The singularity loci of two triangular parallel manipulators

We study the Jacobian of spatial parallel manipulators that have triangular base and top platform architectures with 2-2-2 and 3-2-1 actuator configurations. The Jacobian matrix is formulated using dual quaternion coordinates. The zero locus of the determinant of the resulting Jacobian is what we refer to as the singularity loci. The result is an eighth order polynomial in the dual quaternion coordinates. The loci of singular positions for a fixed orientation for the 2-2-2 actuator arrangement is a cubic surface. The loci of singular positions for the 3-2-1 actuator arrangement is also a cubic surface, which factors into three planes