PMC

4-Bar Spherical Parallel Manipulator (4B-SPM) architecture. The 4B-SPM uses three parallelogram 4-bar substructures. Each substructure has two actuated DoF: pitch and roll. The roll DoF axis of each substructure intersects with the others at a singular point which represents the virtual center of a spherical workspace. The top linkage in each 4-bar substructure is extended to reach a mobile platform that moves tangential to the spherical workspace. Each top linkage is coupled to the mobile platform using a spherical joint ().

IRISS; an isotropic spherical serial mechanism having RCM point developed at UCLA mechatronics and control lab (a) DIAMOND; a spherical compact size mechanism having singularity free workspace with RCM point, developed at K.N Toosi University[] (b) Eye-Rhas; a double parallelogram based gravity balanced RCM mechanism, developed at Eindhoven University of Technology. “Courtesy of Preceyes b.v.”[] (c).

a Typical Miura-ori pattern consisting of parallelogram tessellations. Mountain and valley creases are presented with red and blue lines, the base and height of the parallelogram are both 3 mm, and the acute internal angle α = 60°. b The optimized origami patterns for a spherical shape. c, d Modified origami patterns are folded to approach given spherical and saddle surfaces (the maximum bending energy is given)

Computing the average crossing number. (a) Projection directions that result in one overcrossing are parallel to vectors of the form t_v - t_u, where t_u is on u and t_v on v. (b) Those directions trace a parallelogram P = P₁P₂P₃P₄, where P₁ = v^sp - u^sp, P₂ = v^ep - u^sp, P₃ = v^sp - u^ep and P₄ = v^ep - u^ep. The average crossing number equals the signed area of P projected on S₂ and normalized by half of the area of S₂, which can be computed using tools of Spherical Geometry [28].