### Nonlinear Controls Lab

Most robots are able to assume any position because each joint of the robot arm has a motor or actuator. Some robots have joints that do not have motors. Such devices are called underactuated mechanical systems. It is possible to control underactuated systems, but it is difficult because the mathematics can at times be very complicated. Systems that can be controlled usually have some simplifying feature. The nature of this research is to develop ways of new mathematical approaches so that the control analysis is not as difficult as with existing approaches.

Real world instances of underactuated mechanical systems include unicycles, rockets, hover boards, fixed wing and multi-rotor aircraft, and gantry cranes. To appreciate why a rocket, such as the Apollo moon launch vehicle, is an underactuated system, try putting the end of a broom stick in the palm of your hand and push it straight upward. Unless you move your hand sideways to balance the broom, you will not be successful in moving the broom upward. Another application of underactuated systems control is to quickly load cargo boxes on an ocean going vessel using a gantry crane so that the cargo boxes do not swing as they are moved. Unicycles will not stand up on their own unless some means is found to balance them. Finally, multi-rotor aircraft cannot move forward or sideways unless you tilt them in the direction you want to move.

**Research Goals**

• Develop a mathematical control law that renders the original underactuated, rigid body mechanical device as a stable system that can robustly withstand random disturbances.

• The control law which performs this rendering effectively replaces the original mechanical system in terms of a new mechanical system having the desired, specified properties of the precious bullet.

• The research challenge is in determining the new mechanical system. Once the new system is found, the control law which performs the rendering can also be found. Previous attempts to find the new mechanical system have relied on the solution of differential and partial differential equations. The solution of these equations provide the new distribution of mass and inertia of the rendered system together with its potential energy. The immediate goal is to find the new system mass and inertia through a process involving the solution of algebraic equations.

• The next goal is then to find the new potential energy by solving algebraic equations.

**Additional Research**

Wind turbines offer many control system challenges requiring nonlinear control theory. Two such areas are:

• Operating the wind turbine at maximum efficiency. For a given wind speed, there is a rotor speed that produces a maximum power figure. This type of operation can be achieved through nonlinear control. Research efforts are moving in the direction of simplifying the process.

• Active vibration control. Wind turbines are flexible structures and vibrations can create premature component failure and limit the life of the turbine. Active vibration control using the electrical generator and the blade pitch actuators provide a means of reducing the vibrations in various parts of the turbine. Better ways can always be found. One exciting challenge is the emergence of multi-rotor turbines where four rotors are mounted on a single tower. Using each generator and the blade pitching of each rotor to control vibrations of the entire structure is an exceptional challenge.