Serial manipulator kinematics with dual quaternions and grassmannians

Inverse kinematics (IK) of serial manipulators is related to reflection-free Euclidean spaces where distances and orientations remain invariant. These direct isometries can be derived from simpler forms known as quaternions and dual quaternions. Combining them with Hermann Grassmann's Extension Principle, the IK of serial manipulators is formulated entirely in projective geometry free of any metric. The only rules governing this geometry is the preservation of ratios and incidence. Actually, the holonomicity of these IK problems can be described using incidence relations alone.The algebraic constraints, derived from incident relations, define a manifestly holonomic system as opposed to general holonomic systems that need only satisfy Frobenius' Theorem for integrability using Pfaffian forms. The solution to these algebraic equations will require an introduction to algebraic geometry and commutative algebra. Definitions of basic geometric and algebraic objects along with a study of their respective properties are included. Pertinent theorems are proved and illustrated with simple examples to establish a dictionary between algebra and geometry. By this means, kinematic analysis is conveniently subjected to the theories of algebraic geometry and commutative algebra.More precisely, the inverse kinematics (IK) of a 6R serial manipulator (6Rsm) is formulated in the homogeneous projective space of dual quaternions (DQ). This leads to a robust algorithm because the 8-space of DQ together with the Grassmannian extension ensures that only valid solutions, which satisfy all of the constraint equations, are admitted. Numerical examples, based on two real 6Rsm architectures, are presented to illustrate the efficacy of the new algorithm. Its fiber (solutions) and critical loci (singularities) are described