Theoretical Studies On The Kinematics Of Tensegrities Based On The Quasi-static Assumption

Tensegrity has received extensive attention in multidisciplinary fields in recent years because of its characteristic of generating movement and adjusting structural stiffness simultaneously by length actuation of internal members.Based on the quasi-static assumption,the kinematic path of a tensegrity can be thought of as consisting of continuous stable equilibrium configurations.Some basic problems in the kinematics of tensegrities including form-finding of equilibrium configurations,realization of internal mechanism displacements,tracking the kinematic path considering load-carrying stiffness constraints and structural adaptability,are studied theoretically in this paper.Main works are as follows:(1)A form-finding method of tensegrity equilibrium configurations is proposed from the perspective of the kinematic analysis.The product of the transpose of equilibrium matrix and itself is defined as the state matrix,and it is pointed out that the variation in the zero eigenvalues of the matrix can be used to detect the variations of the self-stress and mechanism characteristics of a tensegrity on a kinematic path.The sensitivity equation describing the relationship between the variations in the eigenvalues and eigenvectors of the state matrix and node displacements is derived.The sensitivity equation can be used to modify the eigenvalues of the state matrix corresponding to the unfeasible configuration on a kinematic path by adjusting the node coordinates to satisfy the stability conditions.This is the form-finding process of feasible(stable equilibrium)configurations on a kinematic path of the tensegrity.(2)The internal mechanism displacement is the most efficient way for the movement of tensegrity.A strategy is proposed to trace the shortest path of internal mechanism displacement towards the target configuration of a tensegrity driven by the length actuation of active members.The internal mechanism displacement closest to the target displacement and their deviation are the components of the target displacement in the null space and row space of the compatibility matrix corresponding to passive members,respectively.A strategy is given for quickly constructing the basis in the row space of the compatibility matrix consisting of passive members after an active member is converted into a passive one.When the number is limited,a one-by-one exclusion method is proposed for the selection of the active members to minimize the deviation.(3)A method for the kinematic analysis of tensegrity considering the load-carrying stiffness is proposed.The linear relationship between the driving elongations of members and the node displacements is established.The analytical incremental expressions for the elastic stiffness matrix and the geometric stiffness matrix with respect to the driving elongations of members are derived.The variation in the load-induced elastic deformation during the movement can thus be evaluated.A basic equation that satisfies both the prescribed movement direction and the load-carrying stiffness constraints is established.A method for determining the minimum number of active members needed for the specified movement of tensegrity is given,and a strategy for tracking the path with the shortest total driving elongations is proposed.(4)The adaptability of the tensegrity is studied from the perspective of maintaining the load-carrying stiffness.The evaluation indexes of the contributions of geometric and elastic stiffness components to the load-carrying stiffness of tensegrities are proposed.By adjusting the structural shape to increase the elastic stiffness and replace the contribution of part of geometric stiffness,the prestress and the elastic potential energy can be reduced while maintaining the load-carrying stiffness of tensegrities.The sensitivity matrix reflecting the relationship between the loaded-induced displacement increment of the designated nodes and the variation of member rest lengths is established.It is pointed out that the increments of element rest lengths constructed by the linear combination of the null space basis vectors of the sensitivity matrix will not cause the change of the load-carrying stiffness,but can adjust the structural shape and prestress.An adjustment strategy is further proposed to reduce the prestress,the contribution of the geometric stiffness and the elastic potential energy.The adaptability of tensegrity is demonstrated by two examples.(5)The major factors in the deformability and adaptability of modular tensegrity structures are analyzed.The ’star’ tensegrity prism with many mechanism modes is applied as the basic unit to construct modular structures.Compared with two types of ordinary prismatic structures with different connection modes,the ’star’ prismatic structure has a higher precision in achieving the target displacement by the internal mechanism displacement,and its deformability and adaptability considering the load-carrying stiffness constraints are greater.It is pointed out that the decrease of the number of modules will lead to the reduction of deformability and adaptability of the modular tensegrity structures.