Mechanism Constraints and Singularities—The Algebraic Formulation

Various mathematical formulations are used to describe mechanism and robot kinematics. This mathematical formulation is the basis for kinematic analysis and synthesis, i.e., determining displacements, velocities and accelerations, on the one hand, and obtaining design parameters on the other hand. Vector/matrix formulation containing trigonometric functions is arguably the most favored approach used in the engineering research community. A less well-known but nevertheless very successful approach relies on an algebraic formulation. This involves describing mechanism constraints with algebraic (polynomial) equations and solving these equation sets that pertain to some given mechanism or robot, with the powerful tools of algebraic and numerical algebraic geometries. In the first section of this chapter, the algebraic formulation of Euclidean displacements using Study coordinates (or dual quaternions) will be recalled. Then it will be shown how constraint equations of different kinematic chains can be derived using either geometric insight, elimination methods or the linear implicitization algorithm (LIA). LIA will be described in detail because it can be used even without deep kinematic and geometric insight into the properties of the kinematic chain. In case the kinematic chain consists of only elementary joints, the constraint equations consist of a set of polynomial equations. In the language of algebraic geometry, the corresponding polynomials form an ideal which can be treated with almost classical methods that might not be well known in the engineering community. Therefore in the third section, these methods will be recalled and it will be shown how they can be used to derive the properties of the kinematic entities. In the last section, we apply the introduced algebraic methods to the complete analysis of the 3-UPU-TSAI parallel manipulator.