# An estimation problem in GPS tracking

Problem: Consider two cars (taken to be point objects), named leader $L$ and follower $F$, both equipped with GPS devices that communicate with each other. The object of $F$ is to follow $L$ as closely as possible as the latter moves arbitrarily on the plane. Given that all GPS devices have a Circular Error Probable (CEP) distribution of error, with a prescribed mean $\mu = (\mu_x,\mu_y)$ and a prescribed covariance matrix $\Sigma_{2\times 2}$.

• Given that $L$ traverses a (piecewise smooth) curve $C$ in the plane, what is the expected curve traversed by $F$? Further, what is the distribution of $F$'s paths?
• What is the optimal way for $F$ to estimate $L$ over a period of time?

Background: This is a practical problem I faced in experimental work, and not homework by any means. I am aware of tools such as Kalman Filtering for optimal state estimation in the face of white noise, but am not sure about exactly how to extend them to this case. I would also like to know of pertinent research literature .

• Because this is a practical problem, it seems worthwhile to point out that the distributions of the errors in the positions of $L$ and $F$ will be strongly positively correlated provided $L$ and $F$ are close, because many of the errors affecting the GPS positions will be common to the two readings. The correlation will decrease as the distance between $L$ and $F$ increases. The answer will therefore depend on that distance as well as on the speed of $L$, its acceleration, and the frequency with which $F$ and $L$ obtain GPS readings. And don't forget the strong temporal correlation... – whuber Nov 27 '11 at 22:16

## 3 Answers

I agree that as posed the question is incomplete. I am also puzzled about the mention of CEP (which is the circle centered at the mean that contains 50% of the distribution. Knowing the mean and covariance matrix would be enough to characterize a bivariate normal distribution. Are you assuming bivariate normal for the GPS accuracy? Maybe circular normal because x and y coordinates are independent. Of course if you know the mean and covariance of a bivariate normal the CEP is then determined. Having worked in the Aerospace industry in the 1980s study the GPS user equipment accuracy based on how many satellites can pick up the signal i know that CEP is a commonly used parameter. What is the mechanism that the follower uses? Perhaps he moves toward the point estimate from his GPS device? In that case he would be moving toward the GPS estimated center for the location of the leader. He would probably follow a straight line until he sees a position update and would then move toward that updated position. In that way he would be following a broken line with the number of chnages in the direction of the line dictated by the frequency of the update.

IMHO, the problem definition is incomplete. The answer would depend on the frequency of communication between L and F, and the speed of travel. If you can calculate the GPS position very frequently, if the readings are independent of each other and the communication frequency is also high, then both vehicles can traverse almost identical path. Also, if the vehicles are traveling very slowly, there would be enough communication between the cars to avoid discrepancy in path.

It also depends on plethora of other parameters, the skewness of the path etc. So this is the way I would go about it. I would simulate the scenario as accurately as possible and estimate the discrepancy using sampling.

Since you say this is a real world problem, you should also consider the fact that there are only specified number of paths (also called "roads") and that would reduce the discrepancy even further.

• I wonder about simulation as a model building tool: the logic seems circular, because the result you get will depend on the simulation you create. If you can simulate the situation, then surely you have (at least implicitly) a model for it that is amenable to analysis, right? – whuber Dec 30 '11 at 19:23
• @whuber I don't think Ganesh is trying to "model". Rather, he is trying to "estimate". Simulation is a perfectly logical solution if estimating something is intractable in closed form. As I mentioned in my post, the problem definition is incomplete. The user should first create a realistic simulation and see what variables are available, sampling frequency etc. – ElKamina Jan 3 '12 at 17:58

This is an incomplete question. For the first question, the control policy or algorithm is necessary. For the second question, the optimal estimate will depend upon whether there is global knowledge (F knows L's observations), and more critically, the metric for optimality. Optimality metrics may emphasize energy consumption, deviation from the leader trajectory, etc.
As a first step, separate the estimation problem from the control problem, and then you can approach simultaneous methods.