Matematyczne podstawy rozszerzonych filtrów Kalman w lokalizacji robota

The Extended Kalman Filter (EKF) is a widely used algorithm in robot localization. It estimates the robot 's position and orientation by combinaing sensor data with a mathetical model of thee robot' s motion. Understanding the mathematical foundations of EKF is essentiail for effectiva implementation andd tuning.

State consigniotion andd Prediction

Te cory of EKF involves presenting thee robot 's state as a vector, typically including ding position and orientation. The prevention step wykorzystuje a nonlinear motion tone model tich contert state forward in time. Thi involves calculating thee Jacobian matrix of thee motion model to linearize thee non linear equations around thee concurt estimate.

Mierzenie Update andLinearyzation

Sensor measurements are measurement are measurement the update step. Since measurements are often nonlinear functions of thee state, EKF linearizes these functions using their ir Jacobians. Thi process addistres the predicted te state based one thee difference between expeed ted andd actual sensor reads.

Równania matematyczne

Te przewidywane równania są następujące:

(zob. pkt 2.2.1.1.1 niniejszego załącznika)

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

(zob. pkt 2.1.1.1 niniejszego załącznika)

Please Please = Flease Please Fause + Qever

where f is the nonlinear motion model, Fconsibis its Jacobian, P is the covariance matrix, and Q is the process noise covariance.

Te update equations are:

Xi1; Xi1; FLT: 0 Xi3; Xi3; Kalman gain: Xi1; Xi1; FLT: 1 Xi3; Xi3;

Khagen = Hahaha (Hahaha Hahaha Hahaha + Rahal)

Xi1; Xi1; FLT: 0 Xi3; Xi3; State update: Xi1; Xi1; FLT: 1 Xi3; Xi3;

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

Xi1; Xi1; FLT: 0 Xi3; Xi3; Covariance update: Xi1; Xi1; FLT: 1 Xi3; Xi3; Xi3;

PTION = (I - KTION HTION) PTION

Here, h is the nonlinear measurement model, Hconsibis its Jacobian, Rconsibis the measurement noise covariance, and zconsignis the actual sensor measurement.