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In particular, if these solutions for and are all nonnegative, eight solutions to the inverse kinematic problem are found. From Figure 1, the coordinates of nodes , , , and can be obtained as follows: 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. Generally, the output workspace can be obtained by analyzing the singular configurations and the corresponding behaviors of the mechanism. Furthermore, the following equations can be derived from (7) and (8): (1)Node is coincident with node . 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 . When some components (rigid rods or springs) are actuated, tensegrity mechanisms can be obtained. Furthermore, nodes and are fixed to the ground and the whole mechanism lies in a horizontal plane. An example of the mechanisms output workspace and singular curves is shown in Figure 3. One has the following. where This research is supported by the National Natural Science Foundation of China (no. In addition, the detailed discussion of the Jacobian will be presented in Section 4.2. As a consequence, the singularity analysis of the planar 4-bar tensegrity mechanism has been completed. For the mechanism shown in Figure 1, the potential energy of the system will reach its minimum when the mechanism is in equilibrium. Eliminating the parameter, , from (5) yields It is composed of three springs, four bars, and two prismatic actuators. The proposed applications of tensegrity mechanisms range from a flight simulator [18], a space telescope [19], and a robot [20] to a sensor [21]. 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 particular, in this case, the mechanism cannot be in equilibrium.

One has the following. (1)Node is coincident with node while node is coincident with node . Two solutions for are given by (19). 51375360) and the Fundamental Research Funds for the Central Universities (no. Moreover, the configuration described in (vi) corresponds to a singular point (m) inside the actuator workspace. When a tensegrity mechanism is in equilibrium, its Jacobian can be defined as

Firstly, the analytical solutions to the forward and inverse kinematic problems are found by using an energy based method. The singularity analysis of a mechanism can be completed by analyzing its Jacobian. Read the winning articles. The analytical solutions to the forward and inverse kinematic problems were found by using an energy based method. Box 188, Xi'an 710071, China, External forces applied in a direction perpendicular to the line joining nodes, External forces applied along a direction perpendicular to the line joining nodes, All the nodes of the mechanism are located on the, External forces applied in a direction parallel to the. Remarkably, because 22 matrices commute thisallows these matrices to be replaced by complex exponentials and the coordinate vectors to be replaced by complex numbers andthe derivations and calculations do not change. It was demonstrated that the finite movements of the actuators can be generated when the end-effector reached the boundaries of the output workspace.

Substituting (14) into (13) yields With the solutions for now known, the solutions to the forward kinematic problem can be found by substituting (8) and (10) into (5). Moreover, the compressive element joining node pairs and can gain an arbitrary rotation with respect to node with actuators being locked. Moreover, external forces applied along a direction parallel to the -axis are resisted by the mechanism with no forces generated in the actuators.(vi). In future work, the authors wish to research the control of the 4-bar tensegrity mechanism. The expressions for and are detailed in Appendix A. Moreover, the expressions for and can be computed from (7) and (8) as follows: This first set details theposition and velocity analysis of afour-bar linkage. Solving (17) for , we obtain For conventional mechanisms, Jacobian is used to describe the relations between input and output velocities. For the 4-bar tensegrity mechanism researched here, the actuator workspace consists of the ranges of variables, and . K. Snelson, Continuous Tension, Discontinuous Compression Structures, USA Patent 31,696,11, February 1965. It seems that he was inspired by some novel sculptures completed by Snelson [2]. Update: December 12, 2017. In (23), it should be noted that . Furthermore, an attractive characteristic was found; that is, the mechanism can be folded in a small volume for transportation purposes. In the following paragraphs, we will find the solutions for . For this reason, the ranges imposed to and can be obtained as follows: Afterwards, a Jacobian was developed and the singular configurations were discussed. Substituting (23) into (26), the solutions for are found.

