Bayes Filter Reminder. Kalman Filter Localization. Properties of Gaussians. Gaussians. Prediction. Correction. σ 2. Univariate. 1 2πσ e.
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1 Kalman Filter Localization Bayes Filter Reminder Prediction Correction Gaussians p(x) ~ N(µ,σ 2 ) : Properties of Gaussians Univariate p(x) = 1 1 2πσ e 2 (x µ) 2 σ 2 µ Univariate -σ σ Multivariate µ Multivariate We stay in the Gaussian world as long as we start with Gaussians and perform only linear transformations 1
2 Introduction to Kalman Filter (1) Two measurements no dynamics Discrete Kalman Filter Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation Weighted least-square with a measurement Finding minimum error After some calculation and rearrangements Another way to look at it weigthed mean Matrix (nxn) that describes how the state evolves from t to t-1 without controls or noise. Matrix (nxl) that describes how the control u t changes the state from t to t-1. Matrix (kxn) that describes how to map the state x t to an observation z t. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance R t and Q t respectively. 6 Kalman Filter Updates in 1D Kalman Filter Updates in 1D 7 8 2
3 Kalman Filter Updates in 1D Kalman Filter Updates 9 10 Linear Gaussian Systems: Initialization Initial belief is normally distributed: Linear Gaussian Systems: Dynamics Dynamics are linear function of state and control plus additive noise:
4 Linear Gaussian Systems: Dynamics Linear Gaussian Systems: Observations Observations are linear function of state plus additive noise: Linear Gaussian Systems: Observations Kalman Filter Algorithm 1. Algorithm Kalman_filter( µ t-1, Σ t-1, u t, z t ): 2. Prediction: Correction: Return µ t, Σ t
5 The Prediction-Correction-Cycle The Prediction-Correction-Cycle Prediction Correction The Prediction-Correction-Cycle Kalman Filter Summary Prediction Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k n 2 ) Optimal for linear Gaussian systems! Most robotics systems are nonlinear! Correction
6 Nonlinear Dynamic Systems Linearity Assumption Revisited Most realistic robotic problems involve nonlinear functions Non-linear Function EKF Linearization (1)
7 EKF Linearization (2) EKF Linearization (3) 25 Depends on uncertainty 26 EKF Linearization: First Order Taylor Series Expansion Prediction: EKF Algorithm 1. Extended_Kalman_filter( µ t-1, Σ t-1, u t, z t ): 2. Prediction: Correction: 5. Correction: Return µ t, Σ t
8 Robot Position Prediction Robot Position Prediction: Example In a first step, the robots position at time step k+1 is predicted based on its old location (time step k) and its movement due to the control input u(k): f: Odometry function Observation The second step it to obtain the observation Z(k+1) (measurements) from the robot s sensors at the new location at time k+1 The observation usually consists of a set n 0 of single observations z j (k+1) extracted from the different sensors signals. It can represent raw data scans as well as features like lines, doors or any kind of landmarks. The parameters of the targets are usually observed in the sensor frame {S}. Therefore the observations have to be transformed to the world frame {W} or the measurement prediction have to be transformed to the sensor frame {S}. This transformation is specified in the function h i (seen later) measurement prediction Observation: Example Raw Date of Laser Scanner Set of discrete line features Extracted Lines Extracted Lines in Model Space 8
9 Measurement Prediction In the next step we use the predicted robot position and the map M(k) to generate multiple predicted observations z t what you should see They have to be transformed into the sensor frame Measurement Prediction: Example For prediction, only the walls that are in the field of view of the robot are selected. This is done by linking the individual lines to the nodes of the path We can now define the measurement prediction as the set containing all n i predicted observations The function h i is mainly the coordinate transformation between the world frame and the sensor frame Need to compute the measurement Jacobian to also predict uncertainties Measurement Prediction: Example The generated measurement predictions have to be transformed to the robot frame {R} Matching Assignment from observations z j (k+1) (gained by the sensors) to the targets z t (stored in the map) For each measurement prediction for which an corresponding observation is found we calculate the innovation: According to the figure in previous slide the transformation is given by and its Jacobian by and its innovation covariance found by applying the error propagation law: The validity of the correspondence between measurement and prediction can e.g. be evaluated through the Mahalanobis distance: 9
10 Matching: Example Matching: Example To find correspondence (pairs) of predicted and observed features we use the Mahalanobis distance with Estimation: Applying the Kalman Filter Kalman filter gain: Update of robot s position estimate: The associate variance Estimation: Example Kalman filter estimation of the new robot position : By fusing the prediction of robot position (magenta) with the innovation gained by the measurements (green) we get the updated estimate of the robot position (red) 10
11 Multihypothesis Tracking 41 11
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