Page - 95 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator is counteracted by the re-sampling step, where a set of a posteriori particlesx+h(k) is randomly generated based on the relative likelihood q¯h of the h states vectors. For a graphic representation of how the particles are moved during one time step, see Figure 5.1. In the re-sampling step, different strategies can be applied to randomly generate a new set of samples x+h(k). Within this work, the re-sampling strategy proposed by Simon is used, [Sim06]. At every time step k, the following procedure has to be applied. First, for every particleN, a random number r is chosen that is uniformly distributed between 0≤ r≤1. Next, starting with the first particleh= 1, the relative likelihoods q¯h(k) are accumulated until the ∑m h=1 q¯h(k)≥ r. The new particle x+1 (k) is now set to x−m(k). This is doneN times at each time stepkuntil all particlesx+h(k) have been assigned. In the theoretical case ofN=∞, the PDF of x+h(k) is equal to the PDF p(z(k)|x(k)). It has tobementionendthatother re-sampling strategies exist thatmightbemoreefficient in terms of computational effort for this application. After the re-sampling step, any statistical measure (e.g. mean or covariance) can be computed for the current time step kwith the a posteriori particles. TheN particles are already distributed according to the PDF p(x(k)|z(k)). For friction potential estimation, the most likely state of xˆ(k) at every time step k, which is of interest, is given by its expected valueE(x(k)|z(k)), which reads xˆ(k) =E(x(k)|z(k)) = 1 N N∑ h=1 x+h(k), (5.9) [Sim06]. Figure 5.1 shows a graphic representation of the steps of the particle filter. It can be seen that at time step k, theN particles are distributed based on the state’s PDF p(x(k−1)) from the last time stepk−1. After calculating the relative likelihoods q¯h based on measurements and a given distribution p(z(k)|x(k), the re-sampling step is conducted. The againN new particles x+h(k) now move toward p(z(k)|x(k), as can be seen by comparing the gray line in Figure 5.1a and the gray particles in Figure 5.1b). As these particles are randomly generated, a few particles also occur in regions with a low probability. This ensures that the algorithm is also able to detect changes in the vehicle state x(k) from one time step to another. Nevertheless, how fast the particle filter can converge depends on how close the measurement distribution is to the prior distribution. 95