The operation of machine tools often demands a highly accurate knowledge of the tool center point's (TCP) position. The displacement of the TCP over time can be inferred from thermal models, which comprise a set of geometrically coupled heat equations. Each of these equations represents the temperature in part of the machine, and they are often formulated on complicated geometries. The accuracy of the TCP prediction depends highly on the accuracy of the model parameters, such as heat exchange parameters, and the initial temperature. Thus it is of utmost interest to determine the influence of these parameters on the TCP displacement prediction. In turn, the accuracy of the parameter estimate is essentially determined by the measurement accuracy and the sensor placement. Determining the accuracy of a given sensor configuration is a key prerequisite of optimal sensor placement. We develop here a thermal model for a particular machine tool. On top of this model we propose two numerical algorithms to evaluate any given thermal sensor configuration with respect to its accuracy. We compute the posterior variances from the posterior covariance matrix with respect to an uncertain initial temperature field. The full matrix is dense and potentially very large, depending on the model size. Thus, we apply a low-rank method to approximate relevant entries, i.e. the variances on its diagonal. We first present a straightforward way to compute this approximation which requires computation of the model sensitivities with with respect to the initial values. Additionally, we present a low-rank tensor method which exploits the underlying system structure. We compare the efficiency of both algorithms with respect to runtime and memory requirements and discuss their respective advantages with regard to optimal sensor placement problems.

翻译：机床运行通常需要精确了解刀具中心点（TCP）的位置。TCP随时间变化的位移可通过热模型推断，该模型由一组几何耦合的热方程组成。每个方程代表机床某部分的温度，且通常在复杂几何区域上建立。TCP预测的准确性高度依赖于模型参数的准确性，例如热交换参数和初始温度。因此，确定这些参数对TCP位移预测的影响至关重要。进而，参数估计的准确性本质上由测量精度和传感器布置决定。确定给定传感器配置的精度是实现最优传感器布置的关键前提。本文针对特定机床建立了热模型。在此基础上，我们提出了两种数值算法来评估任意热传感器配置的精度。我们基于不确定的初始温度场，从后验协方差矩阵计算后验方差。该全矩阵稠密且规模可能极大（取决于模型大小）。因此，我们采用低秩方法近似相关条目，即对角线上的方差。首先，我们提出一种直接计算该近似的方法，该方法需要计算模型对初始值的灵敏度。此外，我们提出一种利用底层系统结构的低秩张量方法。我们比较了两种算法在运行时间和内存需求上的效率，并讨论了它们对最优传感器布置问题的各自优势。