Parallel manipulators are prone to have force-unconstrained configurations. If the resultant. forces together do not span the system of forces to be applied or sustained, the manipulator is degenerate and is force unconstrained. Physically. the mobile platform can have motion even if all the actuated, joints are locked, i.e., the manipulator can instantaneously gain one or more degrees of freedom that are unconstrained by the actuators. Since force-unconstrained configurations are uncontrollable, the identification and elimination of such configurations become critical.

Two methodologies for identifying the force-unconstrained poses are analyzed. The first method involves the differentiation of the nonlinear kinematic constraints of the input and output variables with respect to time. The second method makes use of the reciprocal screws associated with the actuated joints. Force-unconstrained poses of planar manipulators are analyzed depending on their actuation:​

Non Redundant Planar Parallel Manipulators.- Force-unconstrained poses of non-redundantly actuated planar parallel manipulators can be mathematically expressed as a function of the three variables that define the dimensional space of the manipulator. i.e., the location and orientation of the mobile platform. As a consequence, these poses can be plotted as surfaces in the mentioned three dimensional space. i.e., there are two orders of infinity of force-unconstrained poses. Examples of force-unconstrained poses of parallel manipulators are presented:​ 3-RPR, 3-PRR, and 3-RRR, where the underscore indicates the actuated joint. For the 3-RPR manipulator, a comparison and discussion between both methodologies is carried out. For the 3-PRR and 3-RRR manipulators, an efficient technique for identifying their force-unconstrained poses, based upon having joint displacements as known values, is presented.

Planar Parallel Manipulators with In-Branch Redundancy.- Force-unconstrained poses of planar parallel manipulators with actuated joints replacing passive joints lead to conditions of the joint displacements that have to be satisfied. In particular, the RRR - 2RRR. PRR - 2PRR. and RRR - 2RRR layouts are analyzed. In addition, equivalent mechanisms, whose motions describe the path of continuous force-unconstrained poses. are presented. The force-unconstrained poses of the analyzed layouts with in-branch redundancy represent curves in the three dimensional space, i.e., there is one order of infinity of force-unconstrained poses.

Planar Parallel Manipulators with In-Branch Redundancy.- Force-uncon-strained poses of planar parallel manipulators with the inclusion of actuated branches, beyond three. lead to a system of multivariable polynomials. Elimination methods are used to reduce the multivariable polynomials to a single polynomial in terms of one variable. In particular, Gröbner Bases and dialytic elimination methods are employed. The actuation layouts 4-RPR. 4-PRR. and 4-RRR are analyzed. The force-unconstrained poses of the analyzed planar parallel manipulators with additional actuated branches also represent curves in the mentioned three dimensional space, i.e.. there is one order of infinity of force-unconstrained poses.