Jennie Cochran

Nonholonomic Source Seeking With Tuning of Angular Velocity

We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the nonholonomic unicycle. The vehicle does not have the capability of sensing its position or the position of the source but is capable of sensing the scalar signal originating from the source. The signal field is assumed to decay away from the position of the source but the vehicle does not have the knowledge of the functional form of the field. We employ extremum seeking to steer the vehicle to the source. Our control strategy keeps the forward velocity constant and tunes the angular velocity, a setting suitable for most autonomous vehicles, including aerial ones. Because of the constant forward velocity constraint, after it has converged near the source, the vehicle exhibits extremely interesting and complex motions. Using averaging theory, we prove local exponential convergence to an ldquoorbit-likerdquo attractor around the source. We also present a thorough analysis of non-local behaviors and attractors that the vehicle can exhibit near the source. The richness and complexity of behaviors makes only some of them amenable to analysis, whereas others are illustrated through a carefully laid out simulation study.

3-D Source Seeking for Underactuated Vehicles Without Position Measurement

Our past work introduced source seeking methods for GPS-denied autonomous vehicles using only local signal measurement and operating in two dimensions. In this paper, we extend these results to three dimensions. The 3D extensions introduce many interesting challenges, including the choice of vehicle models in 3D, sensor placement to allow probing-based gradient estimation of an unknown signal field in 3D, the question of what type of pattern of vehicle motion can be produced in an underactuated 3D vehicle to allow tuning by single-loop or multiloop extremum seeking, and the shape of attractors, which become very complex in 3D. We present two control schemes that address these questions. The first scheme focuses on vehicles with a constant forward velocity and the ability to actuate pitch and yaw velocities. The second scheme employs vehicles with constant forward and pitch velocities and actuate only the roll velocity. Our results include convergence analysis and simulation results.

Source Seeking for Two Nonholonomic Models of Fish Locomotion

In this paper, we present a method of locomotion control for underwater vehicles that are propelled by a periodic deformation of the vehicle body, which is similar to the way a fish moves. We have developed control laws employing ldquoextremum seekingrdquo for two different ldquofishrdquo models. The first model consists of three rigid body links and relies on a 2-degree-of-freedom (DOF) movement that propels the fish without relying on vortices. The second fish model uses a Joukowski airfoil that has only 1 DOF in its movement and, thus, relies on vortex shedding for propulsion. We achieve model-free and position-free ldquosource seeking,rdquo and, if position is available, navigation along a predetermined path.

Backstepping boundary control of Navier-Stokes channel flow: a 3D extension

We present an extension from 2D to 3D of a boundary control law which stabilizes the parabolic profile of an infinite channel flow. In this 3D case, we include an additional controller in the spanwise direction which, similar to the streamwise and wall normal velocity controllers, actuates solely along one boundary. The controller that acts in the wall normal direction transforms the system into a strict feedback form. The streamwise and spanwise controllers decouple the normal vorticity from the normal velocity and then stabilize the normal velocity. This result guarantees exponential stability in the L2 sense of the linearized Navier-Stokes equations

3D nonholonomic source seeking without position measurement

We consider the three dimensional problem of directing a nonholonomic vehicle to seek the source of a scalar signal without the use of position information. If we assume the signal strength decays with distance from the source then we achieve convergence to the source by making use of the extremum seeking method. In the kinematic vehicle model we employ, the forward velocity is constrained to a constant and the control inputs are the yaw and pitch velocities. We present a control scheme which tunes these angular velocities and prove the local exponential convergence of this scheme. We also provide simulations which illustrate the behavior of the vehicle under different scenarios, such as static and moving sources, signal fields with spherical and elliptical level sets and parameter regimes not covered by theory.

Source seeking with a nonholonomic unicycle without position measurements and with tuning of angular velocity — Part II: Applications

For Part I see ibid. (2007). We present results for autonomous vehicles operating in GPS-denied environments while performing several different tasks. These vehicles employ extensions of extremum seeking to accomplish their goals. Previously, extremum seeking has successfully been applied to vehicles seeking the source of some signal, while operating in such environments. This paper considers the objectives of tracking a diffusive signal, tracing a level set of a signal field, and modification of the algorithm for use on a vehicle with limited movement capabilities. We present each scenario, detail each control scheme and, in addition, present simulation results.

Motion planning and trajectory tracking for three-dimensional Poiseuille flow

We present the first solution to a boundary motion planning problem for the Navier–Stokes equations, linearized around the parabolic equilibrium in a threedimensional channel flow. The pressure and skin friction at one wall are chosen as the reference outputs as they are the most readily measurable ‘wall-restricted’ quantities in experimental fluid dynamics and also because they play a special role as performance metrics in aerodynamics. The reference velocity input is applied at the opposite wall. We find the exact (method independent) solution to the motion planning problem using the PDE (partial differential equation) backstepping theory. The motion planning solution results in open-loop controls, which produce the reference output trajectories only under special initial conditions for the flow velocity field. To achieve convergence to the reference trajectory from other (nearby) initial conditions, we design a feedback controller. We also present a detailed examination of the closed-form solutions for gains and the behaviour of the motion planning solution as the wavenumbers grow or the Reynolds number grows. Numerical results are shown for the motion planning problem.

Source seeking for a Joukowski foil model of fish locomotion

We continue with the study of fish-like underwater source seeking initiated in a companion paper where we considered a three-rigid-link fish model. In this paper we consider a more realistic fish model based on a Joukowski airfoil which has only one degree of freedom in its movement, and thus relies on vortex shedding to move through a perfect fluid. Both the propulsion problem and the source seeking problem are solved by employing the same sinusoidal perturbation input— the flapping of the fishs tail. The fish converges to the signal source despite being unaware of its position, the sources position, and the spatial distribution of the source.

Extremum seeking for motion optimization: From bacteria to nonholonomic vehicles

What do E. coli bacteria and GPS-denied unmanned vehicles have in common? In the absence of position and attitude measurements, both bacteria and autonomous vehicles have to rely on solving real-time optimization problems based on other available signal measurements, such as concentration of chemical agents, thermal or light intensity, or other environmental signals such as acoustic or electomagnetic. We overview extremum seeking, a method for non-model based optimization that we have been developing over the last ten years, and its applications to autonomous vehicles. The use of extremum seeking for navigating vehicles in GPS-denied environments is a rather challenging technical problem because the nonholonomically constrained vehicle kinematics violate the standard assumptions of exponential stability of the plant in the classical theory of extremum seeking. We review applications to nonholonomic vehicles in both two and three dimensions.

GPS denied source seeking for underactuated autonomous vehicles in 3D

Extremum seeking has been successfully applied to source seeking for autonomous vehicles operating in two dimensions. In this paper we extend these results to vehicles operating in three dimensions. The extension is interesting for several reasons. First, there is the choice of vehicle models to consider, and second there is the question of what type of vehicle movement can be actuated. We present two control schemes which address these questions. The first scheme focuses on vehicles with a constant forward velocity and the ability to actuate pitch and yaw velocities. The second scheme explores vehicles which operate with a constant forward velocity and a constant pitch velocity and which are capable of actuating only the roll velocity. We present the vehicle models, details of the control schemes, and simulation results.