Research

Literature

Personal

I am a Ph.D. graduate of the Control and Dynamical Systems department at Caltech. My principal advisor was Professor Joel Burdick of the Mechanical Engineering Department, and my co-advisor was Professor Jerrold Marsden if the Control and Dynamical Systems department.

Interests: nonlinear control and dynamics, geometric mechanics, robotic and biomimetic engineering, coordinated systems.

Thesis: My thesis work has progressed to be on the development of a general averaging theory that works to arbitrary order of approximation. It provides a new paradigm for thinking of averaging theory and systems that can be analyzed using such a technique. The general averaging theory is described via the formulation of a nonlinear Floquet theory and its synthesis with perturbation theory. As such, the theory contains additional infrastructure to determine stability of the original system from stability properties of the averaged system. The concepts have been used to demonstrate a generic and constructive feedback control strategy for systems with and without drift. By appealing to elements of nonlinear Floquet theory, it is possible to demonstrate exponential convergence of the control technique.

The averaging theory is based on an exponential series representation for the flow of nonlinear time-varying vector fields. The exponential representation has important consequences for nonlinear systems. For example, the proof of nonlinear Floquet theory follows precisely that of linear Floquet theory due to the exponential representation. Just as state space linear control ideas are heavily dependent on the properties of the exponential, so are nonlinear control ideas. If derived properly, elements from nonlinear control will nicely correspond to their linear counterparts making the proofs easier, and aiding in the intuitive understanding of nonlinear control systems.

The mathematics has also been applied to nonlinear control systems such as the nonholonomic integrator (both first and second order). The real strength lies in its ability to simplify biomimetic control systems, without requiring nonlinear transformations. Within the realm of biomimetic control, the theory has been applied to the snakeboard, a carangiform fish, a potential-flow amoeba model, and a kinematic biped. Current work seeks to apply the theory to a dissipative snake model, a dissipative eel model, and a salamander model that includes both terrestrial and aquatic locomotion. I am also part of a research group that is seeking to model and understand gymnotiform locomotion. The focus is on the black-ghost knifefish, which navigates via an undulating anal fin.

In general, the goal of this biomimetic research is to demonstrate that although biological systems operate in diverse environments and utilize distinct locomotive strategies, there exists sufficient mathematical similarities regarding their dynamical description to analyze them within a common framework. Nonlinear Floquet theory and the related averaging theory provide this framework, while also allowing for the analysis of stabilizing feedback strategies. Lastly, mathematics from the area of geometric mechanics provides additional descriptive structure for intuitive derivation and understanding of the equations of motion for these systems. All of these concepts can be applied to standard nonlinear control problems.

The averaging and control techniques lead to a linear control model of nonlinear dynamical systems. By incorporating robust control ideas, it should be possible to determine sensitivity to parametric uncertainty of these nonlinear systems. Optimal control ideas can also be used with the discretized model or with the averaged autonomous model, to obtain an approximate optimal control strategy for these locomotive systems. That is the focus of current and future research.

Vela P.A., Morgansen K.M., and Burdick J.W. Second-Order Averaging Methods for Oscillatory Control of Underactuated Mechanical Systems, preprint, 2002. ps.

Vela P.A. and Burdick J.W., A Generalized Averaging Theory via Series Expansions, submitted for publication, 2004. pdf, ps, ps.gz.

Vela P.A. and Burdick J.W., Averaging Methods for Control Part I: Driftless Systems, submitted for publication, 2004. pdf, ps, ps.gz.

Vela P.A. and Burdick J.W. Geometric Homogeneity and Configuration Controllability of Nonlinear Systems, preprint, 2004.

Vela P.A. and Burdick J.W. Control of Underactuated Mechanical Systems with Drift Using Higher-Order Averaging Theory, submitted 2003 IEEE Conference on Decision and Control, 2002. pdf.

Vela P.A. and Burdick J.W. Geometric Homogeneity and Controllability of Nonlinear Systems, submitted 2003 IEEE Conference on Decision and Control, 2002. pdf.

Vela P.A. and Burdick J.W. Control of Biomimetic Locomotion via Averaging Theory, to Appear 2003 IEEE Conference on Robotics and Automation, 2002. ps, pdf.

Vela P.A. and Burdick J.W. Control of Underactuated Driftless Systems Using Higher-Order Averaging Theory, to Appear 2003 American Control Conference, 2002. ps, pdf.

Vela P.A. and Burdick J.W. A General Averaging Theory via Series Expansions, to Appear 2003 American Control Conference, 2002. ps, pdf.

Vela P.A., Morgansen K.M., and Burdick J.W. Underwater Locomotion From Oscillatory Shape Deformations, Proceedings IEEE Conference on Decision and Control, 2002. ps, pdf.

Morgansen K.M., Vela P.A., and Burdick J.W. Trajectory Stabilization for a Planar Carangiform Robot Fish, Proceedings IEEE Conference on Robotics and Automation, 2002. ps.

Vela P.A., Morgansen K.M., and Burdick J.W. Second-Order Averaging Methods for Oscillatory Control of Underactuated Mechanical Systems, Proceedings American Control Conference, 2002. ps.

Vela P.A., Averaging and Control of Nonlinear Systems, 2003. pdf, ps, ps.gz.