Truss Method For Kinematic Analysis Of 3D Overconstrained Linkages And Design Of Transformable Polyhedrons

To overcome the problem that it is very difficult to analysis or design complicated 3D overconstrained linkages and their assemblies with conventional kinematic tools,this dissertation has proposed a novel truss method by applying structural theory to the truss form of 3D linkages to study their kinematic behaviours with the consideration of both the topology and geometry conditions.The method has also been adopted to analyse 3D overconstrained linkages and to design deployable polyhedrons.The major research findings are as follows.First,the new truss method was developed,which converts 3D linkages to their corresponding truss forms while maintaining the kinematic behaviours.Under this method,mobility of complex linkages can be analysed by counting states of independent displacements with Maxwell's rule.Besides that,their motion paths are able to be generated by the displacement updated algorithm based on singular value decomposition(SVD)of equilibrium matrix,and bifurcation position can hence be detected by recording singular values of equilibrium matrix during the motion process.The proposed method has been validated with planar 4R linkage,spherical 4R linkage,and threefold-symmetric Bricard linkage as examples.Next,to eliminate strict overconstrained geometric conditions of linkages so that the tolerance of their fabrication error can be improved,the 3D overconstrained linkages are transformed into their corresponding truss forms.According to Maxwell's rule and rank of the equilibrium matrix,the redundant bars in the truss form of the overconstrained linkage can be detected and removed to obtain a non-overconstrained linkage,while its kinematic equivalence is well kept.Adopting this method,the non-overconstrained forms of Bennett linkage and Myard 5R linkage have been found as RSSR linkage and RRSRR linkages,respectively.Furthermore,discussion on fabrication errors has been carried out to demonstrate the tolerance on the mobility and input-output curve of the non-overconstrained form.And,polyhedral transformation has been realised by a kind of multi-loop linkages with complex topology,which enables large volumetric change amongst Platonic and Archimedean solids.Here,three sets of transformations have been proposed with their corresponding spatial linkages,namely truncated octahedron and cube,truncated tetrahedron and tetrahedron,as well as cuboctahedron and octahedron.Their constructions process and kinematic analysis were investigated in details by using the proposed truss method.Finally,motion analysis indicates that transformation paths are unique without singularity,which are further demonstrated with physical validation models.We envisage that our method could be extended to other paired polyhedrons.Therefore,the truss method opens up a new way to analyse kinematics of 3D linkages.Meanwhile,the resultant non-overconstrained forms of overconstrained linkages and polyhedral transformations with one DOF are of great potential in engineering applications.