Understanding the current state of dynamic systems, from the trajectory of a self-driving car to the position of a satellite, is a complex challenge. These systems are constantly changing, and observations are often imperfect due to noisy sensors and incomplete data. The Extended Kalman Filter (EKF) is a widely applied tool that helps filter noise and provide a more accurate picture of a system’s true state.
The Problem of Estimation
Real-world systems operate unpredictably. For example, a drone’s movement is influenced by wind gusts, and its GPS signal might be weak or inaccurate. Raw sensor data alone often isn’t enough to provide a reliable estimate. A single GPS reading might have an error of several meters. The EKF addresses these challenges by combining predictions based on a system’s known behavior with incoming, noisy measurements, aiming for a more precise and consistent estimate than either source could provide alone.
From Kalman to Extended Kalman
The Kalman Filter (KF), developed by Rudolf E. Kalman in the early 1960s, estimates the state of linear dynamic systems. These systems have variables described by straight lines or simple proportionalities. The KF recursively processes measurements over time, providing an estimate that minimizes error under assumptions like Gaussian noise.
The standard Kalman Filter is limited when dealing with real-world systems, which are often non-linear. For example, equations describing robot arm motion or fluid flow are typically not linear. Applying a linear filter to a non-linear system can lead to inaccurate estimates because the filter’s assumptions about how noise propagates are violated.
The Extended Kalman Filter (EKF) adapts the Kalman Filter for non-linear systems. The “extension” in EKF refers to its method of linearizing the non-linear system around the current estimated state at each time step. This linearization uses Taylor series expansions and Jacobian matrices, which are matrices of partial derivatives. By creating a linear approximation of the non-linear dynamics, the EKF applies the principles of the linear Kalman Filter, though it is not always optimal for highly non-linear systems.
How the Extended Kalman Filter Works Conceptually
The Extended Kalman Filter operates through a continuous, two-step iterative process: a “prediction” step and an “update” step. It essentially makes an educated guess about the system’s future, then corrects that guess based on what it actually observes.
Prediction Step
The prediction step, sometimes called the time update, uses a mathematical model of the system’s dynamics to forecast its state. For instance, if tracking a vehicle, this step predicts its new position and velocity based on its current estimated state and any known inputs, like engine thrust or steering angle. This prediction also includes an estimate of the uncertainty associated with the forecasted state, acknowledging that the model itself is not perfect and there might be unpredicted changes, often referred to as process noise.
Update Step
The update step, also known as the measurement update, incorporates actual measurements from sensors to refine the predicted state. The EKF compares its predicted measurement (what it expected to see based on its prediction) with the actual measurement received from the sensors. The difference between these two, known as the measurement residual, helps the filter determine how much to adjust its initial prediction. The filter then uses a calculated “Kalman gain” to weigh the importance of the new measurement versus its own prediction, ultimately producing a more accurate and refined estimate of the system’s current state.
Where Extended Kalman Filters are Used
The Extended Kalman Filter finds widespread application across various fields due to its ability to handle non-linear dynamics and noisy data.
One prominent area is GPS navigation, where EKFs combine noisy satellite signals with vehicle motion models to provide a more accurate and smooth position estimate, especially in environments with signal interference. This allows for precise localization, fundamental for many modern technologies.
Autonomous vehicles, such as self-driving cars, heavily rely on EKFs for their perception and navigation systems. The filter fuses data from multiple sensors like radar, lidar, and cameras to estimate the vehicle’s own position and velocity, as well as the positions and velocities of other vehicles and pedestrians around it. This sensor fusion enables the car to build a consistent and reliable understanding of its surroundings, which is paramount for safe operation.
In robotics, EKFs are used for simultaneous localization and mapping (SLAM), allowing robots to build maps of unknown environments while simultaneously determining their own location within that map. Aerospace applications also benefit significantly, with EKFs being used for spacecraft navigation and attitude determination, ensuring accurate trajectory control during missions. The filter’s capability to provide robust state estimates in uncertain conditions makes it an adaptable tool across diverse and complex systems.