Producing negative active stiffness in redundantly actuated planar rotational parallel mechanisms

Generally, the Cartesian stiffness matrix of closed-loop mechanisms can be divided into the passive stiffness and the active stiffness. The passive stiffness was determined by the kinematic parameters and joint stiffness, while the active stiffness is sub-divided into the forced active stiffness KE, related to the external wrench, and active stiffness KIG generated by internal force. The passive stiffness is always positive definite, however, the active stiffness can be positive, semi-positive or negative definite. We exemplified it with a redundantly actuated planar rotational parallel mechanism (RAPR-PM), showing negative stiffness criteria for the active stiffness. Whether the elastic limbs are compressed or stretched, once the preload is known, the sign of the active stiffness can be determined from kinematic parameters. The results can be used to design variable stiffness mechanisms. Since some biological structures of animals, like fish, can be represented by a series of joint-linked RAPR-PMs, the conclusions in this paper can be extended to study the inherent stiffness mechanism of animals.