A statically balanced tensegrity mechanism has zero stiffness in the finite affine modes that preserve the length of conventional members (i.e. members that are not zero-free-length springs). A length-preserving affine mode will exist iff all conventional members lie on a projective conic, and the number of independent zero-stiffness modes is then a linear function of the number of unique conventional member directions.
In any affinely deformed configuration there will again exist such a conic for the given set of conventional members, proving the finiteness of the zero-stiffness modes. Unless there are non-unique member directions that 'split up' and hence reduce the number of zero-stiffness modes, or 'merge' and increase the number. How do we exclude this? Members with the same direction are parallel or in-line, and an affine transformation preserves parallellism, provided the transformation is invertible. Therefore, any parallel members will remain parallel (and vice versa). Furthermore, the number of these zero-stiffness modes will remain constant in any configuration, provided that the affine transformation is invertible, i.e. the structure is not pushed into a lower dimension.
These observations provide insights in and have consequences for various aspects of the design and analysis of this type of structure, including finding the range of motion, displacement analysis, and form finding.
In order to establish the range of motion of the tensegrity structure, i.e. the possible subspace of equilibrium configurations, one approach could be to start from an initial configuration, prescribe a nodal displacement and iterate with Newton-Raphson until the new configuration is obtained.
This approach fails as it requires the inverse of the tangent stiffness matrix during the iteration. By the very nature of this type of structure, the tangent stiffness matrix is singular in all of its equilibrium configurations; the dimension of its nullspace, modulo rigid-body motions, being equal to the number of zero-stiffness modes. The use of a pseudo-inverse for the calculation offers a way forward, but is very slow to converge (if at all) due to the flatness of the stiffness function. This effect increases with the number of degrees of freedom.
For each of the equilibrium configurations of a zero stiffness tensegrity structure, the tension coefficients are constant; they are a material property of the zero-free-length springs, and by definition the conventional member lengths remain unchanged from configuration to configuration. This remarkable fact has already surfaced in form-finding literature, under a different guise.
In a common engineering form-finding method, the appropriate force densities are sought in order to achieve the required nullity in the stress matrix. The examples given by Vassart (1999) already show that for a given set of tension coefficients, a wide range of configurations is possible:
A whole family of geometries can be defined with the same set of selfstress coefficients using the redundant nodes.
The zero-stiffness tensegrity structures provide an interpretation as to the underlying cause of this freedom in design, and may also account for possible convergence issues when using a numerical approach to finding equilibrium configurations with the force-density method.
Finding the range of motion of zero stiffness tensegrity structures will require a different approach than the classic Newton-Raphson iteration, due to the singularity of the tangent stiffness matrix of the structure. The use of a pseudo-inverse offers no practical method.
The freedom of configurations for a given set of tension coefficients (or force densities) found in form-finding literature can now be accounted for, from a new perspective.