KalmanFilters.md (1291B)
1 # Kalman Filters 2 3 **Source:** Probabilistic Robotics 4 5 **Chapter:** 3 6 7 **Definition:** Kalman filters are a continuous, uni-modal approach to localization that combines gaussians to predict position and uncertainty. 8 9 ## Gaussian Combination 10 11 1D Gaussians are defined by the pair $(\sigma^2, mu)$, representing the variance and mean respectively. 12 13 With Kalman filters, we want to maintain these two values during our move, sense loop. 14 15 To combine two Gaussians we compute the following where we have $(\sigma^2_a, mu_a)$ and $(\sigma^2_b, \mu_b)$: 16 17 $\mu_c = \frac{\mu_a * \sigma^2_b + \mu_b * \sigma^2_a}{\sigma^2_a + \sigma^2_b}$ 18 19 $\sigma^2_c = \frac{1}{\frac{1}{\sigma^2_a} + \frac{1}{\sigma^2_b}}$ 20 21 ## Higher Dimensional Combinations (Multivariate Gaussians) 22 23 Higher dimensional gaussians can useful for tracking both position and velocity to improve next step predictions. 24 25 Gaussian combination is easy in 1D. The generalization is a bit less so, and uses matrices. 26 27 A few variables in use for KFs in higher dimensions are: 28 29 - F : this matrix is the state transition matrix. This is multiplied by the state vector to predict positions at timestep t+1 30 - H : This is the observation matrix. 31 - X_t : This is the transpose of the state estimation vector. 32 - P: Estimate uncertainty matrix.