The three-probe method for separating the spindle rotation error and the specimen form error is extensively described in the literature. An attractive feature is its application in in-process measurement. However, the resulting uncertainty is studied far less extensively. In this paper, an evaluation and propagation method for the uncertainty, as well as for an uncertainty reduction, is given based on the system transfer function (S-function). First, utilizing the Laplace transform, the measurement system model is developed and expressed by an S-function. Second, the propagation laws of input uncertainties are analytically deduced by computing the partial derivatives of the S-function of roundness. Then, the laws are numerically validated by Monte Carlo simulations. The uncertainty propagation laws show that the uncertainties propagate with varying amplification over the harmonic domain, and moreover, they enable the quantification of both the harmonic and the total uncertainty of roundness. Taking the roundness uncertainty as a decisive parameter, three approaches are proposed for uncertainty reduction: (1) the hybrid 3-PM, where two roundness estimates are combined by taking individual harmonic estimate with the lowest uncertainty, (2) the fusion 3-PM, where the weighted average is taken over the harmonic domain, and (3) the angle optimization, which minimizes the total roundness uncertainty by properly arranging the sensor angles. The angle optimization is applied to the conventional 3-PM, as well as to the hybrid and the fusion 3-PMs. The genetic algorithm is adopted to speed up the optimization process. Finally, practical roundness measurements are performed.

在文献中广泛描述了用于分离主轴旋转误差和样本形状误差的三探针方法。一个有吸引力的功能是它在过程中测量中的应用。但是，对结果不确定性的研究远远不够广泛。在本文中，基于系统传递函数（S函数），给出了不确定性的评估和传播方法以及不确定性的降低方法。首先，利用拉普拉斯（Laplace）变换，开发测量系统模型并通过S函数表示。其次，通过计算圆度S函数的偏导数来解析推导输入不确定性的传播规律。然后，通过蒙特卡洛模拟对定律进行数值验证。不确定度传播定律表明，不确定度在谐波域上随着放大倍数的变化而传播，此外，它们还可以量化谐波和圆度的总不确定度。以圆度不确定度为决定性参数，提出了三种降低不确定度的方法：（1）混合3-PM，其中两个圆度估计是通过采用具有最低不确定度的单个谐波估计来组合的，（2）融合3-PM ，其中加权平均值取自谐波域，以及（3）角度优化，可通过适当安排传感器角度来最大程度地降低总圆度不确定度。角度优化适用于常规3-PM，以及混合和融合3-PM。采用遗传算法加快了优化过程。