Tensegrity mechanisms can be viewed as one alternative solution to conventional mechanisms. 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). For this reason, it is always assumed that the planar 4-bar tensegrity mechanism is in equilibrium. In addition, the singular configuration described in (v) corresponds to a point (m) of the actuator workspace boundary. A static analysis of tensegrity structures was given by Juan and Mirats Tur [10]. (3)Finite movements of nodes ,, , and along a direction parallel to the -axis are possible. Therefore, when the node generates a rotation centered on node with in radius, the end-effector of the mechanism, node , will reach the boundaries of output workspace. By substituting (8) and (10) into (5), we obtain. Moreover, the movement of node is also constrained to a rotation centered on node . Therefore, solving (22), the following is obtained: 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 applications of tensegrity systems can be divided into two main branches. Furthermore, the ranges imposed to and are chosen as. The term tensegrity was created by Fuller [1] as a combination of the words tensional and integrity. where and . Substituting (20) and (21) into (13) and rearranging yields In this paper, the kinematics, singularity, and workspaces of a planar 4-bar tensegrity mechanism have been investigated. In Figure 1, the stiffness of the springs of lengths () is denoted by . In most cases, the output workspace can be obtained by mapping the actuator workspace into the output domain. Moreover, the coordinates of nodes and are chosen to be nonnegative. For the 4-bar tensegrity mechanism considered here, can be written as follows: 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 . One has the following. 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 from (12) and (13): 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 . K505131000087). 2014, Article ID 967251, 10 pages, 2014. https://doi.org/10.1155/2014/967251, 1School of Electro-Mechanical Engineering, Xidian University, P.O. In (43), and are the Cartesian coordinates of node with . [25], virtual zero-free-length spring can be created by extending the actual spring beyond its attachment point. Copyright 2014 Zhifei Ji et al. This paper is organized as follows. This hypothesis is not problematic since, as was explained by Gosselin [24] and Shekarforoush et al. B. Fuller, Tensile-integrity Structures, USA Patent 30,635,21, November 1965. During the past twenty years, considerable research has been performed on the control, statics, and dynamics of class-1 tensegrity mechanisms. Furthermore, the Jacobian was computed and the singularity configurations were obtained in Section 4. Considering the condition shown in (34) and combining (32) with (33), we have Thirdly, the actuator and output workspaces are mapped. The elements of the Jacobian matrix, , can be computed as follows: (3)Infinitesimal movements of node along a line parallel to the -axis cannot be generated. In Section 2, the planar 4-bar tensegrity mechanism was introduced. Solving (15) for , we obtain Under this assumption, the explicit relationships between the input and output variables can be developed. The bars of length are joining node pairs DE, CF, BC, and AD while the springs are joining node pairs CE, EF, and DF. (1)Node is located on the -axis. Moreover, negative solutions for should be eliminated.

However, the boundaries of the actuator workspace are not always corresponding to the boundaries of the output workspace. where (2)Finite movements of node along a direction perpendicular to the line joining nodes and are possible with actuators being locked. 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. The node is coincident with node while node is coincident with node . where . (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. Substituting the expression into (8) yields where. Box 188, Xi'an 710071, China. 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. Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. (4)External forces applied along a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(vii). However, for tensegrity mechanisms, these relationships cannot be established since there are more degrees of freedom than actuators. No related content is available yet for this article.

Four solutions for can be found by (26). Furthermore, the curve viii can be described by. As the complexity of robotic applications in space increases, new demands for lighter and quicker mechanisms arise. Therefore, substituting (18) into (19), two solutions for are found. Furthermore, the use of springs in tensegrity allows them to have the advantage of being deployable. Secondly, the definition of a tensegrity mechanisms Jacobian is introduced. Moreover, the variables () are also detailed in Appendix A. represents four solutions to (15) for . For this reason, tensegrity mechanisms can be viewed as one alternative solution to conventional mechanisms in some applications. From Figure 1, it can be seen that the shape of the mechanism can be determined for the given actuator lengths. The actuator and output workspaces were mapped, respectively, in Section 5. where the expressions for and are shown in (37). From Figure 3, it can be seen that curve i corresponds to the singular configuration (i) described in Section 4.2. 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 mechanisms geometry but also on the internal forces in the springs. For the mechanism considered here, the forward kinematic analysis consists in computing the Cartesian coordinates of node for the given actuator lengths. Furthermore, the singular configurations can be obtained by examining (37) and (40). The determinant of can be derived from (28) as follows: Compared with conventional mechanisms, tensegrity mechanisms can be modeled with greater accuracy since all of their components are axially loaded. Moreover, the force in the spring joining nodes and is equal to zero. (4)External forces applied in a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(iv). Considering the range imposed to , negative solutions for should be eliminated. The forward and inverse kinematic analysis was performed in Section 3. Furthermore, this characteristic brings an advantage to the mechanism. (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. One has the following. From Figure 3, it can be observed that the curve viii does not correspond to any singular configuration described in Section 4.2. 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]. In (A.2), the expressions for and are as follows: In (B.1), the expressions for are as follows: The authors declare that there is no conflict of interests regarding the publication of this paper. Moreover, a review of form-finding methods is given by Tibert and Pellegrino [5]. It can be seen that (22) is an equation of degree 4 in . 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, 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. This problem has been studied by many authors [57]. 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. The expressions of these singular configurations and the corresponding behaviors of the mechanism are described as follows.(i). Tensegrity systems are formed by a set of compressive components and tensile components. (4)External forces applied in a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(ii). (2)Finite movements of node in a direction perpendicular to the line joining nodes and are possible.

