Stiffness modeling is one of the most significant issues in the design of parallel kinematic machine (PKM). This paper presents a semi-analytical approach that enables the stiffness of PKM with complex machine frame geometry to be estimated effectively. This approach can be implemented by three steps: (i) decomposition of the entire system into two sub-systems associated with the parallel mechanism and the machine frame respectively; (ii) stiffness modeling of each sub-system using the analytical approach and the finite element analysis; and (iii) generation of the stiffness model of the entire system by means of linear superposition. In the modeling process of each sub-system, the virtual work princi- ple and overall deflection Jacobian are employed with special attention to the bending rigidity of the constrained passive limb and the interface stiffness of the machine frame. The stiffness distribution of a 5-DOF hybrid robot named TriVariant-B is investigated as an example to illustrate the effectiveness of this approach. The contributions of component rigidities to that of the system are evaluated using global indices. It shows that the results achieved by this approach have a good match to those obtained through finite element analysis and experiments.
WANG YouYuHUANG TianZHAO XueManMEI JiangPingDerek G CHETWYND
Due to the structural complexity, the dynamic modeling and quick performance evaluation for the parallel kinematic machines （PKMs） are still to be remained as two challenges in the stage of conceptual design. By using the finite element method and substructure synthesis, this paper mainly deals with the dynamic modeling and eigenvalue evaluation of a novel 3-DOF spindle head named the A3 head. The topological architecture behind the proposed A3 head is a 3-RPS parallel mechanism, which possesses one translational and two rotational capabilities. The mechanical features of the A3 head are briefly addressed in the first place followed by inverse position analysis. In the dynamic modeling, the platform is treated as a rigid body, the RPS limbs as the continuous uniform beams and the joints as lumped virtual springs. With the combination of substructure synthesis and finite element method, an analytical approach is then proposed to formulate the governing equations of motion of system using the compatibility conditions at interface between the limbs and the platform. Consequently, by solving the eigenvalue problem of the governing equations of motion, the distribution of lower natural frequencies of the A3 head throughout the entire workspace can be predicted in a quick manner. Modal analysis for the A3 head reveals that the distributions of lower natural frequencies are strongly related to the mechanism configuration and are axially symmetric due to system kinematic and structural features. The sensitivity analysis of the system indicates that the dimensional parameters of the 3-RPS mechanism have a slight effect on system lower natural frequencies while the joint compliances affect the distributions of lower natural frequencies significantly. The proposed dynamic modeling method can also be applied to other PKMs and can effectively evaluate the PKM＇s dynamic performance throughout the entire workspace.
