ETA at waypoints using time-series algorithms:

Problem statement: Given the NURBS method for trajectory generation for UAV swarms as described in previous article, the UAV trajectory was independent of in-flight parameters and both the position and velocity profile of planned trajectory could be obtained using the global locations and expected time of arrival at the waypoints. While a single drone can adhere to the planned trajectory, the internal dynamics of the UAV swarm and their effect on the ETA are harder to quantify. A closer tracking of the ETA at waypoints and trajectory deviations is needed for the UAV swarm.

   

Solution:

Consider a closed loop trajectory of UAV swarm. The effects of UAV swarm dynamics are easier to observe along waypoints in the loop because the NURBS trajectory assumes a constant velocity profile. Uncertainty in external variables such as unmodeled wind-field or uncertainty from internal friction between the drone units, can lead to different arrival times. Uncertainty affecting cruise velocity can be modeled using Gaussian independent random variables with covariance but a time-series algorithm does not need any attributes other than the historical collection of ETAs at the waypoints to be able to predict the next ETA. It only looks at scalar value regardless of the type or factors playing into the arrival time of the swarm while weights can be used to normalize the irregularity of distances between waypoints on the trajectory between start to finish. The historical data is utilized to predict an estimation on the arrival time as if the arrival were a scatter plot along the timeline. Unlike other data mining algorithms that involve additional attributes of the event, this approach uses a single auto-regressive method on the continuous data to make a short-term prediction. The regression is automatically trained as the data accrues so there is no need to parameterize or quantify uncertainties. 

Central to the step of fitting the linear regression, is the notion of covariance stationarity which suggests: 

·        The mean is not dependent on t 

·        The standard deviation is not dependent on t 

·        The covariance (Yt, Yt-j) exists and is finite and does not depend on t 

·        This last factor is called jth order autocovariance 

·        The jth order autocorrelation is described as autocovariance divided by the square of standard deviation 

 

The autocovariance measures the direction of the linear dependence between Yt and Yt-j. 
while the autocorrelation measures both the direction and the strength of the linear dependence between the Yt and Yt-j. 
An autoregressive process is defined as one in which the time dependence in the process decays to zero as the random variables in the process get farther and farther apart. It has the following properties: 
E(Yt) = mean 
Var(Yt) = sigma squared 
Cov(Yt, Yt-1) = sigma squared . phi 
Cor(Yt, Yt-1) = phi 

To fit the linear regression for a restricted data set, we determine the values of the random variable from the length p transformations of the time series data set. 

For a given time-series data set, a corresponding nine data sets for length p transformations are created. The p varies from zero to eight for the nine data sets. Each of these transformed datasets is centered and standardized before modeling; that is for each variable we subtract the mean value and divide by the standard deviation. Then we divided the data set into a training set used as input to the learning method and a holdout set to evaluate the model. The holdout set contains the cases corresponding to the last five observations in the sequence.