Page - 94 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator particle filtering compared to Kalman filtering, for example, is the computational effort, which depends on the numberN of the particles that have to be dealt with at every time step k. Theequations thatdescribe theparticlefilteraregivenbelow. Amoredetailedderiva- tion of the particle filter’s equation is given by Simon, [Sim06], and a very descriptive application (although not related to friction potential estimation) is shown in Watzenig, [Wat06, p.65-75]. Applied to the non-linear state model given in Equations 5.1 and 5.2, the steps to be solved within a particle filter for each state variable xl(k) at each time step k according to Simon, [Sim06], read: 1. Time propagation step: The a priori particlesx−h(k) are calculated based on the l-th state model equation flwith the particles x + h(k−1) from the previous time step k−1 and the process noise wh(k−1) by x−h(k) =f(x + h(k−1),wh(k−1)). (5.7) Fork= 1, thefirstN particlesare randomlygeneratedbasedonthePDFp(xl(0)). 2. Relative likelihood: Therelative likelihoodqh iscomputedbasedonthemeasurementequationh(x − h(k)) and the PDF of vh(k). In the case of Gaussian noise, Equation 5.6 applies. It has to be noted that Equation 5.6 does not give a direct relation, but only a propor- tional one (see∝ in Equation 5.6 where = would be expected). Nevertheless, if it is applied to allN particles, the relative likelihood of the states is equal to that of its particles, [Sim06], 3. Normalising relative likelihood: To ensure that the sum of the likelihoods is equal to one, qh are normalised by q¯h= qh∑N h=1qh . (5.8) The next step, the re-sampling step, is skipped in the approach proposed by Ray, [Ray97], who calculates the most likely value of xˆ(k) based on the normalised relative likelihood for each particle by ∑N h=1x − h(k) · q¯h. 4. Re-sampling step: As some state vectors have a small relative likelihood, they do not contribute significantly to an estimate, but still require computational effort, [DGA00]. This 94