Tire Modeling for Low-Speed and High-Speed Calculations

Vehicle dynamics simulations typically use semi-empirical tire models. The input to these models are normal load, sideslip angle and longitudinal slip, and the output are shear forces, aligning moment, and overturning moment. Since the longitudinal speed is in the denominator of both sideslip angle and longitudinal slip, the calculation of sideslip angle and longitudinal slip at very low longitudinal velocities leads to numerical problems. This has not been a particular stumbling point in the past because vehicle dynamics calculations were largely concerned with high speed analysis. In situations wherein the vehicle was braked to a stop, patchwork techniques sufficed for calculations at low speeds. Now, however, with the advent of serious attention to driving simulators, low speed tire modelling has become more important. This paper presents a new formulation which differs from previous work in that both longitudinal slip and slip angle become state variables rather than kinematic functions of wheel spin rate and wheel velocity. The result is a model which yields traditional solutions at high speed, and which comes to a stop in a reasonable way without numerical problems. The calculated longitudinal slip and slip angle can be paired with any semi-empirical tire model. Several challenging examples, including braking-induced spinout ending in a stop on a severely inclined surface, illustrate the power of this new technique.