A neural net (NN) controller for a general serial-link robot arm is developed. The NN has two layers so that linearity in the parameters holds, but the ''net functional reconstruction error'' and robot disturbance input are taken as nonzero. The structure of the NN controller is derived using a filtered error/passivity approach, leading to new NN passivity properties. On-line weight tuning algorithms including a correction term to backpropagation, plus an added robustifying signal, guarantee tracking as well as bounded NN weights. The NN controller structure has an outer tracking loop so that the NN weights are Conveniently initialized at zero, with learning occurring on-line in real-time. It is shown that standard backpropagation, when used for real-time closed-loop control, can yield unbounded NN weights if 1) the net cannot exactly reconstruct a certain required control function or 2) there are bounded unknown disturbances in the robot dynamics, The role of persistency of excitations explored.