By Simon McBeath
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5) – λ ∈ [0, 1] during braking. 29) (Ψ (λ) − Tb ) , λ˙ = − Jω with ω > 0 and J (1 − λ) Fz μ(λ). 29, it is obvious that the equilibrium points are characterised by T b = Ψ (λ). 6, by the intersections between the curve Ψ (λ) and the constant value of the braking torque T b . To summarise the system behaviour, one may note the following. 1. If T b > max Ψ (λ), the system has no equilibrium points (recall that the λ model has been derived under the assumption that λ ∈ [0, 1). ¯ 1 and 2. 6, where λ λ of T b = Ψ (λ).
From then on, the force loop takes care of regulating the braking force and it manages the overall manoeuvre. 3) T b = κ b Fb , where Fb is the braking force and κb ∈ R+ is the proportionality constant. 4) with ωact = 70 rad/s and τ = 10 ms. Note that the delay accounts for both that possibly due to the actuator dynamics and for that due to the signal transmission introduced by the networked vehicle architecture, which manages the data transmission. Of course, the real value of the pure delay must be carefully established in any practical situation.
71, 72]), also known as the magic formula. The name magic formula comes from the fact that the structure of the model equations does not rely on a physical basis and it appears rather complex and with many parameters to be determined. Further, the magic comes from the fact that by eﬀectively tuning the parameters such a model does indeed ﬁt a wide variety of tyres in a large range of operating conditions. This model has been shown to suitably match experimental data, obtained under particular steady-state conditions, which assume a constant value of both linear and angular velocities of the tyre.