The Modified FIM Motion Algorithm

 

Chief Investigator:

Panos Tzanos

ptzano1@uic.edu

 

 

Motivation:

 

Stability analysis of previous information based motion algorithms proved to be intractable.  With the modified FIM motion algorithm, we are able to show that for any concentration function with a maximum at the source location, stability analysis is tractable. For details, see 1 and 2.

 

Through stability analysis, we came to the following conclusions:

 

*  For any concentration function with a maximum at the source location, there exists an equilibrium point.

*  This equilibrium point is unstable.

*  This equilibrium point is unique.

 

The Modified FIM Motion Algorithm:

 

The modified FIM motion algorithm can be outlined as follows:

 

1.      Take a concentration measurement

2.      Compute ML estimates of unknown parameters

3.      Compute the Fisher Information Matrix (FIM)

4.      Construct the modified FIM

5.      Compute the gradient of the modified FIM

6.      Move in the direction opposite to the gradient

7.      Return to 1 if the Cramer-Rao Bound is greater than the pre-determined tolerance level.

 

More details are given in 1.

 

Results:

 

The modified FIM motion algorithm was applied to a source with a concentration function that approximated the concentration function for a circular source in and open environment, and to a circular source in an open environment.  For a robot at location , and a source at location , the approximate concentration function at time , is of the form:

 

Where  is the intensity of the source, is the diffusivity, and  is the time that diffusion begins. Figure 1 compares this concentration function to the concentration function for a circular source:

 

 

The following figures show that in each case, the robot was able to estimate the vapor source location with a high degree of accuracy. 

 

*  Approximate Circular Source Concentration Function:

 

                  *         The trajectory of the robot:             

 

 

 

 

NOTE: Notice that in this case, the trajectory of the robot ends in a limit cycle.

 

                  *        The source location estimates and the Cramer-Rao Bound of the estimates:

 

 

 

 

 

The estimates of the source location converge to the true location, and are estimated with a high degree of accuracy.

 

*  Circular Source Concentration Function

 

                  *        The trajectory of the robot using different estimation techniques.  We also compare our approach to the trajectory obtained using a traditional concentration gradient   approach:

 

 

 

                 

                  *        The source location estimates and the Cramer-Rao Bound of the estimates:

                 

 

 

 

 

 

 

Plotting the gradient for each case (approximate model on the left, circular source on the right) shows that the source location is an unstable equilibrium point since the gradient always points away from the source location.  Also note that in the case of the circular source, the robot will not reach the center of the source, but it will reach the edge of the source.

 

 

 

Notes:

 

*  This motion algorithm can be fully implement in a distributed network scenario. 

*  This approach is superior to previous gradient based approaches since it can be applied to remote sensing scenarios.

*  Environmental factors such as wind, or physical obstacles can be accounted for in the measurement model.

 

                                                                                                        This research is supported by NSF grant CCR-0330342