Innovation, Resources and Economic Growth
The analysis of the interactions between natural resource scarcity, technological innovation and the dynamics of eco- nomic systems has a long-standing tradition in economics. During the 1980s and the early 1990s, a new phase of these interactions initiat
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Introduction
This contribution is devoted to the problem of tire-road friction estimation. The need for such type of studies, steers from the difficulty of direct sensing of tire forces, slip, slip angles and other external factors. Observer algorithms are, in this context, a low cost alternative for sensors. Tire forces information is relevant to problems like: optimization of Anti-look brake systems (ABS), traction system, diagnostic of the road friction conditions, etc. Literature for tire/road friction estimation is numerous. Bakker et al [1] and Burckhardt [4] describe two analytical models for t i r e / r o a d behavior t h a t are intensively used by researchers in the field. In these two models the coefficient of friction, #, or more precisely, the normalized friction force, i.e. F Friction force # - F~ Normal force is mainly determined based on the wheel slip s and some other p a r a m e t e r s like speed and normal load. Fig. 1 shows two curves, obtained from H a r n e d et al [9], t h a t represent typical # versus s behavior. It is current practice to n a m e the ratio between the friction and the normal forces, #, as being the "coefficient" of friction. Under constant normal force conditions, #, is a constant if and only if the Coulomb model is used to describe friction. Nevertheless, the Coulomb model is too simplistic to suitable represent forces between the rubber tire and the road, which are dominated by the elesto-plastic force/displacement characteristics. Therefore, to consiser it as a constant is a pure idealistic view. p should thus
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FIGURE 1. a) Variations between coefficient of road adhesion # and longitudinal slip s for different road surface conditions (left). b) Variations between coefficient of road adhesion # and longitudinal slip s for different vehicle velocities (right). be viewed more as the ratio between friction and normal forces (i.e. the normalized force), which is indeed a (static or dynamic) function of the system state variables. The expression given by Bakker et al [1], and Paceijka and Sharp [14], also known as "magic formula" is derived heuristically from experimental d a t a to produce a good fit. It provides the tire/road coefficient of friction # as a function of the slip s. The expression in Burckhardt [4] is derived with a similar methodology. The final m a p expresses # as a function of s, the vehicle velocity, v and the normal load on the tire F,~. Kiencke [10] presents a procedure for real-time estimation of #. A simplification to the analytical model by Burckhardt [4] is introduced in such a way that the relation between # and s is linear in the parameters. Kiencke [10] uses
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