VANET – Stochastic Path Prediction

Motivations Route Discovery Safety Warning Accident Notifications Strong Deceleration of Tra ffi c Flow Road Hazards (black ice, fallen tree, etc.) Information Distribution Congestion (makes it possible to choose another path ahead of road) Local Tourism Information Relevance of information is most often geographically delimited

Stochastic Path Prediction Benefits

Wireless Communication Capabilities

The Future Vision of VANET

What leads us to choose Path Prediction using Stochastics ? Noisy nature of measurements ◦ Speed Measurement ◦ Location Measurement ◦ Acceleration Measurement ◦ Direction Measurement Enviromental Influences ◦ Weather Conditions ◦ Electoro-Magnetical Distortions

What leads us to choose Path Prediction using Stochastics ? Degradation of Signals ( of Sensors ) Actual state transform model is completely unknown

Common Approach for Stochastic Prediction Kalman Filter Unscented Transform ( UT )

Rudolf Emil Kalman Rudolf Emil Kalman Born 1930 in Hungary BS and MS from MIT PhD 1957 from Columbia Filter developed in Now retired

Kalman Filter - Introduction Set of mathematical equations Implementing predictor-corrector type estimator Optimal in the sense that it minimizes the estimated error covariance Relative Simplicity and robust nature of filter itself Works well for many applications

What is it used for ? Tracking missiles Tracking heads/hands/drumsticks Extracting lip motion from video Economics Navigation

What does a Kalman Filter do, anyway?

A – n x n Matrix relates the state at the previous time step to the state at the current step k ( may change for each time step ). What does a Kalman Filter do, anyway?

Q – Process noise covariance matrix R – Measurement noise covariance matrix

Normal Probability Distribution Normal Distribution is a continuous probability distribution that has a bell-shaped probability density function, known as the Gaussian function or informally the bell curve: μ - Mean or Expectation (location of the peak)

Variance and Covariance The variance is E[ (x-E[x]) 2 ] Covariance matrix is E[(x-E[x])(x-E[x]) T ]

Covariance Matrix Along the diagonal, C ii are variances. Off-diagonal C ij are essentially correlations.

Preparation State Transition Process Noise Covariance Measurement Noise Covariance

Predict  Correct Kalman Filter operates by: Predicting the new state and its uncertainty Correcting with the new measurement

Predict  Correct

A Simple Example The following example will be based upon “voltage reading” from a source. We assume that it has a constant value of aV (volts), but of course we have some noisy readings above and below a volts. We also assume that the standard deviation of the measurement noise is 0.1 V.

A Simple Example

TIME (ms) VALUE (V)

A Simple Example The Time Update (prediction) and Measurement Update (correction) equations.

A Simple Example

K

Results

Results

Extended Kalman Filter (EKF) Extended Kalman filter (EKF) equations which invoke the Jacobians (first-order EKF) and possibly the Hessians (second-order EKF) of the nonlinear system. Depending on the nonlinearities present Computationally heavy

Extended Kalman Filter (EKF)

Unscented Transform ( UT ) The unscented transformation (UT) is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The intuition behind the UT is that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function or transformation.

Unscented Transform ( UT ) In contrast, the EKF approximates a nonlinear function using linearization, and this approximation can be inaccurate when the models have large nonlinearities over short time periods.

Conclusions The Kalman Filter algorithm is an optimal estimator Prediction of the next steps and cleaning noises Easy to implement Computational heavy

References [1] Hannes Hartenstein & Kenneth P. Laberteaux, "VANET: Vehicular Applications and Inter-Networking Technologies",Johm Wiley and Sons Ltd. [2] Greg Welch & Greg Bishop, "An Introduction to the Kalman Filter", University of North Carolina at Chapel Hill [3] Nathan Funk,"A Study of the Kalman Filter applied to Visual Tracking", University of Alberta [4] Rainer Baumann,"Vehicular Ad hoc Networks (VANET)", Swiss Federal Institute of Technology Zurich