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.

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 [

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 [

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 [

Marc Arsenault and Gosselin [

This paper is organized as follows. In Section

A diagram of the planar 4-bar tensegrity mechanism considered here is shown in Figure

Planar 4-bar tensegrity mechanism.

As shown in Figure

In Figure

From Figure

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.

For the mechanism considered here, the forward kinematic analysis consists in computing the Cartesian coordinates of node

From Figure

Since the Cartesian coordinates of node

Furthermore, the lengths of the springs

As shown in Figure

Due to the ranges imposed to

Substituting (

By differentiating

Due to

The inverse kinematic analysis of the mechanism corresponds to the computation of the actuator lengths for the given Cartesian coordinates of node

Eliminating the parameter,

Solving (

By substituting (

Moreover, substituting (

It can be seen that (

In (

Two solutions for

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.

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

The Jacobian,

The singular configurations of a mechanism correspond to situations where the determinant of

The determinant of

Node

Finite movements of node

Infinitesimal movements of node

External forces applied in a direction perpendicular to the line joining nodes

Node

Finite movements of node

Infinitesimal movements of node

External forces applied along a direction perpendicular to the line joining nodes

Node

Finite movements of Node

Infinitesimal movements of node

External forces applied in a direction perpendicular to the line joining nodes

Node

Finite movements of node

Infinitesimal movements of node

External forces applied in a direction perpendicular to the line joining nodes

All the nodes of the mechanism are located on the

Node

Finite movements of nodes

External forces applied in a direction parallel to the

Node

Finite movements of node

Infinitesimal movements of node

External forces applied along a direction perpendicular to the line joining nodes

Finite movements of node

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

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.

For the 4-bar tensegrity mechanism researched here, the actuator workspace consists of the ranges of variables,

Actuator workspace and singular curves for the planar 4-bar tensegrity mechanism with

From Figure

The output workspace corresponds to the ranges of Cartesian coordinates of node

Output workspace and singular curves for the planar 4-bar tensegrity mechanism with

From Figure

Furthermore, curve iii corresponding to the singular configuration (iii) can be described by

From Section

In (

From Figure

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.

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.

Consider the following

In (

Consider the following

In (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the National Natural Science Foundation of China (no. 51375360) and the Fundamental Research Funds for the Central Universities (no. K505131000087).