In the setting of a robot soccer game, the ability to detect merely the locations of objects on the field is often not enough. Like for real soccer players, it is often essential for robots to predict future locations of the ball (or even of the other players). We have used an Extended Kalman filter (EKF) for such a purpose[5]. The Kalman filter is very suitable for such a purpose since the detection of the ball's location is noisy.
The EKF is a recursive estimator for a possibly non-linear system. The goal of the filter is to estimate the state of a system. The state is usually denoted as an n-dimensional vector x. A set of equations is used to describe the behavior of the system, predicting the state of the system as:
where 
is a non-linear function which represents the
behavior of the non-linear system, 
is the external input to the
system and 
is a zero-mean, Gaussian random variable with
covariance matrix 
. 
captures the noise in the system and
any possible discrepancies between the physical system and the
model. The subscript k denotes the value of a variable at time
step k.
The system being modeled is being observed (measured). The observations can also be non-linear:
where
is the vector of observations and 
is the non-linear
measurement function, and 
is another zero-mean, Gaussian random
variable with covariance matrix 
. It captures any noise in the
measurement process.
The EKF involves a two-step iterative process, namely update and
propagate. The current best estimate of the system's state
and its error covariance is computed on each
iteration. During the update step, the current observations are used
to refine the current estimate and recompute the covariance. During
the propagate step, the state and covariance of the system at the next
time step are calculated using the system's equations. The process
then iteratively repeats, alternating between the update and the
propagate steps.
Through a careful adjustment of the filter parameters modelling the system, we were able to achieve successful tracking and, in particular prediction of the ball trajectory, even when sharp bounces occur.
Our vision processing approach worked perfectly during the RoboCup-97 games. We were able to detect and track 11 objects (5 teammates, 5 opponents and a ball). The prediction provided by the EKF allowed the goalkeeper to look ahead in time and predict the best defending position. During the game, no goals were suffered due to miscalculation of the predicted ball position.