Variance-Minimization Iterative Matching Method for Free-Form Surfaces—Part I: Theory and Method

Free-form surface matching that aligns measured points with a design model is a common problem in manufacturing automation. In this paper, an iterative variance-minimization matching (VMM) method is proposed to address measured points that have measuring defects, such as uneven/open point distributions and measuring noise. The basic idea is that the objective function is defined as the variance of the closest distance from each measured point to the design model, and the measuring defects are considered by incorporating an average distance item into the objective function. Using the defined average distance item, a strategy for analyzing the effect of measuring defects on VMM and existing methods is presented. It is shown that the VMM method does not easily become trapped in a local optimum when measuring defects exist. To consider convergence speed and convergence stability, a new distance based on the first-order point-to-point distance and point-to-tangent distance is developed and used in the objective function. To demonstrate the availability of the proposed method, quadratic convergence and positive definiteness are theoretically analyzed. The proposed method is efficient and insensitive to measuring defects and is useful for shape matching tasks involving free-form surface features. Note to Practitioners-This paper is motivated by the problem of matching measured points with a design model to automate manufacturing processes such as geometric inspection, workpiece localization, and allowance distribution. Measured points are obtained by applying a scanning device where measuring defects usually appear. Existing matching methods suffer from the drawback that the measured points may incline toward dense data and become trapped in a local optimum, due to measuring defects. To address this practical issue, this paper proposes a new method called variance-minimization matching (VMM), in which the objective function is optimized to weaken the effect of measuring defects. By examining the differences between VMM and existing methods, it is found that VMM can achieve quadratic convergence speed. Most importantly, the method is insensitive to uneven/open point distributions. In summary: 1) this method allows us to improve the matching accuracy in the presence of measuring defects; 2) there is no need to obtain a high-quality scan of the entire workpiece, potentially reducing scanning difficulty and improving scanning efficiency; and 3) the requirement of uniform sampling for measured points is reduced.