Stiffness modelling and comparison of the 5-UPS/PRPU parallel machine tool with its non-redundant counterpart

Abstract

This article presents a derivation of the stiffness matrix of a general redundantly actuated parallel mechanism based on the overall Jacobian. The Jacobian of the constraints and actuations is derived using reciprocal screw theory. Based on the mapping relationship between constraint, actuated and external forces combined with the principle of virtual work, a compatibility equation for the deformation of all of the limbs is achieved, and the stiffness model of the general redundantly actuated parallel mechanism is derived. The 5-UPS/PRPU redundantly actuated parallel machine tool is used to illustrate this method. The parallel machine tool comprehensively reveals the effect of the elastic deformation of activepassive joints and some basic transmission parts. The stiffness model is further validated by experimental data. Moreover, the global stiffness matrix of the general redundantly actuated parallel mechanism can be separated into two parts via matrix decomposition. The first part is the stiffness matrix of the corresponding non-redundant parallel mechanism, and the second part is the stiffness matrix of the redundantly actuated limbs (actuators). The redundantly actuated 5-UPS/PRPU parallel machine tool is also investigated for further analysis. The different stiffness characteristics of the machine tool and its corresponding non-redundant 5-UPS/PRPU parallel machine tool are compared. Actuation redundancy is found to improve the stiffness performance of the machine tool efficiently.

A novel Schönflies-motion generator parallel manipulator with decoupled rotational freedom

“…The combination of peripheral limbs with a central kinematic chain, or similar topologies, has been well used in the development of multi-axis machine tools. 43 The active kinematic joints are underlined and are conveniently mounted near the fixed platform. The peripheral limbs are connected to the fixed and moving platforms through universal and spherical joints, respectively.…”

Stiffness and Elastic Deformation of 4-DoF Parallel Manipulator with Three Asymmetrical Legs for Supporting Helicopter Rotor

“…Since 푑푎 v |푑푐 1 is satisfied, both 1 and 2 are the elastic rotational differential deformations of 2 , an equation of force-deformation for the 2SPS + RRPR PM is derived from Equation (8) to Equation (16) as follows:…”

“…Yang et al [7] studied elastostatic stiffness modeling of over constrained PMs. Zhou et al [8] derived the stiffness matrix of a redundantly actuated parallel mechanism based on the overall Jacobian. Based on strain energy and Castigliano's theorem, Enferadi and Tootoonchi [9] obtained mathematical model of the manipulator stiffness matrix.…”

Vibration modelling analysis of defective parallel robots as machine tools for aerospace parts machining

“…18,19 As an initial trending research topic of vibration analysis, the vibration analysis of the Stewart platform was successful, and its studies have gradually transitioned from theoretical vibration analysis to a wide range of practical applications. 20,21 Since both the number of DOFs and the number of branches of the Stewart platform are equal to six, its Jacobian matrix, stiffness matrix and damping matrix are sixth-order matrices, and since the six branches are identical and only the axis stiffness is considered, the derivation of the vibration equation of the six-DOFs vibration system is feasible. However, when a parallel mechanism is used as a machine tool, the DPR is more popular due to the associated cost savings and simple control [22][23][24][25][26] ; however, relatively few vibration studies have been conducted on DPRs.…”