Distance-based formulations have successfully been used to obt
ain closure polynomials
for planar mechanisms without relying, in most cases, on variab
le eliminations. The methods re-
sulting from previous attempts to generalize these techniques t
o spatial mechanisms exhibit some
limitations such as the impossibility of incorporating orient
ation constraints. For the first time, this
paper presents a complete satisfactory generalization. As an e
xample, it is applied to obtain a clo-
sure polynomial for the the general triple-arm parallel robot (that is
, the 3-RP
S 3-DOF robot). This
polynomial, not linked to any particular reference frame, is obtai
ned without variable eliminations
or tangent-half-angle substitutions.