Robots and other similar automation machines have been widely used in various industries, such as automotive and semiconductor industries to improve productivity, quality, and safety in manufacturing processes. However, an unforeseen robot shutdown has the potential to cause an interruption in the entire production line, resulting in significant unplanned downtime, economic, production losses, and even work injuries. Thus, it is of high interest to detect incipient faults in industrial robots before they totally shut down or otherwise fail. A challenge for fault detection in industrial robots is the difficulty to obtain sufficient labeled training data under normal and abnormal health conditions. Thus, unsupervised machine learning algorithms are desired. In this work, a Gaussian mixture model-based unsupervised fault detection framework is proposed to effectively detect the faults in industrial robots using current signals. Signal preprocessing is first performed to clean the measured raw current signals. Then, motion-insensitive fault features chosen based on a system physics model that can reflect the deterioration of the industrial robots are extracted and fed into unsupervised learning algorithms for effective fault detection. The effectiveness and high accuracy of the proposed method are validated by experimental data obtained from industrial robot systems.