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.


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][6][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][12][13][14][15][16][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].[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.

Marc Arsenault and Gosselin
This paper is organized as follows.In Section 2, 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.

Mechanism Description
A diagram of the planar 4-bar tensegrity mechanism considered here is shown in Figure 1.It is composed of three springs, four bars, and two prismatic actuators.The bars of length  are joining node pairs DE, CF, BC, and AD while the springs are joining node pairs CE, EF, and DF.Moreover, the prismatic actuators are used to vary the distances between node pairs AC and BD.
As shown in Figure 1, the bars, springs, and prismatic actuators are connected to each other at each node by 2D frictionless rotational joints.Furthermore, nodes  and  are fixed to the ground and the whole mechanism lies in a horizontal plane.From Figure 1, it can be seen that joints  and  have restrained translational DOF in and -axes but free rotational DOF.In addition, the angle between the x-axis and the bar joining nodes AD is defined as  while the angle between the horizontal and the bar joining nodes BC is defined as .Moreover, the angle between the x-axis and the bar joining nodes CF is defined as  while the angle between the horizontal and the bar joining nodes DE is defined as .From Figure 1, it can also be observed that the distance between nodes  and  is .
In Figure 1, the stiffness of the springs of lengths ℓ  ( = 1, 2, 3) is denoted by .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 ( 1 and  2 ) are chosen as the mechanism's input variables while the Cartesian coordinates of node  ( and ) 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  coordinates of nodes  and  are chosen to be nonnegative.For this reason, the ranges imposed to  and  can be obtained as follows: Furthermore, the ranges imposed to  and  are chosen as From Figure 1, it can be observed that the mechanism is unstable in the case of DE and CF being parallel.In particular, in this case, the mechanism cannot be in equilibrium.For this reason,  +  ̸ = .

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.

Forward Kinematic Analysis.
For the mechanism considered here, the forward kinematic analysis consists in computing the Cartesian coordinates of node  for the given actuator lengths.From Figure 1, it can be seen that when the actuator lengths ( 1 and  2 ) are specified, the triangle formed by nodes , , and  and the triangle formed by nodes , , and  are determined.As a consequence, the movement of node  is confined to a rotation centered on node .Moreover, the movement of node  is also constrained to a rotation centered on node .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  and .
From Figure 1, the coordinates of nodes , , , and  can be obtained as follows: With the coordinates of nodes  and  now known, the coordinates of nodes  and  can be written in the following form: ] , Since the Cartesian coordinates of node  are chosen as the output variables, we therefore write Furthermore, the lengths of the springs CE, EF, and FD can be easily calculated according to (3) and ( 4).Then, the potential energy of the mechanism takes the form As shown in Figure 1, a cosine law for the triangle formed by nodes , , and  can be written as Moreover, a cosine law for the triangle formed by nodes , , and  can be written as Due to the ranges imposed to  and , the expressions for sin  and sin  can be derived as follows: Substituting ( 7), ( 8), (9), and ( 10) into ( 6), the potential energy of the mechanism can be expressed as By differentiating  with respect to  and  separately and equating the results to zero, the following equations are generated: Due to  +  ̸ = , the following equation can be obtained by eliminating the expression sin( + ) from ( 12) and ( 13): Substituting ( 14) into (13) yields where  1 = tan(/2).The expressions for  0 and  1 are detailed in Appendix A. It is apparent that ( 15) is an equation of degree 4 in  1 .Solving (15) for  1 , we obtain where  1 = ±1.Moreover, the variables   ( = 1, . . ., 5) are also detailed in Appendix A.  1 represents four solutions to (15) for  = 1, 2. Computing the arctangent of  1 generates a unique solution for  due to the range imposed to .Moreover, negative solutions for  1 should be eliminated.
With the solutions for  now known, the solutions to the forward kinematic problem can be found by substituting ( 8) and ( 10) into (5).

Inverse Kinematic Analysis.
The inverse kinematic analysis of the mechanism corresponds to the computation of the actuator lengths for the given Cartesian coordinates of node .
It can be seen that ( 22) is an equation of degree 4 in  3 .Therefore, solving (22), the following is obtained: where and where Substituting the expression  3 = tan(/2) into (7), we obtain In (23), it should be noted that  2 = ±1.Moreover,  3 represents four solutions to (22) for  = 1, 2. Considering the range imposed to , negative solutions for  3 should be eliminated.Substituting (23) into (26), the solutions for  2 are found.
Two solutions for  1 are given by (19).Four solutions for  2 can be found by (26).In particular, if these solutions for  1 and  2 are all nonnegative, eight solutions to the inverse kinematic problem are found.

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 where  = [, ] T and  = [ 1 ,  2 ] T .For the 4-bar tensegrity mechanism considered here, J can be written as follows: Substituting ( 8) and ( 10) into (5), we obtain The elements of the Jacobian matrix, J, can be computed as follows: Then, in the following paragraphs, we will derive the expression for /  for  = 1, 2. Substituting ( 7), ( 8), (9), and (10) into ( 12) and (13) yields Furthermore, the following equations can be derived from ( 7) and ( 8): Considering the condition shown in (34) and combining (32) with (33), we have where Moreover, the expressions for / 2 and / 1 can be computed from ( 7) and ( 8) as follows: 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 endeffector (node ) of the mechanism.In addition, the detailed discussion of the Jacobian will be presented in Section 4.2.

