This paper develops a geometric method to estimate the error space of 3-DOF planar mechanisms with the Minimum Volume Ellipsoid Enclosing (MVEE) approach. Both the joint clearances and actuator errors are considered in this method. Three typical planar parallel mechanisms are used to demonstrate. Error spaces of their serial limbs are analyzed. Thereafter, limb-error-space-constrained mobility of the manipulator, namely, the manipulator error space is analyzed. The MVEE method has been applied to simplify the constraint modeling. A closed-form expression for the manipulator error space is derived. The volume of the manipulator error space is numerically estimated. The approach in this paper is to develop a geometric error analysis method of parallel mechanisms with clear algebraic expressions. Moreover, no forward kinematics computations have been performed in the proposed method, in contrast to the widely used interval analysis method. Although the estimated error space is larger than the actual one, because the enclosing ellipses enlarge the regions of limb error space, the method has an attractive advantage of high computational efficiency.

In this paper, we present a novel design framework to connect linkage synthesis with dynamics performance of the linkage. The aim of the design framework is to improve the dynamics performance of the mechanism through linkage design, instead of improving manufacturing accuracy or changing driving strategy. Specifically, the design framework is to complete motion generation of four-bar linkage, considering clearance joints and dynamics performance. The constraint model of motion generation and the dynamics model of four-bar linkage are established, respectively. The coordinates of four joints of four-bar linkage are divided into two parts, one of parts is the parameters to improve the dynamics performance of the linkage and is selected as the optimization variables. The other parts of joint coordinates is to satisfy the kinematics requirements and is obtained by solving constraint equations of motion generation. Through optimization calculation, we can obtain the optimal configuration of the four-bar linkage that achieves specified task positions with high motion accuracy and low wear extent of clearance joint. Finally, a numerical example is proposed to demonstrate the novel design framework.

The synthesis of four-bar linkage has been extensively researched, but for a long time, the problem of motion generation, path generation, and function generation have been studied separately, and their integration has not drawn much attention. This paper presents a numerical synthesis procedure for four-bar linkage that combines motion generation and function generation. The procedure is divided into two categories which are named as dependent combination and independent combination. Five feasible cases for dependent combination and two feasible cases for independent combination are analyzed. For each of feasible combinations, fully constrained vector loop equations of four-bar linkage are formulated in a complex plane. We present numerical examples to illustrate the synthesis procedure and determine the defect-free four-bar linkages.

A parallel-mechanism-equivalent model for repeatability analysis of an overconstrained kinematic coupling is proposed. An overconstrained Kelvin-type coupling with one additional support is introduced and used for method illustration. Contact forces of the an overconstrained coupling under a preload are computed with the Moore-Penrose inverse, and the deformations are obtained using the Hertz theory. The coupling is equivalently modeled as a 7-SPS parallel mechanism, the spherical joints of which represent centers of supporting balls and contact points, and prismatic joints are used to simulate the deformations. Therefore, the pose error of the coupling due to a preload is analyzed using the well-appraised incremental method for forward kinematics analysis of parallel mechanisms. The uncertainties of the preload are discussed and a boundary-sampling method is proposed for repeatability analysis. The main contribution of this study lies in the greatly simplified repeatability analysis of overconstrained kinematic couplings by the proposed parallel-mechanism-equivalent model and the boundary-sampling method. Finally, the proposed methods are validated by a case study.