What is a Kalman filter?

In robotics, we use the Kalman filter to estimate the state of a moving robot while considering the uncertainties of the position of a robot. In short, we predict the state and covariance and then update it based on the measurements. Let us simplify this concept with the help of an example.

Example

Consider an example in which a robot is moving in a straight line. There can be obstacles on the way to the robot that can add noise to the motion of a robot. If we are interested in finding the position of a robot using the sensor measurements, there can be uncertainty in the sensor while measuring the position. To rectify this problem, we use the Kalman filter to estimate the robot's correct position.

Mathematical intuition

Initially, the position of the robot is $x$and the actual measurement is represented by $z$. Mathematically, the Kalman filter is applied in the following steps.

• We can find the robot's position based on its movement. In our case, the robot is moving in a straight line. So the linear motion of a robot is given by the equation shown below. In the equation, $x_k$ is the estimated or predicted position at the time step $k$ and $x_{k-1}$is the previous predicted position of the robot. Here, $v_k$ is the noise of the sensor during measurement.

• An alternate method to predict the robot's position is by using covariance, which is based on the uncertainty of the movement model.

• After predicting the state, it is time to update the position. Based on the scenario, we can predict the Kalman Gain $K_k$based on the uncertainty of the prediction and the measurement.

• With the help of Kalman Gain, the update on the estimated position is given below.

• We keep iterating the update and prediction steps until we have better estimates of the robot's position, given the noise in the sensor.

Applications of the Kalman filter

There are several applications of the Kalman filter. Some of the applications of the Kalman filter are illustrated below.

Conclusion

When measured with noisy sensors, the Kalman filter helps to estimate the position of moving objects. This Answer allows us to understand how it helps us predict and then update the position by looking at the mathematical intuition. The Kalman filter finds its application where the estimation of the position of the moving object is required.