One has the following. 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 . Kinematics, Singularity, and Workspaces of a Planar 4-Bar Tensegrity Mechanism, School of Electro-Mechanical Engineering, Xidian University, P.O. However, there are few articles relating to class-2 tensegrity mechanisms, especially on the study of them. It is apparent that (15) is an equation of degree 4 in . In this paper, the kinematics, singularity, and workspaces of a planar 4-bar tensegrity mechanism were presented. 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. From Figure 1, it can also be observed that the distance between nodes and is . (4)External forces applied along a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(iii). Moreover, a cosine law for the triangle formed by nodes , , and can be written as. From Figure 1, it can be seen that joints and have restrained translational DOF in - and -axes but free rotational DOF. (1)Node is located on the -axis.

One has the following. (1)Node is coincident with node while node is coincident with node . Two solutions for are given by (19). 51375360) and the Fundamental Research Funds for the Central Universities (no. Moreover, the configuration described in (vi) corresponds to a singular point (m) inside the actuator workspace. When a tensegrity mechanism is in equilibrium, its Jacobian can be defined as

Firstly, the analytical solutions to the forward and inverse kinematic problems are found by using an energy based method. The singularity analysis of a mechanism can be completed by analyzing its Jacobian. Read the winning articles. The analytical solutions to the forward and inverse kinematic problems were found by using an energy based method. Box 188, Xi'an 710071, China, External forces applied in a direction perpendicular to the line joining nodes, External forces applied along a direction perpendicular to the line joining nodes, All the nodes of the mechanism are located on the, External forces applied in a direction parallel to the. Remarkably, because 22 matrices commute thisallows these matrices to be replaced by complex exponentials and the coordinate vectors to be replaced by complex numbers andthe derivations and calculations do not change. It was demonstrated that the finite movements of the actuators can be generated when the end-effector reached the boundaries of the output workspace.

Substituting (14) into (13) yields With the solutions for now known, the solutions to the forward kinematic problem can be found by substituting (8) and (10) into (5). Moreover, the compressive element joining node pairs and can gain an arbitrary rotation with respect to node with actuators being locked. Moreover, external forces applied along a direction parallel to the -axis are resisted by the mechanism with no forces generated in the actuators.(vi). In future work, the authors wish to research the control of the 4-bar tensegrity mechanism. The expressions for and are detailed in Appendix A. Moreover, the expressions for and can be computed from (7) and (8) as follows: This first set details theposition and velocity analysis of afour-bar linkage. Solving (17) for , we obtain For conventional mechanisms, Jacobian is used to describe the relations between input and output velocities. For the 4-bar tensegrity mechanism researched here, the actuator workspace consists of the ranges of variables, and . K. Snelson, Continuous Tension, Discontinuous Compression Structures, USA Patent 31,696,11, February 1965. It seems that he was inspired by some novel sculptures completed by Snelson [2]. Update: December 12, 2017. In (23), it should be noted that . Furthermore, an attractive characteristic was found; that is, the mechanism can be folded in a small volume for transportation purposes. In the following paragraphs, we will find the solutions for . For this reason, the ranges imposed to and can be obtained as follows: Afterwards, a Jacobian was developed and the singular configurations were discussed. Substituting (23) into (26), the solutions for are found.

Tensegrity mechanisms can be viewed as one alternative solution to conventional mechanisms. 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). For this reason, it is always assumed that the planar 4-bar tensegrity mechanism is in equilibrium. In addition, the singular configuration described in (v) corresponds to a point (m) of the actuator workspace boundary. A static analysis of tensegrity structures was given by Juan and Mirats Tur [10]. (3)Finite movements of nodes ,, , and along a direction parallel to the -axis are possible. Therefore, when the node generates a rotation centered on node with in radius, the end-effector of the mechanism, node , will reach the boundaries of output workspace. By substituting (8) and (10) into (5), we obtain. Moreover, the movement of node is also constrained to a rotation centered on node . Therefore, solving (22), the following is obtained: 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 applications of tensegrity systems can be divided into two main branches. Furthermore, the ranges imposed to and are chosen as. The term tensegrity was created by Fuller [1] as a combination of the words tensional and integrity. where and . Substituting (20) and (21) into (13) and rearranging yields In this paper, the kinematics, singularity, and workspaces of a planar 4-bar tensegrity mechanism have been investigated. In Figure 1, the stiffness of the springs of lengths () is denoted by . In most cases, the output workspace can be obtained by mapping the actuator workspace into the output domain. Moreover, the coordinates of nodes and are chosen to be nonnegative. For the 4-bar tensegrity mechanism considered here, can be written as follows: 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 . One has the following. 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 from (12) and (13): 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 . K505131000087). 2014, Article ID 967251, 10 pages, 2014. https://doi.org/10.1155/2014/967251, 1School of Electro-Mechanical Engineering, Xidian University, P.O. In (43), and are the Cartesian coordinates of node with . [25], virtual zero-free-length spring can be created by extending the actual spring beyond its attachment point. Copyright 2014 Zhifei Ji et al. This paper is organized as follows. This hypothesis is not problematic since, as was explained by Gosselin [24] and Shekarforoush et al. B. Fuller, Tensile-integrity Structures, USA Patent 30,635,21, November 1965. During the past twenty years, considerable research has been performed on the control, statics, and dynamics of class-1 tensegrity mechanisms. Furthermore, the Jacobian was computed and the singularity configurations were obtained in Section 4. Considering the condition shown in (34) and combining (32) with (33), we have Thirdly, the actuator and output workspaces are mapped. The elements of the Jacobian matrix, , can be computed as follows: (3)Infinitesimal movements of node along a line parallel to the -axis cannot be generated. In Section 2, the planar 4-bar tensegrity mechanism was introduced. Solving (15) for , we obtain Under this assumption, the explicit relationships between the input and output variables can be developed. The bars of length are joining node pairs DE, CF, BC, and AD while the springs are joining node pairs CE, EF, and DF. (1)Node is located on the -axis. Moreover, negative solutions for should be eliminated.

