Euler angles are commonly used as the orientation representation of most two degrees of freedom（2-DOF） rotational parallel mechanisms（RPMs）,as a result,the coupling of two angle parameters leads to complexity of kinematic model of this family of mechanisms.While a simple analytical kinematic model with respect to those parameters representing the geometrical characteristics of the mechanism,is very helpful to improve the performance of RPMs.In this paper,a new geometric kinematic modeling approach based on the concept of instantaneous single-rotation-angle is proposed and used for the 2-DOF RPMs with symmetry in a homo-kinetic plane.To authors＇ knowledge,this is a new contribution to parallel mechanisms.By means of this method,the forwards kinematics of 2-DOF RPMs is derived in a simple way,and three cases i.e.4-4R mechanism（Omni-wrist III）,spherical five-bar one,and 3-RSR1-SS one demonstrate the validity of the proposed geometric method.In addition,a novel 2-DOF RPM architecture with virtual center-of-motion is presented by aid of the same method.The result provides a useful tool for simplifying the model and extending the application of the RPMs.
Type synthesis of lower-mobility parallel mechanisms (PMs) has drawn extensive interests, particularly two main approaches were established by using the reciprocal screw system theory and Lie group theory, respectively. Although every above approach provides a universal framework for structural design of general lower-mobility PMs, type synthesis is still a comparably difficult task for the PMs with particular geometry or required to fulfill some specified tasks. This paper aims at exploring a simple and effective synthesis method for lower-mobility parallel mechanisms with orthogonal arrangement (OPMs), and the applied mathematical tool is established in the displacement group theory. For this purpose, the concept of the Cartesian DOF-characteristic matrix, originated from canonical displacement subgroup and displacement submanifold, is proposed. A new approach based on combination of the atlas of Cartesian DOF-characteristic matrix and displacement group-theoretic method is addressed for both exhaustive classification and type synthesis of OPMs. Type synthesis for some representatives of 3-DOF OPMs verifies effectiveness of the proposed approach.
