Abstract
General, complex geometry forms of RRR regional structures are often avoided due to the presence of inner boundaries within the workspace which tend to complicate robot guidance. Despite the added complexity, certain RRR geometries may have useful applications as they contain large workspace regions where four alternate configurations may be used to reach a given spatial location. Cusp points often appear on the workspace boundaries of general RRR regional structures, and determining their precise location may be useful for both design and guidance purposes.
A twelfth degree polynomial equation in the outer joint variable is derived which defines the location of non-trivial cusps in the workspace.
A new closed form workspace boundary equation is derived in the outer joint variable and x coordinate of the toroidal surface generated by rotation of the two outer revolutes. If the outer joint variable is incremented, a quadratic in x is formed at each step which enables a very efficient determination of the workspace boundaries while also providing the coordinates of the boundary on the toroidal surface.
The General Equation of a Geodesic on a Surface of Revolution applied to the Sphere
