In existing robot fault detection schemes, sensed values of the joint status (position, velocity, etc.) are typically compared against expected or desired values, and if a given threshold is exceeded, a fault is inferred. The thresholds tend to be empirically determined and held constant over a wide range of trajectories. This leads to false alarms when the threshold is too small to counter the error-inducing effects model inaccuracy and to undetected faults when the threshold is too large for the given situation. This paper presents new methods for adaptively choosing fault detection thresholds, subject to sensing and modeling inaccuracies and the changing status of the robot. Our approach chooses optimal thresholds based on a Singular Value Decomposition (SVD) of a specialized error regressor format of the dynamics to minimize the possibility of false alarms or undtected failures. The thresholds vary dynamically with the changing trajectory and configuration of the robot and with the robot's failure status. Examples of the fault detection scheme for a non-planar 3 DOF robot are given.