Analytic formulation of the 6-3 fully-parallel manipulator's singularity determination

When a parallel manipulator reaches a singular configuration (singularity), the end effect (platform) pose cannot be controlled any longer, and infinite active forces must be applied in the actuated joints to balance the loads exerted on the platform. Therefore, these singularities must be avoided during motion. The first step to avoid them is to locate all the platform poses (singularity locus) making the manipulator singular. Hence, the availability of a singularity locus equation, explicitly relating the manipulator geometric parameters to the singular platform poses, greatly facilitates the design process of the manipulator. The problem of determining the platform poses, that make the 6-3 fully-parallel manipulator (6-3 FPM) singular, will be addressed. A simple singularity condition will be written. This singularity condition consists in equating to zero the mixed product of three vectors, that are easy to be identified on the manipulator, and it is geometrically interpretable. The presented singularity condition will be transformed into an equation (singularity locus equation) explicitly containing the geometric parameters of the manipulator and the platform pose parameters making the 6-3 FPM singular. Eventually, the singularity locus equation will be reduced to a polynomial equation by using the Rodrigues parameters to parameterize the platform orientation. This polynomial equation is cubic in the platform position parameters and one of sixth degree in the Rodrigues parameters.