Non-linear frictional dynamics reduce the tracking performance of machine control systems involving high-precision, low-velocity tasks. We present an adaptive non-linear friction compensation scheme for a friction model, which captures problematic friction effects such as Stribeck effect, hysteresis, stick-slip limit cycling, pre-sliding displacement and rising static friction. We show that without robust adaptation, frictional dynamics and other modelling uncertainties can cause an adaptive friction compensation scheme to become unstable. We extend robust adaptive theory to include a new type of error model with a non-linear regression vector and Lipschitz disturbances. By using persistent excitation in the desired trajectory, our controller achieves stable adaptation for friction force effects due to static, Coulomb and viscous components, as well as for inertia and the Stribeck effects, while remaining robust to perturbations in friction force due to frictional lag and frictional memory. Although the Stribeck parameter occurs non-linearly, stable adaptation is achieved by exploiting properties of convexity or concavity. The theory also provides the analytical framework to understand why adding dither, which is an oscillation about the desired trajectory, has been effective in controlling systems with friction. Simulation comparisons show that although both the adaptive and non-adaptive compensators overcome the specific problems caused by non-linear friction dynamics such as stick-slip limit cycling, hunting about a set point, and hysteresis, after transients, adaptation can achieve superior tracking with lower control usage, especially for tasks where persistent excitation occurs naturally and the friction model parameters are detuned by 50%.