Analysis And Design Of Spatial Manipulators : An Exact Algebraic Approach Using Dual Numbers And Symbolic Computation
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This thesis presents a uniﬁed framework for the analysis of instantaneous kinematics and statics of spatial manipulators. The proposed formulation covers the entire range of kinematic behavior, with kinematic singularity and isotropy appearing as special cases. An analogous treatment of statics is also presented. It is established that the formulations presented are capable of generating exact solutions in closed form for several interesting problems in manipulator analysis. Several such results have been obtained via extensive usage of symbolic computation tools developed for this purpose. The proposed approach is applicable to manipulators of diﬀerent architectures. However, the focus is on the parallel and hybrid manipulators, as their analysis presents more challenges than their serial counterparts. The theory of screws has been adopted as the basis of the formulation. Instantaneous kinematics and statics are studied in terms of the principal bases of the space of twists, se(3), and the space of wrenches, se* (3), respectively. A dual number parameterisation of the motion space is adopted to make the formulation compact and dimensionally consistent. The properties of the dual combination obviate the need for an explicit scaling between the linear and angular velocities or the forces and moments. Hence the results obtained from the formulation are purely geometrical. The analysis of the twists is performed via the dual velocity Jacobian matrix. The principal basis of se(3) is obtained from the eigenproblem of a symmetrical dual matrix associated with the Jacobian. The notion of a dual eigenproblem is introduced in this context. Solutions are provided for the general case, as well as a few special cases. The computations involve the solution of at most a cubic equation for any arbitrary degree-of-freedom of rigid-body motion, and closed form results are therefore ensured. The results of the eigen-analysis lead to a decoupling of the rotational and the pure translational components of a rigid-body motion. This is termed as the partitioning of degrees-of-freedom. They also motivate an interesting classiﬁcation of the manipulators based on the instantaneous partition of its degrees-of-freedom. This notion is further extended to analyse the eﬀects of a singularity on the motion characteristics of a manipulator. Due to the duality of se(3) and se*(3), the formulation of statics is completely analogous, and involves, in essence, only the substitution of the dual wrench-transformation matrix for the dual Jacobian. A similar partitioning of the wrench system is introduced based on the eigen-decomposition in the context of statics. It is shown that the principal screws associated with either a system of twists or wrenches can be obtained from a generalised eigenproblem of two symmetric real matrices arising out of the symmetric dual matrix mentioned above. The general 2-and 3-screw systems are analysed in closed form via the generalised characteristic polynomial. Several special screw systems are described in terms of algebraic equations in terms of the coefﬁcients of this polynomial. Principal bases for 4-and 5-systems are obtained in a novel fashion without deriving their reciprocal systems explicitly. Using the same approach based on the analysis of the characteristic polynomial, compact algebraic conditions for singularity and isotropy are derived as the special cases of a single formulation. The above formulations establish the existence of exact closed-form results. However, to implement them symbolically for a real application problem, capabilities in existing computer algebra systems do not suﬃce in general. Therefore simpliﬁcation and computational algorithms are developed for dealing with large expressions with algebraic and trigonometric terms typically appearing in kinematics and statics. Three canonical forms of such expressions and the corresponding simpliﬁcation schemes are presented. The theoretical developments are illustrated with examples of serial, parallel and hybrid manipulators throughout the thesis. However, the most important applications of these are in the kinematic and static analysis of a semi-regular Stewart platform manipulator (in which the top and bottom platforms are semi-regular hexagons). Using the degeneracy of the wrench transformation matrix as the singularity criterion, the singularity manifold of the manipulator is obtained via extensive application of the symbolic simpliﬁcation algorithms. The constant-orientation singularity manifold is derived in a compact closed form, and a complete geometric characterisation and explicit parameterisation of the same are presented. The constant-position singularity manifold is also obtained in closed form. On the other hand, families of conﬁgurations of the manipulator for combined kinematic or static isotropy for a given architecture are derived in closed form. Also, architectural designs are obtained for the manipulator such that it exhibits combined kinematic or static isotropy at a given conﬁguration.