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Singular configurations of parallel manipulators (PMs) are special poses in which the manipulators cannot maintain their inherent infinite rigidity. These configurations are very important because they prevent the manipulator from being controlled properly, or the manipulator could be damaged. A geometric approach is introduced to identify singular conditions of planar parallel manipulators (PPMs) in this paper. The approach is based on screw theory, Grassmann–Cayley Algebra (GCA), and the static Jacobian matrix. The static Jacobian can be obtained more easily than the kinematic ones in PPMs. The Jacobian is expressed and analyzed by the join and meet operations of GCA. The singular configurations can be divided into three classes. This approach is applied to ten types of common PPMs consisting of three identical legs with one actuated joint and two passive joints.
Force control is important in robotics research for safe operation in the interaction between a manipulator and a human operator. The elasticity center is a very important characteristic for controlling the force of a manipulator, because a force acting at the elasticity center results in a pure displacement of the end-effector in the same direction as the force. Similarly, a torque acting at the elasticity center results in a pure rotation of the end-effector in the same direction as the torque. A stiffness synthesis strategy is proposed for a desired elasticity center for three-degree-of-freedom (DOF) planar parallel mechanisms (PPM) consisting of three revolute-prismatic-revolute (3RPR) links. Based on stiffness analysis, the elasticity center is derived to have a diagonal stiffness matrix in an arbitrary configuration. The stiffness synthesis is defined to determine the configuration when the elasticity center and the diagonal matrix are given. The seven nonlinear system equations are solved based on one reference input. The existence and the solvability of the nonlinear system equations were analyzed using reduced Gröbner bases. A numerical example is presented to validate the method.
Redundant actuation for the parallel kinematic machine (PKM) is a well-known technique for overcoming general drawbacks of the PKM by helping it to avoid singularity and enhance stiffness characteristics, among others. Torque distribution plays a critical role in redundant actuation because this actuation causes the PKM to consume too much energy or put a substantial amount of stress on joints and links. This paper proposes a new torque distribution method for reducing the maximum torque of the actuator of a planar PKM. Here the main idea behind the proposed method is the use of superposition of a particular solution for a non-redundant case and an optimized null-space solution for a redundant case with a constant coefficient. The optimal value of a null-space solution can be easily determined by checking only the intersection points of the profile of the actuator's torque as the coefficient varies. We consider three cases of planar PKMs—2-, 3-, and 4-RRR PKMs—and present a detailed procedure for deriving a kinematic solution for the 2-RRR PKM based on Screw theory. We compare the proposed method with the minimum-norm pseudo-inverse method and assess a limitation of the proposed method. The torque distribution algorithm can be used to determine the number of actuators in an efficient manner and to reduce energy consumption.
In this paper, a novel optimal torque distribution method for a redundantly actuated parallel robot is proposed. Geometric analysis based on screw theory is performed to calculate the stiffness matrix of a redundantly actuated 3-RRR parallel robot. The analysis is performed based on statics focusing on low-speed motions. The stiffness matrix consisting of passive and active stiffness is also derived by the differentiation of Jacobian matrix. Comparing two matrices, we found that null-space vector is related to link geometry. The optimal distribution torque is determined by adapting mean value of minimum and maximum angles as direction angles of null-space vector. The resulting algorithm is validated by comparing the new method with the minimum-norm method and the weighted pseudo-inverse method for two different paths and force conditions. The proposed torque distribution algorithm shows the characteristics of minimizing the maximum torque.
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