Potential field methods are rapidly gaining popularity in obstacle avoidance applications for mobile robots and manipulators. While the potential field principle is particularly attractive because of its elegance and simplicity, substantial shortcomings have been identified as problems that are inherent to this principle. Based upon mathematical analysis, this paper presents a systematic criticism of the inherent problems. The heart of this analysis is a differential equation that combines the robot and the environment into a unified system. The identified problems are discussed in qualitative and theoretical terms and documented with experimental results from actual mobile robot runs.