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Kalman Filters Made Simple

Updated: Jun 10, 2023

Imagine you're driving down the highway with your GPS guiding your way. Have you ever wondered what goes on behind the screen of your navigation device? How does it accurately estimate your location, even when there's a bit of inaccuracy in the GPS signal? Enter the world of Kalman Filters!


The Kalman Filter, a trusted companion in the realm of control systems and data analysis, is akin to a GPS for data. Invented by Rudolf E. Kalman, it's a mathematical marvel that takes a series of measurements observed over time and estimates unknown variables in a dynamic system.


Getting to Know the Kalman Filter

Picture this. You're throwing a ball for your dog in the park. You want to predict where it will land so your dog can catch it. But your throw isn't perfect, and neither is your prediction. This is where the Kalman Filter comes into play! It's like a superpower that predicts the ball's actual position based on your imperfect throw and the previous throws you've made.


This amazing tool works in two simple steps:


1. Prediction:

It uses the dynamic model of the system (the way you throw the ball) to predict the next state and error covariance (where the ball might land).

2. Update:

It refines this prediction using new measurements (the actual trajectory of the ball).


While this might seem like magic, it's actually some neat mathematics at work. However, the standard Kalman Filter assumes the system to be linear and the noise to be Gaussian. This doesn't always hold true in real life - not everything moves in a straight line, and not all uncertainties are Gaussian! That's where the Extended and Unscented Kalman Filters come to the rescue.


Venturing into the Extended Kalman Filter (EKF)


Imagine trying to navigate a winding mountain road with a map that only shows straight highways - that's not going to work well! The Extended Kalman Filter (EKF) comes into play when the system is nonlinear. It's like a map that can handle curves. It linearizes the system dynamics and measurement functions about the current estimate, and then applies the standard Kalman filter update.


The EKF is a powerful tool, but it's not perfect. If the system is highly nonlinear, it might lead us off track. Also, the need for analytic Jacobians (a type of derivative) can be a tough requirement for some complex systems.


Unraveling the Unscented Kalman Filter (UKF)


The Unscented Kalman Filter (UKF) is like a supercharged GPS for nonlinear systems. Instead of trying to straighten out the curves, it embraces them! The UKF uses a deterministic sampling approach called the Unscented Transformation to approximate the probability distribution of the state, providing a more accurate estimate for highly nonlinear systems.


In our winding mountain road example, the UKF would be like a GPS that adjusts its predictions based on the actual twists and turns of the road, providing a more accurate navigation route.


An Example


Now, let's delve into an example. Imagine I'm tracking a man on a bicycle riding in a straight line. I want to estimate his position and speed. The process is pretty straightforward:


1. I observe the man's position - this is my measurement.

2. I use the Kalman filter to predict the man's position and speed - this is my prediction.

3. I compare my prediction with my next observation, adjust accordingly, and voila! I have a more accurate estimate.


This example may seem simple, but it encapsulates the essence of these powerful tools. Whether it's tracking a man on a bicycle, guiding a self-driving car through a bustling city, or even predicting stock prices in the unpredictable world of finance, these principles hold.


The Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) would come in handy if our bicyclist suddenly began a game of dodgeball, zigzagging erratically instead of moving in a straight line. The EKF would attempt to straighten out this zigzag path, while the UKF would embrace its erratic nature, providing a more accurate prediction.


In the more complex world of spacecraft navigation, I'd be using similar principles. For instance, if I were trying to dock a spacecraft to the International Space Station (ISS), I'd be dealing with a myriad of variables such as relative position, velocity, gravitational forces, and atmospheric drag. These complex dynamics are often better handled by the Unscented Kalman Filter.


However, it's crucial to remember that while these filters are powerful tools, they aren't foolproof. They rely on the noise being Gaussian and the models being accurate representations of reality, which isn't always the case. But when used judiciously, they prove to be an invaluable tool in my arsenal.


I've come to appreciate the power and versatility of Kalman filters. From tracking a man on a bicycle to guiding spacecraft, these tools have proven to be invaluable assets. So next time you look at your GPS, remember that there's a little bit of Kalman magic working behind the scenes, turning a complex, uncertain world into precise, predictable coordinates.


Now, isn't that something to appreciate?


Cite this article as:

Y. Kumar, “https://www.spacenavigators.com/post/kalman-filters-made-simple.” [Online]. Available: https://www.spacenavigators.com/post/kalman-filters-made-simple

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