Closure conditions for spatial mechanisms: a non-recursive formulation

 In closed spatial single-degree-of-freedom mechanisms six dependent joint variables must be expressed in terms of a single independent variable. The same problem arises when six joint variables of a serial robot must be expressed in terms of six independent coordinates describing the position and the angular orientation of the robot hand relative to the robot base. Such kinematics analyses start out from closure conditions. There are systematic ways of formulating closure conditions, e.g. in terms of scalar products of body-fixed vectors. In the existing literature considerable labour is devoted to the conversion of such vector equations into scalar equations in terms of Denavit–Hartenberg parameters. In one way or other, two-step recursive procedures are used which require lengthy and sometimes non-obvious formula manipulations. In the present paper a straight-forward non-recursive procedure is presented which is applicable to arbitrary spatial mechanisms. Its efficiency is illustrated by applications to the mechanisms RRSRR, 5R-2P, 4R-3P and 7R.