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Using Bayesian Kalman Filter to predict positions of moving particles / objects in 2D (in R)

In this article, we shall see how the Bayesian Kalman Filter can be used to predict positions of some moving particles / objects in 2D.

This article is inspired by a programming assignment from the coursera course Robotics Learning by University of Pennsylvania, where the goal was to implement a Kalman filter for ball tracking in 2D space. Some part of the problem description is taken from the assignment description.

The following equations / algorithms are going to be used to compute the Bayesian state updates for the Kalman Filter.



For the first set of experiments, a few 2D Brownian Motion like movements are simulated for a particle.

  • The noisy position measurements of the particle are captured at different time instants (by adding random Gaussian noise).
  • Next, Kalman Filter is used to predict the particle’s position at different time instants, assuming different position, velocity and measurement uncertainty parameter values.
  • Both the actual trajectory and KF-predicted trajectory of the particle are shown in the following figures / animations.
  • The positional uncertainty (as 2D-Gaussian distribution) assumed by the Kalman Filter is also shown as gray / black contour (for different values of uncertainties).


    kfout.png

    motion1.1motion1.2motion1.3

The next set of figures / animations show how the position of a moving bug is tracked using Kalman Filter.

  • First the noisy measurements of the positions of the bug are obtained at different time instants.
  • Next the correction steps and then the prediction steps are applied, after assuming some uncertainties in position, velocity and the measurements with some Gaussian distributions.

    motion3

  • The position of the bug as shown in the figure above is moving in the x and ydirection randomly in a grid defined by the rectangle [-100,100]x[-100,100].
  • The next figure shows how different iterations for the Kalman Filter predicts and corrects the noisy position observations. The uncertain motion model p(x_t|x_{t-1}) increases the spread of the contour.  We observe a noisy position estimate z_tThe contour of the corrected position p(x_t) has less spread than both the observation p(z_t|x_t) and the motion p(x_t|x_{t-1})  adjusted state.
  • motion3

Next the GPS dataset from the UCI Machine Learning Repository is used to get the geospatial positions of some vehicles at different times.

  • Again some noisy measurement is simulated by adding random noise to the original data.
  • Then the Kalman Filter is again used to predict the vehicle’s position at different time instants, assuming different position, velocity and measurement uncertainties.
  • The position and measurement uncertainties (σ_p,  σ_m) are in terms of latitude / longitude values, where uncertainty in the motion model is σ_v.
  • Both the actual trajectory and KF-predicted trajectory of the vehicle are shown in the following figures / animations.
  • As expected, the more the uncertainties in the position / motion model, the more the actual trajectory differs from the KF-predicted one.

    kfout1kfout2kfout3

    motion2.2motion2.3motion2.4motion2.1.gif

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