System identification of the sinusoidal steady-state response of the Phantom Omni using a local linear model revealed that friction has a non-negligible effect on the accuracy of a global linear model, particularly at low frequencies. Some of the obvious errors observed with the global linear model at low frequencies were (i) the response amplitude was lower; (ii) local linear model coefficients became physically impossible (e.g., negative) at low frequencies; and (iii) low frequency inputs resulted in a greater nonlinearity in the response compared to higher frequency inputs. While standard friction models such as Coulomb friction could be used to model the nonlinearity, there is a desire to create a friction model that is not only accurate for sinusoidal steady-state responses, but can also be generalized to any input response. One measure that is universally present in dynamical systems is energy, and in this paper the relationship between a generalized measure of energy and damping for modeling the effect of friction is developed. This paper introduces the “α-invariant” as a means of generalizing the friction behavior observed with sinusoidal steady-state responses to other waveforms. For periodic waveforms, the α-invariant is shown to be equivalent to the energy dissipated in each cycle, which demonstrates the physical significance of this quantity. The α-invariant nonlinear model formulation significantly outperforms the linear model for both sinusoidal steady state and step responses, demonstrating that this method accurately represents the physical mechanisms in the Phantom Omni. Overall, the α-invariant provides an efficient way of capturing nonlinear dynamics with a small number of parameters and experiments.
