Motion Group Based On Dual Method

The dissertation has investigated the problem of expressing the motion group by dual numbers, dual vectors, dual matrices, attitude quaternion and dual quaternions. With the tools of outer-product-operators and Hamilton-operators, we have depicted precisely the algebric structure of translation group, rotation group and screw group. Moreover, the quaternion differential and dual quaternion differential equations have been discussed in the dissertation.In Chaper two dual numbers, dual vectors, quaternions and dual quaternions are introduced. also the operations of each as well as the primative properties are included. What's more, we have discussed the examples of their applications of the abstract algebraic structures in motion group and some useful concepts, such as dual angles, plucker lines, rotation quaternions and screw dual quaternions. A series of conclusions and method are obtained, for example, equivalence of the rotation transfer matrix, Euler angles and unit quaternions, solving attitude quaternions by eigenvalue algorithm:we prove that the attitude quaternion can be obtained, whose rotation vector is the only unit eigenvector related to the eigenvalue, 1 ,of the rotation transfer matrix, and whose rotation angle can be solved according the trace of the rotation transfer matrix. Also dual vectors are confined to represent uniquely the lines in the space. In the last section of Chaper 2 the precise relationship is established between common rigid motion and dual quaternions, and expressing dual quaternions by method of dual angles and screw axis is obtained, meanwhile the three motion group, transition group, rotation group and screw group, are depicted. Moreover, the structure theorem of unite dual quaternions are useful to to research the inner product and outer product between lines.In Chapter three, the outer-product-operators are firstly introduced, also their basic properties are discussed. We introduce the Hamilton-operators, whose algebraic properties can be investigated by outer-product-operators. In essential, both outer-product-operators and Hamilton-operators are tools to translate the new operations into operations among matrices. We can say that the central work of Chapter 2 is to depict the single motion and its aim is to depict them as vividly as possible and in Chaper 3 the central one is to depict the motion group as a whole, in perspective of homomorphism, emphasizing the operations between two elements. In the first section we proved that the rotation group is isomorphic to factor group, the unit quaternions ball by±1 . And we get the structural theorem of orthornormal dual matrices, thus gaine the algorithm of solving screw quaternions by eigenvalue. We also prove another important theorem,that screw group is isomorphic to another factor group, the unit dual quaternions ball by±1 . Also screw group is isomorphic to the special orthornormal dual matrices group with degree of three. The most important result is that M(4)≌SO(?)(3)≌H(?)(1)/{±1}.In the second and third section of Chaper 3, quaternion differential equation and dual quaternion equation are researched and obtained,...