Singular Configurations.
The singular configurations of a mechanism correspond to situations where the determinant of J is zero, goes to infinity, or is indeterminate.
The determinant of J can be derived from (28) as follows: Substituting ( 36) and ( 38) into (39), we obtain where the expressions for Φ 2 and Φ 3 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.
(2) Finite movements of node  in a direction perpendicular to the line joining nodes  and  are possible while finite movements of node  in a direction perpendicular to the line joining nodes  and  are also possible with actuators being locked.(3) Infinitesimal movements of node  along a direction parallel to the line joining nodes  and  cannot be generated.Moreover, infinitesimal movements of node  in a direction parallel to the line that joins nodes  and  cannot be generated either with actuators being locked.(4) External forces applied in a direction perpendicular to the line joining nodes  and  cannot be resisted by the actuators.
(2) Finite movements of node  along a direction perpendicular to the line joining nodes  and  are possible with actuators being locked.(3) Infinitesimal movements of node  in a direction parallel to the line that joins nodes  and  cannot be generated with actuators being locked.(4) External forces applied along a direction perpendicular to the line joining nodes  and  cannot be resisted by the actuators.
(1) Node  is located on the -axis.
(2) Finite movements of Node  along a direction perpendicular to the line joining nodes  and  are possible with actuators locked.Moreover, finite movements of node  in a direction parallel to the -axis are also possible with actuators being locked.(3) Infinitesimal movements of node  along a line parallel to the -axis cannot be generated.(4) External forces applied in a direction perpendicular to the line joining nodes  and  cannot be resisted by the actuators.
(1) Node  is located on the -axis.
(2) Finite movements of node  along a direction perpendicular to the line joining nodes  and  are possible while finite movements of node  in a direction parallel to the -axis are also possible with actuators being locked.(3) Infinitesimal movements of node  along a line parallel to the -axis cannot be generated.(4) External forces applied in a direction perpendicular to the line joining nodes  and  cannot be resisted by the actuators.
(1) All the nodes of the mechanism are located on the -axis.(2) Node  is coincident with node  while node  is coincident with node B. (3) Finite movements of nodes ,, , and  along a direction parallel to the -axis are possible.(4) External forces applied in a direction parallel to the -axis cannot be resisted by the actuators.Moreover, external forces applied along a direction parallel to the -axis are resisted by the mechanism with no forces generated in the actuators.
(1) Node  is coincident with node  while node  is coincident with node .
(2) Finite movements of node  in a direction perpendicular to the line joining nodes  and  are possible.Moreover, the compressive element joining node pairs  and  can gain an arbitrary rotation with respect to node  with actuators being locked.(3) Infinitesimal movements of node  in a direction parallel to the line joining nodes and  cannot be generated.(4) External forces applied along a direction perpendicular to the line joining nodes  and  cannot be resisted by the actuators.
(1) Finite movements of node  in a direction perpendicular to the line joining nodes  and  are possible while finite movements of node  in a direction perpendicular to the line joining nodes  and  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 the -axis with all the mechanism's nodes located on the -axis.Moreover, an external force parallel to the 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.

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.

Actuator Workspace.
For the 4-bar tensegrity mechanism researched here, the actuator workspace consists of the ranges of variables,  1 and  2 .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.
From Figure 2, 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 ( 1 =  2 = 20 m) of the actuator workspace boundary.Moreover, the configuration described in (vi) corresponds to a singular point ( 1 =  2 = 10 m) inside the actuator workspace.

Output Workspace.
The output workspace corresponds to the ranges of Cartesian coordinates of node .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.
From Figure 3, 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 Furthermore, curve iii corresponding to the singular configuration (iii) can be described by From Section 4.2, we know that the mechanism becomes uncontrollable when the singular configuration (vi) occurred.The node  is coincident with node  while node  is coincident with node .Moreover, the force in the spring joining nodes  and  is equal to zero.The only possible deformation of the mechanism is a rotation with respect to node  of the strut joining nodes  and .Therefore, the expression for the singular configuration (vi) in the output workspace can be written as where In (43),   and   are the Cartesian coordinates of node  with  1 =  2 = .
From Figure 3, 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  is a rotation with respect to node  while the movement of node  is a rotation with respect to node .Therefore, when the node  generates a rotation centered on node  with 2 in radius, the end-effector of the mechanism, node , will reach the boundaries of output workspace.Furthermore, the curve viii can be described by 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.

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.

Figure 2 :
Figure 2: Actuator workspace and singular curves for the planar 4bar tensegrity mechanism with  = 10 m.

Figure 3 :
Figure 3: Output workspace and singular curves for the planar 4bar tensegrity mechanism with  = 10 m.