A New Decoupling Method for Explicit Stiffness Analysis of Kinematically Redundant Planar Parallel Kinematic Mechanism

Optimization and control of stiffness for parallel kinematic mechanisms (PKM) are critical issues because stiffness is directly related to the precision and response characteristics of the end-effector of PKMs. Unlike nonredundant PKMs, redundant PKMs have additional actuators exceeding their essential degrees-of-freedom (DOF), resulting in an increase in the redundancy of control. The stiffness of redundant PKMs is divided into passive and active stiffness. Active stiffness is changeable even in cases of fixed kinematic parameters and end-effector posture. However, it is not easy and intuitive to control the active stiffness of redundant PKMs for the complexity of Hessian matrix operations. This paper describes a new decoupling method for explicit stiffness analysis of redundant PKM with the well-known two-DOF and one-redundant planar five-bar PKM. Three actuating joints are decoupled to three groups containing two actuating joints. With this mathematical configuration, the stiffness matrix for one-redundant actuation is also divided into three stiffness matrices for nonredundant actuation, and the contribution of each actuator can be intuitively investigated. Stiffness matrices for the original and decoupled cases are compared in detail. In particular, this decoupling method is applicable to redundant PKMs with many passive joints. Finding optimal joints for one- or two-redundant actuation with various candidates is more intuitive with this decoupling method.