Force Analysis of the Overconstrained Mechanisms Based on Equivalent Stiffness Considering Limb Axial Deformation

Overconstrained mechanism has the advantages of large bearing capacity and high motion reliability,but its force analysis is complex and difficult because the mechanism system contains overconstraints.Considering the limb axial deformation,taking typical 2SS+P and 7-SS passive overconstrained mechanisms,2SPS+P and 7-SPS active overconstrained mechanisms,and 2SPS+P and 7-SPS passive-input overconstrained mechanisms as examples,a new force analysis method based on the idea of equivalent stiffness is proposed.The equivalent stiffness matrix of passive overconstrained mechanism is derived by combining the force balance and deformation compatibility equations with consideration of axial elastic limb deformations.The relationship between the constraint wrench magnitudes and the external force,limb stiffness is established.The equivalent stiffness matrix of active overconstrained mechanism is derived by combining the force balance and displacement compatibility equations.Here,the relationship between the magnitudes of the actuated wrenches and the external force,limb stiffness is investigated.Combining with the equivalent stiffness of the passive overconstrained mechanism,an analytical relationship between the actuated forces of passive-input overconstrained mechanism and the output displacement,limb stiffness is explored.Finally,adaptability of the equivalent stiffness to overconstrained mechanisms is discussed,and the effect of the limb stiffness on overconstrained mechanisms force distribution is revealed.The research results provide a theoretical reference for the design,research and practical application of overconstrained mechanism.

Mobility Analysis of Altmann Overconstrained Linkages by Modified Grübler-Kutzbach Criterion

Developing a general mobility method/formula is a hot topic lasting for more than 150 years in kinematics. It is necessary to apply any mobility method to puzzling overconstrained mechanisms for verification of its generality. Altmann linkages are such recognized puzzling mechanisms that their mobility analysis is of important significance. A necessary condition for judging a general mobility method is that the method can be fit for Altmann linkages. Firstly, this study classes Altmann linkages into 17 types in terms of the numbers and types of kinematic pairs, and then Altmann overconstrained linkages are further classified into 4 types. Secondly, the mobility of Altmann overconstrained linkages is systematically analyzed by the Modified Grübler-Kutzbach criterion based on screw theory, where passive freedoms are defined as limb passive freedoms and mechanism passive freedoms. In addition, the full-cycle mobility is judged, which overcomes the shortcoming of instantaneous property of screw theory. It is shown that Modified Grübler-Kutzbach criterion not only obtains the correct numerical mobility, but also gives the mobility character by resolving reciprocal screws for the constraint system. This study lays the foundation of verification for the generality of Modified Grübler-Kutzbach criterion. Besides, Altmann overconstrained linkages almost comprise all kinds of modern parallel mechanisms and some classical mechanisms, which provides an important reference for mechanism mobility calculation.

Motion-force Transmissibility Characteristic Analysis of a Redundantly Actuated and Overconstrained Parallel Machine

This paper presents a novel 1T2R three degrees of freedom redundantly actuated and overconstrained ■parallel machining head(■denotes the active prismatic joint), which can construct 5-axis hybrid machine to complete high speed freedom surface milling for large complex structural components in aerospace. Firstly, based on the screw theory, the mobility of the proposed parallel manipulator is briefly analysed. Secondly, the kinematic inverse position and the parasitic motion of the parallel manipulator are explicitly expressed. Furthermore, motion-force transmission performance evaluation indices are derived in detail via an alternative approach based on the screw theory. More importantly, a simple method for quickly solving the maximum virtual power coefficient is proposed, and the motion-force transmission performance evaluation index is greatly improved. To evaluate the kinematic performance, its workspace is calculated. With numerical examples, performance distribution atlases of the manipulator are depicted visually. The corresponding results illustrate that the proposed parallel manipulator has better orientation workspace and superior motion-force transmission performance than the 2 PRU-PRS parallel manipulator, which proves the validity and applicability of applying this manipulator as a machining head.

Mobility of the Myard 5R Linkage Involved in “Gogu Problem”

Since the traditional Griibler-Kutzbach criterion fails in many overconstrained mechanisms, developing a general mobility formula is a hot topic lasting for more than 150 years in mechanisms. GOGU systematically investigated various mobility methods, and pointed that the methods were not fit for two kinds of paradoxical overconstrained mechanisms. The mobility on the two kinds of mechanisms is regarded as ＂Gogu problem＂. The Modified Griibler-Kutzbach criterion has solved the mobility of the second kind of mechanisms in ＂Gogu problem＂, and has developed into a systematic mobility methodology. Myard 5R linkage is one of the single-loop mechanisms involved in ＂Gogu problem＂, its joint axes are distributed in space with special geometric conditions, which increases the difficulty of mobility analysis. The study is to calculate the global mobility of the Myard 5R linkage using the mobility methodology. Firstly, the mobility methodology based on screw theory is briefly introduced. Secondly, some homogeneous transforms are performed according to the D-H parameters and the invariance of the linkage plane symmetry is revealed, which provides an idea to judge a plane-symmetric loop. The special geometric features of the axes distribution are discussed as well. Finally, the global mobility of the Myard 5R linkage is determined by the Modified Grubler-Kutzbach criterion. The results show that the methodology can be applied to more paradoxical mechanisms.

A novel line-symmetric Goldberg 6R linkage with bifurcation property

This paper proposes a novel reconfigurable Goldberg 6R linkage,conformed to the construction of variant serial Goldberg 6R linkage,while simultaneously satisfying the line-symmetric Bricard qualifications.The isomeric mechanism of this novel reconfigurable mechanism is obtained in combination with the isomerization method.The geometrically constrained conditions result in variable motion branches of the mechanism.Based on the singular value decomposition of the Jacobian matrix,the motion branches and branch bifurcation characteristics are analyzed,and the schematics of bifurcations in joint space is derived.This novel 6R linkage features one Goldberg 6R motion branch,two line-symmetric Bricard 6R motion branches,and one Bennett motion branch.With regards to the line-symmetric Bricard 6R motion branches,a similar function for the disassembly and recombination process can be achieved by reconstructing an intermediate configuration through bifurcation.Then,the isomerized generalized variant Goldberg 6R linkage is explicated in a similar way.Acting as a bridge,reconfigurability connects two families of overconstrained mechanisms.