Forward Statics of Tensegrity Robots With Rigid Bodies Using Homotopy Continuation

Tensegrity robots are composed of simple members (rigid bodies and tensile cables) and have features like lightweight, compliance, and robustness, thus representing a promising alternative to soft and rigid robots. Forward statics, which determines the robot pose in static equilibrium, is very helpful for the design of tensegrity robots. However, existing approaches are not efficient or not general enough for Class-
$k$
tensegrity robots, where up to
$k$
rigid bodies can be connected. This letter proposes a general approach for static modeling of tensegrity robots with rigid bodies of arbitrary shapes, using natural coordinates. Introducing an additional set of variables transforms the static equation into a polynomial system with four kinds of parameters: cable rest lengths, deformations, stiffness coefficients, and external forces. Then, the forward statics problem appears as a path-following problem of a parameter homotopy, which is efficiently solved by the homotopy continuation method. Simulations illustrate how the forward statics can be used to evaluate the achievable range and the strength of a 2D tensegrity manipulator and a 3D tensegrity spine. Finally, we conduct hardware experiments on a tensegrity manipulator prototype, validating the accuracy of the proposed approach.