Research Page


Eglin MAV with 21 inch wingspan My current research is in the area of Sensitivity Analysis. I am interested in the design, development and implementation of computational methods for sensitivity calculations. Some logical questions you might want to ask are:
  1. What is a sensitivity?
    Mathematicians and other scientists model physical systems using a set of mathematical equations --- these might be algebraic equations or ordinary differential equations or even partial differential equations. These equations will describe how some STATE VARIABLE such as temperature, velocity, pressure or population changes over time. Inevitably, there are also PARAMETERS floating around in the models. These parameters might describe physical properties such as thermal conductivity, permeability, porosity, birth and death rates, parameters that describe the shape of an airfoil or an airplane wing, parameters that describe the position of sensors and actuators within a control system on a MAV, etc. The sensitivity describes how small changes in a parameter value affect the state variables. In short, a sensitivity is a derivative (math word); it's the rate of change of the state variable per unit change in the parameter value.
  2. How does one compute a sensitivity?
    There are many ways that scientists attempt to compute sensitivities, and it varies largely with the application that one is considering. My current research focuses on the use of Continuous Sensitivity Equation Methods (CSEMs). This approach begins by "differentiating" the mathematical model in order to derive an equation (or a set of equations) for which the sensitivity is a solution. This equation is called the SENSITIVITY EQUATION, and deriving, solving, analyzing and approximating sensitivity equations has been the focus of my research for the last several years. (more than I care to count at this point)
  3. How might one use a sensitivity?
    Sensitivities provide for a direct gradient calculation in some optimal design problems where the constraints are defined by PDEs. They can also be used for parameter prioritization in model analysis, and some of my colleagues have used them in the quanitification of uncertainty in models. For the most part, I am referring to models of a large variety of engineering systems.

DESCRIPTION OF MY CURRENT RESEARCH PROJECT
I am currently funding two post doctoral Researchers -- Dr. Faranak Pahlevani and Dr. John Singler. And I am working with one Ph.D. candidate, Ms. Jennifer Thorenson.
My current research project is funded by a grant from the AFOSR (Air Force Office of Scientific Research) through the DEPSCoR (Defense Experimental Program to Stimulate Competitive Research) program. It is entitled "Sensitivity Analysis for the Optimal Design and Control of Advanced Guidance Systems" (yep, that's a mouthful). I am working on basic research that applies my knowledge of CSEMs to the process of designing feedback control systems for improved flight stability of MAVs (Micro Air-Vehicles). Unfortunately, I don't have a lab where I get to design, fly, test (and generally play with) these MAVs. (after all, I'm a mathematician... we stick to function spaces and simple software tools) However, there are some really great projects going on across the country where such design processes are currently underway. The pictures on this page give examples of such designs. The picture at the top of the page shows a MAV with a 21-inch wingspan flying autonomously. That MAV was developed at Eglin Air Force Base. There are other groups working on these vehicles including a group at the University of Florida. MAV with 6inch wingspan
MAV with 24inch wingspan The image above and the one to the left are the designs of a group at the University of Florida, and you can find out more about their projects at the home page of Michael C. Nechyba in the Dept. of Electrical and Computer Engineering. His web site on Vision-Guided Flight for MAVs has lots of cool pictures and some movies.
You might ask why anyone would be interested in such miniscule aircraft. The hope is that these things can be used in urban environments to protect humans from going into very dangerous places in order to carry out tasks. One of the goals is to design such an aircraft so that it can fly autonomously (without the aid of RC--remote control) through an urban environment. This means that it will need to be very agile and have good flight stability characteristics. Those researchers "in the know" tell us that it will definitely not be as simple as scaling the design of a 747 down to a toy airplane with a wingspan of 6 inches.
It turns out that the aerodynamics that these MAVs must deal with while flying at 25mph and roughly a hundred feet off the ground are much different than those of a 747 flying at 30,000ft. My research focuses on answering some mathematical questions about the optimal placement of sensors and actuators on such vehicles in order give them as much flight stability as possible under a variety of flight conditions. And there are lots of possible design options that we want to consider for the project. For example, different kinds of sensors and actuators will affect our system dynamics in different ways. It would be nice to know which types of sensors and actuators yield the most stability for a variety of system disturbances, and of course, where is the best place to locate these objects on the MAV, etc.


Stanley