UC San Diego

Two different problems are discussed: Motion planning and trajectory generation for the 3D Navier Stokes system and then source seeking with nonholonomic vehicles that have no position information. Taking ideas from robotics, motion planning for the Navier Stokes system of PDEs is examined. In the context of a channel flow, the problem consists of finding the reference velocity input for all time, for each point in space at one wall that guarantees desired output reference skin friction and pressure trajectories (also for all time and for each point in space) at the other wall. In addition to the open loop reference velocity input that depends on a specific initial condition, a feedback component that augments the open loop reference velocity input and stabilizes the system about the entire reference state trajectory is also designed. This controller, which was developed using PDE backstepping, is different from previously developed controllers in that it was developed for the full infinite dimensional system instead of a discretized version of the system and it is explicit with symbolically computed gains. For use in environments where position information is unavailable, the extremum seeking method is applied to autonomous vehicles as a means of navigating to find the source of some signal which the vehicles can measure locally. The signal is maximum at the source and decreases with distance away from the source. This work is distinct from previous work in that the vehicles have no position information, no communication and are nonholonomic. Detailed convergence analysis and full characterization of vehicle behavior for the method applied to nonholonomic 2D and 3D vehicles are provided. All vehicles are based on the 2D unicycle with constant forward velocity and actuated angular velocity. Using these ideas, the extremum seeking method is also applied to vehicles which are modeled as either three-link fish or Joukowski airfoils. A control law, based on extremum seeking, guides the biomimetic vehicle to seek the source of a signal, to move to a point in space and to follow a predetermined path