Dynamics of Tensegrity Systems: Compact Forms

Rather than the traditional vector differential equation, this paper introduces rigid body dynamics in a new form, as a matrix differential equation. We focus on axisymmetric rigid bodies which are adequate to describe a large class of problems, including tensegrity systems. For a system of beta rigid bodies, the forces are characterized in terms of network theory, and the kinematics are characterized in terms of the bar vectors (directed connections between two nodes attached to a rigid body). The dynamics are characterized by a second order differential equation in a 3 times 2beta configuration matrix. The first contribution of the paper is the dynamic model of a broad class of systems of rigid bodies, characterized in a compact form, requiring no inversion of a variable mass matrix. The second contribution is the derivation of all equilibria as a linear algebra problem in the control variables. The third contribution is the derivation of a linear model of the system of rigid bodies. One significance of these equations is the exact characterization of the statics and dynamics of all class 1 tensegrity structures, where rigid bar lengths are constant and the string force densities are control variables, which appear linearly. This will offer a significant advantage in control design tasks