However, the boundaries of the actuator workspace are not always corresponding to the boundaries of the output workspace. where (2)Finite movements of node along a direction perpendicular to the line joining nodes and are possible with actuators being locked. 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. The node is coincident with node while node is coincident with node . where . (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. Substituting the expression into (8) yields where. Box 188, Xi'an 710071, China. 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. Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. (4)External forces applied along a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(vii). However, for tensegrity mechanisms, these relationships cannot be established since there are more degrees of freedom than actuators. No related content is available yet for this article.

Four solutions for can be found by (26). Furthermore, the curve viii can be described by. As the complexity of robotic applications in space increases, new demands for lighter and quicker mechanisms arise. Therefore, substituting (18) into (19), two solutions for are found. Furthermore, the use of springs in tensegrity allows them to have the advantage of being deployable. Secondly, the definition of a tensegrity mechanisms Jacobian is introduced. Moreover, the variables () are also detailed in Appendix A. represents four solutions to (15) for . For this reason, tensegrity mechanisms can be viewed as one alternative solution to conventional mechanisms in some applications. From Figure 1, it can be seen that the shape of the mechanism can be determined for the given actuator lengths. The actuator and output workspaces were mapped, respectively, in Section 5. where the expressions for and are shown in (37). From Figure 3, it can be seen that curve i corresponds to the singular configuration (i) described in Section 4.2. 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 mechanisms geometry but also on the internal forces in the springs. For the mechanism considered here, the forward kinematic analysis consists in computing the Cartesian coordinates of node for the given actuator lengths. Furthermore, the singular configurations can be obtained by examining (37) and (40). The determinant of can be derived from (28) as follows: Compared with conventional mechanisms, tensegrity mechanisms can be modeled with greater accuracy since all of their components are axially loaded. Moreover, the force in the spring joining nodes and is equal to zero. (4)External forces applied in a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(iv). Considering the range imposed to , negative solutions for should be eliminated. The forward and inverse kinematic analysis was performed in Section 3. Furthermore, this characteristic brings an advantage to the mechanism. (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. One has the following. From Figure 3, it can be observed that the curve viii does not correspond to any singular configuration described in Section 4.2. 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]. In (A.2), the expressions for and are as follows: In (B.1), the expressions for are as follows: The authors declare that there is no conflict of interests regarding the publication of this paper. Moreover, a review of form-finding methods is given by Tibert and Pellegrino [5]. It can be seen that (22) is an equation of degree 4 in . 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, 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. This problem has been studied by many authors [57]. 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. The expressions of these singular configurations and the corresponding behaviors of the mechanism are described as follows.(i). Tensegrity systems are formed by a set of compressive components and tensile components. (4)External forces applied in a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(ii). (2)Finite movements of node in a direction perpendicular to the line joining nodes and are possible.

One has the following. 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 . Kinematics, Singularity, and Workspaces of a Planar 4-Bar Tensegrity Mechanism, School of Electro-Mechanical Engineering, Xidian University, P.O. However, there are few articles relating to class-2 tensegrity mechanisms, especially on the study of them. It is apparent that (15) is an equation of degree 4 in . In this paper, the kinematics, singularity, and workspaces of a planar 4-bar tensegrity mechanism were presented. 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. From Figure 1, it can also be observed that the distance between nodes and is . (4)External forces applied along a direction perpendicular to the line joining nodes and cannot be resisted by the actuators.(iii). Moreover, a cosine law for the triangle formed by nodes , , and can be written as. From Figure 1, it can be seen that joints and have restrained translational DOF in - and -axes but free rotational DOF. (1)Node is located on the -axis.

Es sieht so aus, als ob wir nicht das finden konnten, wonach du gesucht hast. Möglicherweise hilft eine Suche.