A Geometric Approach for Error Space Estimation of Planar Linkage

This article develops a geometric method to estimate the clearances-induced error space of any planar linkage. The error space discussed here represents the unconstrained mobility of the end-effector when actuators of the mechanism are locked, and is expressed by a connected geometry in 3-dimensional Euclidean frame {x, y, θ}. First, error space of the planar mechanism is modeled and closed-form expressions are derived. Then, levels of joints in error propagation analysis are defined and illustrated with an example of a eight-bar linkage, following which error propagation path among closed-loop structures is given. The modeling of error propagation and accumulation is introduced in detail. Moreover, a simplification technique is discussed for simple expression of the error space propagated from previous joints. This study provides a way to have a deep insight into the accuracy performance of any planar linkage and the proposed error space evaluation method is validated by case study of error space estimation of a four-bar linkage and a six-bar linkage. For the four-bar linkage, the structure with optimal accuracy is obtained. And for the six-bar linkage, the error space of the end-effector is expressed in closed form and visualized in the 3-dimensional frame. Finally, this work is concluded and advances of the proposed method are emphasized.