On the Stiffness Analysis and Elastodynamics of Parallel Kinematic Machines

Abstract

Accurate models to describe the elasticity of robots are becoming essential for those applications involving high accelerations or high precision to improve quality in positioning and tracking of trajectories. Stiffness analysis not only involves the mechanical structure of a robot but even the control system necessary to drive actuators. Strategies aimed to reduce noise and dangerous bouncing effects could be implemented to make control systems more robust to flexibility disturbances, foreseing mechanical interaction with the control system because of regenerative and modal chattering (1). The most used approaches to study elasticity in the literature encompass: the Finite Element Analysis (FEA), the Matrix Structural Analysis (MSA), the Virtual Joint Method (VJM), the Floating Frame of Reference Formulation (FFRF) and the Absolute Nodal Coordinates Formulation (ANCF). FEA is largely used to analyze the structural behavior of a mechanical system. The reliability and precision of the method allow to describe each part of a mechanical system with great detail (2). Applying FEA to a robotic system implies a time-consuming process of re-meshing in the pre-processing phase every time that the robot posture has changed. Optimization all over the workspace of a robot would require very long computational time, thus FEA models are often employed to verify components or subparts of a complex robotic system. The MSA includes some simplifications to FEA using complex elements like beams, arcs, cables or superelements (3–5). This choice reduces the computational time and makes this method more efficient for optimization tasks. Some authors recurred to the superposition principle along with the virtual work principle to achieve the global stiffness model (6–8). Others considered the minimization of the potential energy of a PKM to find the global stiffness matrix (9), while some approaches used the total potential energy augmented adding the kinematic constraints by means of the Lagrange multipliers (10; 11). The first papers on VJM are based on pseudo-rigid body models with “virtual joints” (12–14). More recent papers include link flexibility and linear/torsional springs to take into account bending contributes (15–19). These approaches recur to the Jacobian matrix to map the stiffness of the actuators of a PKM inside its workspace; especially for PKMs with reduced mobility, it implies that the stiffness is limited to a subspace defined by the dofs of its end-effector. Pashkevich et al. tried to overcome this issue by introducing a multidimensional lumped-parameter model with localized 6-dof virtual springs (20). Finally, the FFRF and ANCF are powerful and accurate formulations, based on FEM and continuum mechanics, to study any flexible mechanical system (21). The FFRF is suitable 5