Snapshot 1: the filtered trajectory as well as it's velocity are reconstructed in real time.

Snapshot 2: at high measurement noise, the accuracy of the filter deteriorates.

Snapshot 3: it is possible to adjust the magnitude of the gravitational potential to be repulsive (negative), as well as an optional constant deacceleration.

The object undergoes motion in a gravitational field, experiencing an acceleration

Here,

is the vector from the planet Mars to the object,

is the strength of the potential, adjustable by the user to be either positive (attractive) or negative (repulsive), and

is a constant acceleration in the direction opposite the velocity of the object, also adjustable by the user. In the case of a collision between the object and Mars, the object is stopped on the planet's surface, while the walls of the box are reflective.

The Kalman filter is used to reconstruct the position and velocity of the object from noisy position measurements. For a detailed description of the Kalman filter, see e.g. [1,3,4]. Following the notation in [1], the model for the object's discrete time evolution can be expressed as

where

denotes the state vector, and

denotes the noisy position measurements made,

is the process noise,

is the measurement noise, and

;

The measurement noise covariance matrix is assumed to be known

where

is the two dimensional identity matrix and

is the measurement noise variance set by the user. The process noise covariance is given by

where

is the variance of the

component of the acceleration, and similarly for

. The variances are determined from a sample of the true accelerations experienced by the object (with sample size equal to the variable number of data points), although in practice this information is not available and other methods must be employed (see e.g. [2]).

To determine the performance of the filter, two measures are updated in real time: (i) the MSE,

, and (ii) the trace of the state covariance matrix.

[1] Wikipedia, the free encyclopedia. Kalman filter. https://en.wikipedia.org/wiki/Kalman_filter.

[2] Fernando V. Lima and Murali R. Rajamani. Autocovariance Least-Squares (ALS) Toolbox. http://jbrwww.che.wisc.edu/software/als/index.html.

[3] Cuevas, E., Zaldivar, D., and Rojas, R, "Kalman filter for vision tracking." 2005.

[4] Kalman, R. E. "A New Approach to Linear Filtering and Prediction Problems." Trans. ASME Ser. D. J. Basic Eng. 82, 35-45, 1960.